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<title>Solving Equations Quadratic in Form</title>
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<div class="section" id="fwk-redden-ch06_s03" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.3</span> Solving Equations Quadratic in Form</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch06_s03_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch06_s03_o01" numeration="arabic">
<li>Develop a general strategy for solving quadratic equations.</li>
<li>Solve equations that are quadratic in form.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch06_s03_s01" version="5.0" lang="en">
<h2 class="title editable block">General Guidelines for Solving Quadratic Equations</h2>
<p class="para editable block" id="fwk-redden-ch06_s03_s01_p01">Use the coefficients of a quadratic equation to help decide which method is most appropriate for solving it. While the quadratic formula always works, it is sometimes not the most efficient method. If given any quadratic equation in standard form,</p>
<p class="para block" id="fwk-redden-ch06_s03_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0619" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>
</span>
where <em class="emphasis">c</em> = 0, then it is best to factor out the GCF and solve by factoring.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch06_s03_s01_p04">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0620" display="inline"><mrow><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p05">In this case, <em class="emphasis">c</em> = 0 and we can solve by factoring out the GCF 3<em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0621" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p07">Then apply the zero-product property and set each factor equal to zero.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0622" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mtext>4</mtext><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p09">Answer: The solutions are 0 and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0623" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s03_s01_p10">If <em class="emphasis">b</em> = 0, then we can solve by extracting the roots.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch06_s03_s01_p11">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0624" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p12">In this case, <em class="emphasis">b</em> = 0 and we can solve by extracting the roots. Begin by isolating the square.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p13"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0625" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>8</mn><mn>5</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p14">Next, apply the square root property. Remember to include the ±.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p15"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0626" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mfrac><mn>8</mn><mn>5</mn></mfrac></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>R</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><msqrt><mrow><mo>−</mo><mn>4</mn><mo>⋅</mo><mn>2</mn></mrow></msqrt></mrow><mrow><mpadded height="1.4em"><msqrt><mn>5</mn></msqrt></mpadded></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mstyle color="#007fbf"><mpadded height="2.2em"><msqrt><mn>5</mn></msqrt></mpadded></mstyle></mrow><mrow><mstyle color="#007fbf"><msqrt><mn>5</mn></msqrt></mstyle></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><mn>2</mn><mi>i</mi><msqrt><mn>2</mn></msqrt><mo>⋅</mo><msqrt><mn>5</mn></msqrt></mrow><mn>5</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><mn>2</mn><mi>i</mi><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>5</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p16">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0627" display="inline"><mrow><mo>±</mo><mfrac><mrow><mn>2</mn><mi>i</mi><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>5</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch06_s03_s01_p17">When given a quadratic equation in standard form where <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> are all nonzero, determine the value for the discriminant using the formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0628" display="inline"><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow><mo>.</mo></math></span></p>
<ol class="orderedlist editable block" id="fwk-redden-ch06_s03_s01_o01" numeration="loweralpha">
<li>If the discriminant is a perfect square, then solve by factoring.</li>
<li>If the discriminant is not a perfect square, then solve using the quadratic formula.</li>
</ol>
<p class="para editable block" id="fwk-redden-ch06_s03_s01_p18">Recall that if the discriminant is not a perfect square and positive, the quadratic equation will have two irrational solutions. And if the discriminant is negative, the quadratic equation will have two complex conjugate solutions.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch06_s03_s01_p19">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0629" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p20">Begin by rewriting the quadratic equation in standard form.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0630" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>21</mn><mi>x</mi><mo>+</mo><mn>15</mn><mi>x</mi><mo>+</mo><mn>35</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mi>x</mi><mo>+</mo><mn>35</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p22">Substitute <em class="emphasis">a</em> = 9, <em class="emphasis">b</em> = 30, and <em class="emphasis">c</em> = 25 into the discriminant formula.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0631" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>30</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>25</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>900</mn><mo>−</mo><mn>900</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p24">Since the discriminant is 0, solve by factoring and expect one real solution, a double root.</p>
<p class="para" id="fwk-redden-ch06_s03_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0632" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s01_p26">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0633" display="inline"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s03_s01_p27">It is good to know that the quadratic formula will work to find the solutions to all of the examples in this section. However, it is not always the best solution. If the equation can be solved by factoring or by extracting the roots, you should use that method.</p>
</div>
<div class="section" id="fwk-redden-ch06_s03_s02" version="5.0" lang="en">
<h2 class="title editable block">Solving Equations Quadratic in Form</h2>
<p class="para block" id="fwk-redden-ch06_s03_s02_p01">In this section we outline an algebraic technique that is used extensively in mathematics to transform equations into familiar forms. We begin by defining <span class="margin_term"><a class="glossterm">quadratic form</a><span class="glossdef">An equation of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0634" display="inline"><mrow><mi>a</mi><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>u</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span> where <em class="emphasis">a</em>, <em class="emphasis">b</em> and <em class="emphasis">c</em> are real numbers and <em class="emphasis">u</em> represents an algebraic expression.