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7cff5e44-b610-4179-b36f-d3735aaad773.html
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<h4>Activity </h4>
<p>For questions 1 - 2, consider the expression \(x^2 + bx + 100\). (Note the leading coefficient is 1.)</p>
<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">
<li>Create a table with all pairs of factors of +100 that would give positive values of \(b\).</li>
</ol>
<p>For each pair, state the \(b\) value they produce. (Use as many rows as needed.)</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (positive)
</th>
</tr>
</thead>
<tbody>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answers:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (positive)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>10</p>
</td>
<td>
<p>10</p>
</td>
<td>
<p>20</p>
</td>
</tr>
<tr>
<td>
<p>20</p>
</td>
<td>
<p>5</p>
</td>
<td>
<p>25</p>
</td>
</tr>
<tr>
<td>
<p>25</p>
</td>
<td>
<p>4</p>
</td>
<td>
<p>29</p>
</td>
</tr>
<tr>
<td>
<p>50</p>
</td>
<td>
<p>2</p>
</td>
<td>
<p>52</p>
</td>
</tr>
<tr>
<td>
<p>100</p>
</td>
<td>
<p>1</p>
</td>
<td>
<p>101</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Create a table with all pairs of factors of +100 that would give negative values of \(b\).</li>
</ol>
<p>For each pair, state the \(b\) value they produce. (Use as many rows as needed.)</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (negative)
</th>
</tr>
</thead>
<tbody>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Compare your answers:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (negative)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>–10</p>
</td>
<td>
<p>–10</p>
</td>
<td>
<p>–20</p>
</td>
</tr>
<tr>
<td>
<p>–20</p>
</td>
<td>
<p>–5</p>
</td>
<td>
<p>–25</p>
</td>
</tr>
<tr>
<td>
<p>–25</p>
</td>
<td>
<p>–4</p>
</td>
<td>
<p>–29</p>
</td>
</tr>
<tr>
<td>
<p>–50</p>
</td>
<td>
<p>–2</p>
</td>
<td>
<p>–52</p>
</td>
</tr>
<tr>
<td>
<p>–100</p>
</td>
<td>
<p>–1</p>
</td>
<td>
<p>–101</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<p>For questions 3–5, consider the expression \(x^2 + bx − 100\). (Note the leading coefficient is 1.)</p>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="3">
<li>Create a table with all pairs of factors of –100 that would result in positive values of \(b\).</li>
</ol>
<p>For each pair of factors, state the \(b\) value they produce. (Use as many rows as needed.)</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (positive)
</th>
</tr>
</thead>
<tbody>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal3">
<p>Compare your answers:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (positive)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>25</p>
</td>
<td>
<p>–4</p>
</td>
<td>
<p>21</p>
</td>
</tr>
<tr>
<td>
<p>20</p>
</td>
<td>
<p>–5</p>
</td>
<td>
<p>15</p>
</td>
</tr>
<tr>
<td>
<p>50</p>
</td>
<td>
<p>–2</p>
</td>
<td>
<p>48</p>
</td>
</tr>
<tr>
<td>
<p>100</p>
</td>
<td>
<p>–1</p>
</td>
<td>
<p>99</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal4" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="4">
<li>Create a table with all pairs of factors of –100 that would give negative values of \(b\).</li>
</ol>
<p>For each pair, state the \(b\) value they produce. (Use as many rows as needed.)</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (negative)
</th>
</tr>
</thead>
<tbody>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal4">
<p>Compare your answers:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\)(negative)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>–25</p>
</td>
<td>
<p>4</p>
</td>
<td>
<p>–21</p>
</td>
</tr>
<tr>
<td>
<p>–20</p>
</td>
<td>
<p>5</p>
</td>
<td>
<p>–15</p>
</td>
</tr>
<tr>
<td>
<p>–50</p>
</td>
<td>
<p>2</p>
</td>
<td>
<p>–48</p>
</td>
</tr>
<tr>
<td>
<p>–100</p>
</td>
<td>
<p>1</p>
</td>
<td>
<p>–99</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal5" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="5">
<li>Create a table with all pairs of factors of –100 that would result in a zero value of \(b\).</li>
</ol>
<p>For each pair, state the \(b\) value they produce. (Use as many rows as needed.)</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (zero)
</th>
</tr>
</thead>
<tbody>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal5">
<p>Compare your answers:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factor 1
</th>
<th scope="col">
Factor 2
</th>
<th scope="col">
\(b\) (zero)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>–10</p>
</td>
<td>
<p>10</p>
</td>
<td>
<p>0</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<p>For problems 6–9, write each expression in factored form. Note that the leading coefficient for each expression is 1.</p>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e6f72e97-9ec3-4b2c-ba58-64e6df94ba21" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>\(x^2 − 25x + 100\)</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the expression in factored form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> \((x − 20)(x − 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="7cc13240-5fa1-437a-93fe-8e24d966fe5f" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="7">
<li>\(x^2 + 15x − 100\)</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the expression in factored form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> \((x + 20)(x − 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c30ee140-b975-43f4-bfd3-8aa9ee2959b0" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="8">
<li>\(x^2− 15x − 100\)</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the expression in factored form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\((x − 20)(x + 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="84b2d014-87f7-4bd8-98ba-5f65d8b41776" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="9">
<li>\(x^2 + 99x − 100\)</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the expression in factored form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> \((x + 100)(x − 1)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<h4>Video: Analyzing factors of 100 and –100</h4>
<p>Watch the following video to learn more about analyzing the factors of 100 and –100 and applying them in factoring quadratic expressions.</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/270f058fe550b32cb28545a231874e87896e014f">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/dfe27fff44654188ca206882048b6dc328ac4c3e" srclang="en_us">https://k12.openstax.org/contents/raise/resources/87200bcaaba7836d36bc214d69281dc1be0e0abb
</video></div>
</div>
<br>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
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<h4>Extending Your Thinking</h4>
<p>How many different integers \(b\) can you find so that the expression \(x^2 + 10x + b\) can be written in factored form?</p>
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<p>Enter the number of different integers \(b\).</p>
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<p>Compare your answer:</p>
<p>Infinitely many. The number \(b\) needs to have two factors that add up to 10. Here are some examples:</p>
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<li> \((x + 4)(x + 6) = x^2 + 10x + 24\) </li>
<li> \((x + 2)(x + 8) = x^2+ 10x + 16\) </li>
<li> \((x + 11)(x − 1) = x^2 + 10x − 11\) </li>
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<p>There are infinitely many examples because there are infinitely many pairs of integers whose sum is 10.</p>
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