# What is the the vertex of y = (x+6)(x+4) ?

Jan 4, 2016

The vertex is the point $\left(x , y\right) = \left(- 5 , - 1\right)$.

#### Explanation:

Let $f \left(x\right) = \left(x + 6\right) \left(x + 4\right) = {x}^{2} + 10 x + 24$.

One approach is to just realize that the vertex occurs halfway between the $x$-intercepts of $x = - 4$ and $x = - 6$. In other words, the vertex is at $x = - 5$. Since $f \left(- 5\right) = 1 \cdot \left(- 1\right) = - 1$, this means the vertext is at $\left(x , y\right) = \left(- 5 , - 1\right)$.

For a more general approach that works even when the quadratic function has no $x$-intercepts, use the method of Completing the Square :

$f \left(x\right) = {x}^{2} + 10 x + 24 = {x}^{2} + 10 x + {\left(\frac{10}{2}\right)}^{2} + 24 - 25 = {\left(x + 5\right)}^{2} - 1$.

This puts the quadratic function in "vertex form", which allows you to see that its minimum value of $- 1$ occurs at $x = - 5$.

Here's the graph:

graph{(x+6)(x+4) [-20, 20, -10, 10]}