</span></span>,</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0635" display="block"><mrow><mi>a</mi><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>u</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p03">Here <em class="emphasis">u</em> represents an algebraic expression. Some examples follow:</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0636" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>8</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mover><mstyle color="#007fbf"><mo>⇒</mo></mstyle><mrow><mi>u</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow></mover></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>u</mi><mo>+</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mover><mstyle color="#007fbf"><mo>⇒</mo></mstyle><mrow><mi>u</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mover></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>u</mi><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>7</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mover><mstyle color="#007fbf"><mo>⇒</mo></mstyle><mrow><mi>u</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mover></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>u</mi><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p05">If we can express an equation in quadratic form, then we can use any of the techniques used to solve quadratic equations. For example, consider the following fourth-degree polynomial equation,</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0637" display="block"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p07">If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0638" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0639" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></math></span> and we can write</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0640" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>⇒</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><msup><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="center"><mrow><mn>4</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="left"><mrow><mn>32</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><msup><mi>u</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="center"><mrow><mn>4</mn><mi>u</mi></mrow></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="left"><mrow><mn>32</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p09">This substitution transforms the equation into a familiar quadratic equation in terms of <em class="emphasis">u</em> which, in this case, can be solved by factoring.</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0641" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mtext> </mtext><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>u</mi><mo>−</mo><mn>32</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mi>u</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mn>8</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mo>−</mo><mn>4</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p11">Since <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0642" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> we can back substitute and then solve for <em class="emphasis">x</em>.</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0643" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>u</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>u</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mo> ↓</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mn>8</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>4</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>2</mn><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p13">Therefore, the equation <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0644" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mo>=</mo><mn>0</mn></mrow></math></span> has four solutions <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0645" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mo>±</mo><mn>2</mn><mi>i</mi></mrow><mo>}</mo></mrow></mrow></math></span>, two real and two complex. This technique, often called a <span class="margin_term"><a class="glossterm"><em class="emphasis">u</em>-substitution</a><span class="glossdef">A technique in algebra using substitution to transform equations into familiar forms.</span></span>, can also be used to solve some non-polynomial equations.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n01">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p14">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0646" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p15">This is a radical equation that can be written in quadratic form. If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0647" display="inline"><mrow><mi>u</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow></math></span> then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0648" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mi>x</mi></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p16"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0649" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>−</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mtext> </mtext><mn>2</mn><mi>u</mi><mtext> </mtext><mtext> </mtext><mo>−</mo><mtext> </mtext><mtext> </mtext><mn>8</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p17">Solve for <em class="emphasis">u</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0650" display="block"><mtable columnspacing="0.1em"><mtr><mtd><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>u</mi><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mi>u</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p19">Back substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0651" display="inline"><mrow><mi>u</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow></math></span> and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0652" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mi>x</mi></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msqrt><mi>x</mi></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p21">Recall that squaring both sides of an equation introduces the possibility of extraneous solutions. Therefore we must check our potential solutions.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p22"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0653" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em" columnlines="solid none"><mtr columnalign="left"><mtd columnalign="center"><mrow><mstyle color="#007f3f"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007f3f"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right" style="border-right:1pt solid black"><mrow><mtable columnalign="center" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007f3f"><mn>16</mn></mstyle><mo>−</mo><mn>2</mn><msqrt><mrow><mstyle color="#007f3f"><mn>16</mn></mstyle></mrow></msqrt><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>16</mn><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>16</mn><mo>−</mo><mn>8</mn><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007f3f"><mn>4</mn></mstyle><mo>−</mo><mn>2</mn><msqrt><mstyle color="#007f3f"><mn>4</mn></mstyle></msqrt><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>−</mo><mn>4</mn><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#ff0000"><mo>✗</mo></mstyle></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p23">Because <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0654" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow></math></span> is extraneous, there is only one solution, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0655" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>16</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p24">Answer: The solution is 16.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n02">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p25">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0656" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p26">If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0657" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0658" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0659" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtext> </mtext><mtext> </mtext><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><msup><mi>u</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mo>−</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>3</mn><mi>u</mi><mtext> </mtext><mtext> </mtext><mo>−</mo><mtext> </mtext><mtext> </mtext><mn>10</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p28">Solve for <em class="emphasis">u</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p29"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0660" display="block"><mtable columnspacing="0.1em"><mtr><mtd><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>u</mi><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mi>u</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>5</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p30">Back substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0661" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math></span> and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0662" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>125</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p32">Check.