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5164d4a0-76a1-45bf-a98d-d527a5c2a6a2.html
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<h4>Activity</h4>
<p><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">Completing the square</span> can be used to solve quadratic equations.</p>
<p>Example: Solve \(x^2+8x-2=0\).</p>
<p>To complete the square to solve an equation, follow these steps:</p>
<p><strong>Step 1</strong> - Write the equation in the form \(ax^2+bx=c\).<br>
\(x^2+8x=-2\)</p>
<p><strong>Step 2</strong> - Complete the square by finding \((\frac{b}{2})^2\).<br>
\((\frac{b}{2})^2=(\frac{8}{2})^2=16\)</p>
<p><strong>Step 3</strong> - Add the value to both sides of the equation.<br>
\(x^2+8x+16=-2+16\)</p>
<p><strong>Step 4</strong> - Factor the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">perfect square</span> trinomial and simplify the constants.<br>
\((x+4)^2=14\)</p>
<p><strong>Step 5</strong> - Take the square root of each side (remember the plus-minus sign with the constant).<br>
\(\sqrt{(x+4)^2}=\sqrt{14}\)<br>
\(x+4= \pm \sqrt{14}\)</p>
<p><strong>Step 6</strong> - Write two separate equations and solve each.<br>
\(x + 4 = \sqrt{14}\) or \(x + 4 =\; - \;\sqrt{14}\)<br>
\(x = -4 + \sqrt{14}\) or \(x =\; -\;4\; –\; \sqrt{14}\)</p>
<p>Solve each equation by completing the square. In part a, select the value of \((\frac{b}{2})^2\) that will complete the square. In part b, choose the <strong>two </strong>roots.</p>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="0f2d6070-64b7-419a-870e-316e16b8bab6" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#1a-->
<div class="os-raise-ib-pset-problem" data-content-id="008d3d4f-b998-4f27-83f7-aca8328cfb6b" data-problem-comparator="integer" data-problem-type="input" data-solution="1">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<li>\((x−3)(x+1)=5\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the correct value of \((\frac{b}{2})^2\) that will complete the square?</li>
</ol>
<p>Enter the value:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \((\frac{b}{2})^2 = 1\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is \((\frac{b}{2})^2 = 1\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#1b-->
<div class="os-raise-ib-pset-problem" data-content-id="79029430-6d86-490c-a7d2-3acf7a23fa03" data-problem-type="multiselect" data-solution='["\\(x = -2\\)", "\\(x = 4\\)"]' data-solution-options='["\\(x = -2\\)", "\\(x = 1\\)", "\\(x = 8\\)","\\(x = 4\\)", "\\(x = 9\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li><strong>Select two</strong> roots that solve the quadratic equation.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(x = -2\), \(x = 4\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answers are: \(x = -2\), \(x = 4\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Text Entry Interaction Start -->
<br>
<!--Q#2a-->
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>\(x^2+\frac{1}{2}x=\frac{3}{16}\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the correct value of \((\frac{b}{2})^2\) that will complete the square?</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal1" data-schema-version="1.0">
<p>Compare your answer:</p>
<p>Compare your answer:</p>
<p>\((\frac{b}{2})^2= \frac{1}{16}\)</p>
</div>
<!--Interaction End -->
<br>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="1af0d484-709e-435d-9bd4-9241adccac81" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#2b-->
<div class="os-raise-ib-pset-problem" data-content-id="76c1f248-4edb-4a02-9d24-dab54110bcd7" data-problem-type="multiselect" data-solution='["\\(x=\\frac14\\)", "\\(x=-\\frac34\\)"]' data-solution-options='["\\(x = \\frac12\\)", "\\(x=\\frac14\\)", "\\(x=-\\frac34\\)","\\(x = -\\frac14\\)", "\\(x = 1\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li><strong>Select two </strong>roots that solve the quadratic equation.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(x=\frac14\), \(x=-\frac34\) </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answers are: \(x=\frac14\), \(x=-\frac34\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Text Entry Interaction Start -->
<br>
<!--Q#3a-->
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3a" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="3">
<li>\(x^2+3x+\frac{8}{4}=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the correct value of \((\frac{b}{2})^2\) that will complete the square?</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal3a" data-schema-version="1.0">
<p>Compare your answer:</p>
<p>\((\frac{b}{2})^2= \frac94\)</p>
</div>
<!--Interaction End -->
<br>
<!--Interaction End -->
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="43d165b9-fd10-4fcf-912e-cf8e85993c51" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#3b-->
<div class="os-raise-ib-pset-problem" data-content-id="d20a0059-beb2-4112-8fb1-4ccc23cc7f14" data-problem-type="multiselect" data-solution='["\\(x = -2\\)", "\\(x = -1\\)"]' data-solution-options='["\\(x = -2\\)", "\\(x=\\frac{1}{16}\\)", "\\(x = 9\\)","\\(x = \\frac14\\)", "\\(x = -1\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li><strong>Select two</strong> roots that solve the quadratic equation.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(x = -2\), \(x = -1\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answers are: \(x = -2\), \(x = -1\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q4a-->
<div class="os-raise-ib-pset-problem" data-content-id="4bac0056-6a2a-4f4a-ae8f-753e654ff4ae" data-problem-comparator="integer" data-problem-type="input" data-solution="25">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="4">
<li>\((7−x)(3−x)+3=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the correct value of \((\frac{b}{2})^2\) that will complete the square?</li>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \((\frac{b}{2})^2= 25\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is \((\frac{b}{2})^2= 25\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#4b-->
<div class="os-raise-ib-pset-problem" data-content-id="abf2d852-e418-46c5-85cb-845a6773c2b7" data-problem-type="multiselect" data-solution='["\\(x = 6\\)", "\\(x = 4\\)"]' data-solution-options='["\\(x = -2\\)", "\\(x = 6\\)", "\\(x = 21\\)","\\(x = 4\\)", "\\(x = 9\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li><strong>Select two</strong> roots that solve the quadratic equation.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(x = 6\), \(x = 4\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answers are: \(x = 6\), \(x = 4\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#5a-->
<div class="os-raise-ib-pset-problem" data-content-id="e8b33f01-81e0-4dfc-910c-6fc0f6987a44" data-problem-comparator="float" data-problem-type="input" data-solution="0.64">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="5">
<li>\(x^2+1.6x+0.63=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the correct value of \((\frac{b}{2})^2\) that will complete the square?</li>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \((\frac{b}{2})^2=0.64\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is \((\frac{b}{2})^2=0.64\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q#5b-->
<div class="os-raise-ib-pset-problem" data-content-id="c95204d9-28c9-4d83-a860-91c4b31263a6" data-problem-type="multiselect" data-solution='["\\(x = -0.7\\)", "\\(x = -0.9\\)"]' data-solution-options='["\\(x = -0.7\\)", "\\(x = 0.4\\)", "\\(x = 0.27\\)","\\(x = 0.7\\)", "\\(x = -0.9\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li><strong>Select two</strong> roots that solve the quadratic equation.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(x = -0.7\), \(x = -0.9\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answers are: \(x = -0.7\), \(x = -0.9\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<br>
<h4>Video: Solve by Completing the Square</h4>
<p>Watch the following video to learn more about solving to complete the square.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/288db4630304a967aa6e11fd7c31f5115d5f67ad">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/93eb60f23ed4edcf7fbd9d4ae51631981b16557a" srclang="en_us">https://k12.openstax.org/contents/raise/resources/288db4630304a967aa6e11fd7c31f5115d5f67ad
</video></div>
</div>