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<h4>Activity </h4>
<p>Let’s begin by reviewing how to convert between standard and factored form using expressions like the ones we have seen before.</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>Each row of the table below has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.</li>
</ol>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factored form
</th>
<th scope="col">
Standard form
</th>
</tr></thead><tbody>
<tr>
<td>
<p>\((x + 5)(x + 6)\)</p>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>
<p>\(x^2+13x +30\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x − 3)(x − 6)\)</p>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>
<p>\(x^2−11x +18\)</p>
</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">
Factored form
</th>
<th scope="col">
Standard form
</th>
</tr></thead><tbody>
<tr>
<td>
<p>\((x + 5)(x + 6)\)</p>
</td>
<td>
<p>\(x^2 + 11x + 30\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x + 10)(x + 3)\)</p>
</td>
<td>
<p>\(x^2 + 13x + 30\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x − 3)(x − 6)\)</p>
</td>
<td>
<p>\(x^2 − 9x + 18\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x − 9)(x − 2)\)</p>
</td>
<td>
<p>\(x^2 − 11x + 18\)</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<p>In the warm up, you examined factors that had different signs. Two numbers’ signs affect the sum when they are combined and the product when they are multiplied. Use your observations from the warm up to explore these expressions that are in some ways unlike the ones we have seen before.</p>
<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>Each row of the table below has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.</li>
</ol>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Factored form
</th>
<th scope="col">
Standard form
</th>
</tr></thead><tbody>
<tr>
<td>
<p>\((x + 12)(x − 3)\)</p>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>
<p>\(x^2 − 9x − 36\)</p>
</td>
</tr>
<tr>
<td> </td>
<td>
<p>\(x^2 − 35x − 36\)</p>
</td>
</tr>
<tr>
<td> </td>
<td>
<p>\(x^2 + 35x − 36\)</p>
</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">
Factored form
</th>
<th scope="col">
Standard form
</th>
</tr></thead><tbody>
<tr>
<td>
<p>\((x + 12)(x − 3)\)</p>
</td>
<td>
<p>\(x^2 + 9x − 36\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x − 12)(x + 3)\)</p>
</td>
<td>
<p>\(x^2 − 9x − 36\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x − 36)(x + 1)\)</p>
</td>
<td>
<p>\(x^2 − 35x − 36\)</p>
</td>
</tr>
<tr>
<td>
<p>\((x + 36)(x − 1)\)</p>
</td>
<td>
<p>\(x^2 + 35x − 36\)</p>
</td>
</tr>
</tbody>
</table>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="d0ee6fbb-b03a-4f03-b17d-dcdd0bb645ac" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>Name some ways that the expressions in the second table are different from those in the first table (aside from the fact that the expressions use different numbers).</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your explanation.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> In the second table, the expressions in standard form all have a negative constant term. In the first table, they are all positive and the numbers are smaller.</p>
<p>The factored expressions in the first table are either both sums or both differences (same operation in both binomials). In the second table, they all consist of a sum and a difference (different operations in the binomials).</p>
</div>
</div>
<br>
<!--Interaction End -->
<div class="os-raise-student-reflection">
<p class="os-raise-student-reflection-title">Why Should I Care?</p>
<img src="https://k12.openstax.org/contents/raise/resources/9551a37a37e8de3027a388ddb792a6675924846a" width="300"/>
<p>Matteo's mom is a landscape architect. One day, she took Matteo to see the large succulent garden that she had designed for a new hospital. She told him that the garden's area was determined using a single quadratic function. </p>
<p>Then, she introduced Matteo to Lucy, the accountant for the plant store where she bought the succulents. Lucy used quadratic functions to model a demand curve and determine how to maximize profits. This allowed them to find the most fantastic desert plants that would make the plant store the most money.</p>
<p>Both Matteo's mom and Lucy use quadratic functions to excel at their jobs.</p>
</div>