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p33"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0663" display="block"><mrow><mtable columnalign="left" columnlines="none solid none none" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="center"><mrow><mstyle color="#007f3f"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>125</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="center"><mrow><mstyle color="#007f3f"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>8</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mtable columnalign="center" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>125</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>125</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>5</mn><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><msup><mn>5</mn><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mn>5</mn><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>25</mn><mo>−</mo><mn>15</mn><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtable columnalign="center" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>8</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>8</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>⋅</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>+</mo><mn>6</mn><mo>−</mo><mn>10</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p34">Answer: The solutions are −8, 125.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n03">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p35">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0664" display="inline"><mrow><mn>3</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>7</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p36">If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0665" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0666" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p37"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0667" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>3</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>7</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>7</mn><mi>u</mi><mtext> </mtext><mtext> </mtext><mo>−</mo><mtext> </mtext><mtext> </mtext><mn>6</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p38">Solve for <em class="emphasis">u</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p39"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0668" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>u</mi><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mn>3</mn><mi>u</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p40">Back substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0669" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> and solve for <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p41"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0670" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p42">The original equation is actually a rational equation where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0671" display="inline"><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> In this case, the solutions are not restrictions; they solve the original equation.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p43">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0672" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mtext> </mtext><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n04">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p44">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0673" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>8</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p45">If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0674" display="inline"><mrow><mi>u</mi><mo>=</mo><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0675" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p46"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0676" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd columnalign="center"><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd columnalign="center"><mo>+</mo></mtd><mtd columnalign="center"><mn>8</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow><mo>)</mo></mrow></mtd><mtd columnalign="center"><mo>+</mo></mtd><mtd columnalign="center"><mn>7</mn></mtd><mtd columnalign="center"><mo>=</mo></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"></mtd><mtd columnalign="center"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd columnalign="center"><msup><mi>u</mi><mn>2</mn></msup></mtd><mtd columnalign="center"><mo>+</mo></mtd><mtd columnalign="center"><mn>8</mn><mi>u</mi></mtd><mtd columnalign="center"><mo>+</mo></mtd><mtd columnalign="center"><mn>7</mn></mtd><mtd columnalign="center"><mo>=</mo></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p47">Solve for <em class="emphasis">u</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p48"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0677" display="block"><mtable columnspacing="0.1em"><mtr><mtd><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>u</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>=</mo><mo>−</mo><mn>7</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p49">Back substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0678" display="inline"><mrow><mi>u</mi><mo>=</mo><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow></math></span> and solve for <em class="emphasis">t</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p50"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0679" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mfrac><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mi>t</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>7</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>t</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>7</mn><mi>t</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>t</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>8</mn><mi>t</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p51">Answer: The solutions are −1, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0680" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>.</mo></math></span> The check is left to the reader.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p52"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0681" display="inline"><mrow><mn>12</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>16</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p53">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0682" display="inline"><mrow><mfrac><mn>6</mn><mn>5</mn></mfrac></mrow></math></span>, 2.</p>
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</div>
</div>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p55">So far all of the examples were of equations that factor. As we know, not all quadratic equations factor. If this is the case, we use the quadratic formula.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n05">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p56">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0683" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>23</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> Approximate to the nearest hundredth.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p57">If we let <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0684" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0685" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p58"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0686" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>23</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd></mtr><mtr><mtd><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>u</mi><mo>+</mo><mn>23</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p59">This equation does not factor; therefore, use the quadratic formula to find the solutions for <em class="emphasis">u</em>. Here <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0687" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0688" display="inline"><mrow><mi>b</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0689" display="inline"><mrow><mi>c</mi><mo>=</mo><mn>23</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p60"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0690" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>u</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow><mo>±</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>23</mn></mrow><mo>)</mo></mrow></mrow></msqrt></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>10</mn><mo>±</mo><msqrt><mn>8</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>10</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>±</mo><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p61">Therefore, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0691" display="inline"><mrow><mi>u</mi><mo>=</mo><mn>5</mn><mo>±</mo><msqrt><mn>2</mn></msqrt></mrow><mo>.</mo></math></span> Now back substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0692" display="inline"><mrow><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p62"><span class="informalequation">
