<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-completing-the-square">Completing the Square Exercise</a></p><p><span style="color:#27ae60;">Completing the Square</span> is a technique we can apply to a quadratic equation in order to get into a more useful form. It's probably best to illustrate this with an example.</p>
<p>Imagine we have the quadratic below. Notice that it is NOT easily factorabe. If it were, we would simply factor it.</p>
<p>$$y = x^2 + 6x + 10$$</p>
<p><strong>Step 1</strong>: Halve and then square the $x$-term's coefficient.</p>
<p>$$\frac{6}{2} = 3$$</p>
<p>$$3^2 = 9$$</p>
<p><strong>Step 2</strong>: Add $9$ to both sides of the equation and group the three terms together, leaving the $10$ on the outside.</p>
<p>$$y + 9 = (x^2 + 6x + 9) + 10$$</p>
<p><strong>Step 3</strong>: Simplify.</p>
<p>$$y+9 = (x+3)^2 + 10$$</p>
<p><strong>Step 4</strong>: Move the $9$ back over.</p>
<p>$$y = (x+3)^2 +1$$</p>
<p><strong><span style="color:#8e44ad;">Why is this form helpful?</span></strong></p>
<p>It immediately allows us to see the minimum possible $y$ value. Zero out the $(x+3)^2$ term with $x=-3$ and we can see that the smallest possible $y$ value is $1$.</p>