Finding the Minimum Value

<p><span style="font-size:22px"><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-finding-the-minimum-value">Finding the Minimum Value Exercise</a></span></p><p>You can <span style="color:#27ae60;">Find the Minimum Value</span> of a quadratic equation in a couple of ways.</p> <p><strong><span style="color:#8e44ad;">Way 1: If possible, factor the quadratic equation.</span></strong></p> <p>If it has one $x$-intercept, then that is literally the minimum value. See the example below:</p> <p style="text-align: center;">$$y=x^2+6x+9$$</p> <p style="text-align: center;">$$y=(x+3)^2$$</p> <p style="text-align: center;"><span style="color:#e74c3c;">So we know the minimum point is $(-3, 0)$</span></p> <p>If it has two $x$-intercepts, the $x$-value exactly between those two values can be used to find the minimum.</p> <p>$$y = (x+5)(x-3)$$</p> <p>The two $x$-intercepts are $-5$ and $3$. The $x$-value exactly in the middle is $x = -1$. Plug that into the original equation to find the minimum.</p> <p>$$y = (4)(-4) = -16$$</p> <p style="text-align: center;"><span style="color:#e74c3c;">So we know the minimum point is $(-1, -16)$</span></p> <p><strong><span style="color:#8e44ad;">Way 2: Complete the square (arguably the easier method).</span></strong></p> <p>If you complete the square of a quadratic, the minimum point is fairly obvious. See the example below:</p> <p>$$y = x^2 + 10x + 2$$</p> <p>$$y + 25 = x^2 +10x + 25 + 2$$</p> <p>$$y + 25 = (x^2 +10x + 25) + 2$$</p> <p>$$y + 25 = (x+5)^2 + 2$$</p> <p>$$y = (x+5)^2 - 23$$</p> <p style="text-align: center;"><span style="color:#e74c3c;">So we know the minimum point is $(-5, -23)$</span></p>