<math xml:id="fwk-redden-ch06_m0693" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>u</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>−</mo><msqrt><mn>2</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>u</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>−</mo><msqrt><mn>2</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>5</mn><mo>−</mo><mpadded height="1.2em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>5</mn><mo>+</mo><mpadded height="1.2em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p63">Round the four solutions as follows.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p64"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0694" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msqrt><mrow><mn>5</mn><mo>−</mo><mpadded height="1.3em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt><mo>≈</mo><mo>−</mo><mn>1.89</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msqrt><mrow><mn>5</mn><mo>+</mo><mpadded height="1.3em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt><mo>≈</mo><mo>−</mo><mn>2.53</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>5</mn><mo>−</mo><mpadded height="1.3em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt><mo>≈</mo><mn>1.89</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>5</mn><mo>+</mo><mpadded height="1.3em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></msqrt><mo>≈</mo><mn>2.53</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p65">Answer: The solutions are approximately ±1.89, ±2.53.</p>
</div>
<p class="para block" id="fwk-redden-ch06_s03_s02_p66">If multiple roots and complex roots are counted, then the <span class="margin_term"><a class="glossterm">fundamental theorem of algebra</a><span class="glossdef">If multiple roots and complex roots are counted, then every polynomial with one variable will have as many roots as its degree.</span></span> implies that every polynomial with one variable will have as many roots as its degree. For example, we expect <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0695" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn></mrow></math></span> to have three roots. In other words, the equation</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p67"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0696" display="block"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math>
</span>
should have three solutions. To find them one might first think of trying to extract the cube roots just as we did with square roots,</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p69"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0697" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mn>8</mn><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p70">As you can see, this leads to one solution, the real cube root. There should be two others; let’s try to find them.</p>
<div class="callout block" id="fwk-redden-ch06_s03_s02_n06">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch06_s03_s02_p71">Find the set of all roots: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0698" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p72">Notice that the expression <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0699" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn></mrow></math></span> is a difference of cubes and recall that <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0700" display="inline"><mrow><msup><mi>a</mi><mn>3</mn></msup><mo>−</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Here <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0701" display="inline"><mrow><mi>a</mi><mo>=</mo><mi>x</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0702" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn></mrow></math></span> and we can write</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p73"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0703" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p74">Next apply the zero-product property and set each factor equal to zero. After setting the factors equal to zero we can then solve the resulting equation using the appropriate methods.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p75"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0704" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>±</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msqrt></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mo>±</mo><msqrt><mrow><mo>−</mo><mn>12</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mo>±</mo><mn>2</mn><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo>±</mo><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch06_s03_s02_p76">Using this method we were able to obtain the set of all three roots <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0705" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>±</mo><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span>, one real and two complex.</p>
<p class="para" id="fwk-redden-ch06_s03_s02_p77">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0706" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>±</mo><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch06_s03_s02_p78">Sometimes the roots of a function will occur multiple times. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0707" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span> has degree three where the roots can be found as follows:</p>
<p class="para block" id="fwk-redden-ch06_s03_s02_p79"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0708" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch06_s03_s02_p80">Even though <em class="emphasis">g</em> is of degree 3 there is only one real root {2}; it occurs 3 times.</p>
<div class="key_takeaways editable block" id="fwk-redden-ch06_s03_s02_n07">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch06_s03_s02_l01" mark="bullet">
<li>The quadratic formula can solve any quadratic equation. However, it is sometimes not the most efficient method.</li>
<li>If a quadratic equation can be solved by factoring or by extracting square roots you should use that method.</li>
<li>We can sometimes transform equations into equations that are quadratic in form by making an appropriate <em class="emphasis">u</em>-substitution. After solving the equivalent equation, back substitute and solve for the original variable.</li>
<li>Counting multiple and complex roots, the fundamental theorem of algebra guarantees as many roots as the degree of a polynomial equation with one variable.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s03_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd01">
<h3 class="title">Part A: Solving Quadratic Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch06_s03_qs01_p01"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0709" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0710" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0711" display="inline"><mrow><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0713" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0715" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>90</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0717" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0719" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0721" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0723" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0725" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0727" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0729" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0731" display="inline"><mrow><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>28</mn><mi>t</mi><mo>+</mo><mn>49</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0733" display="inline"><mrow><mn>25</mn><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>t</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0735" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>u</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0737" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>u</mi><mo>−</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0739" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>11</mn><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0741" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0742" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>6</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0744" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0746" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0748" display="inline"><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02">
