332db8c6-7248-4cf1-b101-7427237c63b5.html

<h4>Deriving the Quadratic Formula Using an Example</h4>
<p>Here is another example that goes through the process of deriving the <span class="os-raise-ib-tooltip" data-schema-version="&ldquo;1.0&rdquo;" data-store="glossary-tooltip">quadratic formula</span>:</p>
<p>Original equation<br>
  \(x^2+7x+4=0\)</p>
<p><strong>Step 1</strong> - Multiply each side by 4. This helps us avoid some tricky fractions. (In general, multiply each side of the equation by 4 times the coefficient of the \(x^2\) term. In this case, that is just 4 times 1, which is 4.)<br>
  \(4x^2+28x+16=0\)</p>
<p><strong>Step 2</strong> - Subtract 16 from each side.<br>
  \(4x^2+28x=-16\)</p>
<p><strong>Step 3</strong> - Rewrite.<br>
  \(4x^2\) as \((2x)^2\) and \(28x\) as \(14(2x)\)<br>
  \((2x)^2+14(2x)=-16\)</p>
<p><strong>Step 4</strong> - Use \(P\) as a placeholder for \(2x\).<br>
  \(P^2+14P=-16\)</p>
<p><strong>Step 5</strong> - Add to complete the square.<br>
  \(P^2+14P+7^2=-16+7^2\)</p>
<p><strong>Step 6</strong> - Write the left side as a squared factor.<br>
  \((P+7)^2=-16+7^2\)</p>
<p><strong>Step 7</strong> - Find the square root of the expression on the right.<br>
  \(P+7= \pm \sqrt{-16+7^2}\)</p>
<p><strong>Step 8</strong> - Subtract 7 from both sides to isolate.<br>
  \(P=-7 \pm \sqrt{-16+7^2}\)</p>
<p><strong>Step 9</strong> - Rearrange the expression under the square root sign.<br>
  \(P=-7 \pm \sqrt{7^2-16}\)</p>
<p>Step 10<br>
  Re-substitute \(2x\) for \(P\).<br>
  \(2x=-7 \pm \sqrt{7^2-16}\)</p>
<p>Step 11<br>
  Divide each side by 2 to isolate \(x\).<br>
  \(x=\frac{-7 \pm \sqrt{7^2-16}}{2}\) </p>
<p>Notice the original equation, \(y=x^2+7x+4\), has \(a=1\), \(b=7\), \(c=4\), and the last step is of the form \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), which is the quadratic formula.</p>
<h4>Try It: Deriving the Quadratic Formula Using an Example</h4>
<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">
    <p>Jaiden was deriving the quadratic formula from the quadratic \(x^2+11x+5=0\). What is the step after the step shown below?</p>
    <p>\((x+\frac{11}{2})^2=\frac{11^2-4(1)(5)}{4(1)}\)</p>
    <p>\((x+\frac{11}{2})^2=\frac{121-20}{4}\)</p>
  </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-schema-version="1.0" data-wait-for-event="Reveal1">
  <p>Here is how to find the next step:</p>
  <p>To isolate \(x\), take the square root of both sides:</p>
  <p>\(\sqrt{(x+\frac{11}{2})^2}=\sqrt{\frac{121-20}{4}}\)</p>
</div>