# What is the the vertex of y=x^2+7x +12 ?

Jan 29, 2016

$\left(- \frac{7}{2} , - \frac{1}{4}\right)$

#### Explanation:

Re-express in vertex form by completing the square:

$y = {x}^{2} + 7 x + 12$

$= {x}^{2} + 7 x + {\left(\frac{7}{2}\right)}^{2} - {\left(\frac{7}{2}\right)}^{2} + 12$

$= {\left(x + \frac{7}{2}\right)}^{2} - \frac{49}{4} + \frac{48}{4}$

$= 1 {\left(x - \left(- \frac{7}{2}\right)\right)}^{2} + \left(- \frac{1}{4}\right)$

The equation:

$y = 1 {\left(x - \left(- \frac{7}{2}\right)\right)}^{2} + \left(- \frac{1}{4}\right)$

is in vertex form:

$y = a {\left(x - h\right)}^{2} + k$

with multiplier $a = 1$ and vertex $\left(h , k\right) = \left(- \frac{7}{2} , - \frac{1}{4}\right)$