<h3 class="title">Part B: Solving Equations Quadratic in Form</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02_qd01" start="23">
<p class="para" id="fwk-redden-ch06_s03_qs01_p46"><strong class="emphasis bold">Find all solutions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0750" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>72</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0753" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>17</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0756" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>13</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0757" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>17</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0759" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0760" display="inline"><mrow><mi>x</mi><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0761" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>5</mn><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0762" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>6</mn><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0763" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0764" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>35</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0765" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0767" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0769" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>9</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0771" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>8</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0773" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>14</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>15</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0775" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>24</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0777" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mi>x</mi></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0779" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0781" display="block"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0783" display="block"><mrow><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>13</mn><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0785" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0787" display="inline"><mrow><mn>16</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0788" display="inline"><mrow><mn>30</mn><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>15</mn><msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0790" display="inline"><mrow><msup><mi>y</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0791" display="inline"><mrow><mn>81</mn><msup><mi>y</mi><mn>4</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa48">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0793" display="block"><mrow><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa49">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0795" display="block"><mrow><mn>12</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>11</mn><mrow><mo>(</mo><mrow><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0797" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>19</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0799" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0800" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>5</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>50</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0801" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>7</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0803" display="inline"><mrow><msup><mi>x</mi><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>13</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0804" display="inline"><mrow><msup><mi>y</mi><mn>4</mn></msup><mo>−</mo><mn>14</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>46</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0807" display="inline"><mrow><msup><mi>x</mi><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0808" display="inline"><mrow><mn>2</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0809" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0811" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>17</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0813" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p123"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0815" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0817" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>6</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02_qd02" start="63">
<p class="para" id="fwk-redden-ch06_s03_qs01_p127"><strong class="emphasis bold">Find all solutions. Round your answers to the nearest hundredth.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0818" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0819" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0820" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>14</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0821" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>31</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0822" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0823" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>30</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02_qd03" start="69">
<p class="para" id="fwk-redden-ch06_s03_qs01_p140"><strong class="emphasis bold">Find the set of all roots.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0824" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0826" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0828" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>27</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0830" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0832" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0834" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p153"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0836" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0838" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0840" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><msup><mi>q</mi><mn>3</mn></msup></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0841" display="inline"><mrow><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p159"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0843" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0844" display="inline"><mrow><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02_qd04" start="79">
<p class="para" id="fwk-redden-ch06_s03_qs01_p161"><strong class="emphasis bold">Find all solutions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0846" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>7</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0849" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0852" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>28</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>27</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0855" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>16</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>64</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0857" display="block"><mrow><mrow><mo>|</mo> <mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow> <mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0860" display="block"><mrow><mrow><mo>|</mo> <mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow> <mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0862" display="block"><mrow><mrow><mo>|</mo> <mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow> <mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0865" display="block"><mrow><mrow><mo>|</mo> <mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi></mrow> <mo>|</mo></mrow><mo>=</mo><mn>6</mn></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd02_qd05" start="87">
