# What is the vertex form of y=(2x+1)(x +6) + 6x?

Jul 8, 2016

$y = \textcolor{g r e e n}{2} {\left(x - \left(\textcolor{red}{- \frac{19}{4}}\right)\right)}^{2} + \left(\textcolor{b l u e}{- \frac{313}{8}}\right)$

with vertex at $\left(\textcolor{red}{- \frac{19}{4}} , \textcolor{b l u e}{- \frac{313}{8}}\right)$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{b r o w n}{\left(2 x + 1\right) \left(x + 6\right)} + 6 x$

(first converting into standard form):
$\textcolor{w h i t e}{\text{XXX}} \Rightarrow y = \textcolor{b r o w n}{2 x \left(x + 6\right) + 1 \left(x + 6\right)} + 6 x$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow y = \textcolor{b r o w n}{2 {x}^{2} + 12 x + x + 6} + 6 x$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow y = 2 {x}^{2} + 19 x + 6$

Remember the vertex form is $y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Extract the $\textcolor{g r e e n}{m}$ factor
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} \left({x}^{2} + \frac{19}{2} x\right) + 6$

Complete the square
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} \left({x}^{2} + \frac{19}{2} x + {\left(\frac{19}{4}\right)}^{2}\right) + 6 - 2 \cdot {\left(\frac{19}{4}\right)}^{2}$

Simplify into vertex form
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} {\left(x - \left(\textcolor{red}{- \frac{19}{4}}\right)\right)}^{2} + \left(\textcolor{b l u e}{- \frac{313}{8}}\right)$

Since $- \frac{19}{4}$ is about $- 5$
and $- \frac{313}{8}$ is about $- 39$
the graph below of the original equation supports this answer.
graph{(2x+1)(x+6)+6x [-19.8, 12.24, -40.05, -24.01]}