<p class="para" id="fwk-redden-ch06_s03_qs01_p178"><strong class="emphasis bold">Find a quadratic function with integer coefficients and the given set of roots. (Hint: If <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0867" display="inline"><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0868" display="inline"><mrow><msub><mi>r</mi><mn>2</mn></msub></mrow></math></span> are roots, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0869" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><msub><mi>r</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span>)</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p179"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0870" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><mn>3</mn><mi>i</mi></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p181"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0872" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><mi>i</mi><msqrt><mn>5</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p183"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0874" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><msqrt><mn>3</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p185"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0876" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>±</mo><mn>2</mn><msqrt><mn>6</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p187"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0878" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>1</mn><mo>±</mo><msqrt><mn>3</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p189"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0880" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>2</mn><mo>±</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p191"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0882" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>1</mn><mo>±</mo><mn>6</mn><mi>i</mi></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p193"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0884" display="inline"><mrow><mrow><mo>{</mo><mrow><mn>2</mn><mo>±</mo><mn>3</mn><mi>i</mi></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s03_qs01_qd03_qd01" start="95">
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p195">On a note card, write out your strategy for solving a quadratic equation. Share your strategy on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch06_s03_qs01_p196">Make up your own equation that is quadratic in form. Share it and the solution on the discussion board.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s03_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p03_ans">0, 9</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p07_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0712" display="inline"><mrow><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></math></span>, 0</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0716" display="inline"><mrow><mo>±</mo><mn>3</mn><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa07_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0720" display="block"><mrow><mo>±</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac><mi>i</mi></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa09_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0724" display="block"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa11_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0728" display="block"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>23</mn></mrow></msqrt></mrow><mn>2</mn></mfrac><mi>i</mi></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p27_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0732" display="inline"><mrow><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p31_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0736" display="inline"><mrow><mn>2</mn><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p35_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0740" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa19_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0743" display="block"><mrow><mo>±</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa21_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0747" display="block"><mrow><mfrac><mrow><mn>2</mn><mo>±</mo><msqrt><mn>5</mn></msqrt></mrow><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="23">
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa23_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p48_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0751" display="inline"><mrow><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0752" display="inline"><mrow><mo>±</mo><mn>3</mn><mi>i</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p52_ans">±2, ±3</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p56_ans">1</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p60_ans">4, 9</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p64_ans">−27, −8</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa33_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0766" display="block"><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa34_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p72_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0770" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, 5</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa36_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa37_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0774" display="block"><mrow><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>,</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa38_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa39_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0778" display="block"><mrow><mo>±</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa40_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa41_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0782" display="block"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa42_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa43_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0786" display="block"><mrow><mo>±</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa44_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p92_ans">0, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0789" display="inline"><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa46_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa47_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0792" display="block"><mrow><mo>±</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mo>±</mo><mfrac><mi>i</mi><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa48_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa49_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p100_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0796" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>, 6</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa50_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa51_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p104_ans">1, 16</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa52_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p108_ans">−1, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0802" display="inline"><mrow><mfrac><mn>1</mn><mn>512</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa54_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s03_qs01_qa55_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s03_qs01_p112_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0805" display="inline"><mrow><mo>±</mo><msqrt><mrow><mn>7</mn><mo>−</mo><msqrt><mn>3</mn></msqrt></mrow></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0806" display="inline"><mrow><mo>±</mo><msqrt><mrow><mn>7</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></msqrt></mrow></math></span></p>