What is the vertex form of y= 2x^2 + 8x+4 ?

Jan 7, 2018

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

Explanation:

The general vertex form is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} y = 2 {x}^{2} + 8 x + 4$

Extract the $\textcolor{g r e e n}{m}$ factor (in this case $\textcolor{g r e e n}{2}$ from the first two terms
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} \left({x}^{2} + 4 x\right) + 4$

Supposing that the $\left({x}^{2} + 4 x\right)$ are part of a squared binomial ${\left(x - \textcolor{red}{a}\right)}^{2} = \left({x}^{2} - 2 \textcolor{red}{a} x + {\textcolor{red}{a}}^{2}\right)$
then $- 2 \textcolor{red}{a} x$ must be equal to $4 x$
$\rightarrow \textcolor{red}{a} = \frac{4}{- 2} = \textcolor{red}{- 2}$
and in order to "complete the square" an extra ${\textcolor{red}{a}}^{2} = {\left(- 2\right)}^{2} = \textcolor{m a \ge n t a}{4}$ will need to be added inside the brackets to the $\left({x}^{2} + 4 x\right)$ we already have.
Note that if we insert this $\textcolor{m a \ge n t a}{+ 4}$ because of the factor $\textcolor{g r e e n}{m} = \textcolor{g r e e n}{2}$
we will really be adding $\textcolor{g r e e n}{2} \times \textcolor{m a \ge n t a}{4} = \textcolor{b r o w n}{8}$ to the expression.

To maintain equality if we add $\textcolor{b r o w n}{8}$ we will also need to subtract it:
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} \left({x}^{2} + 4 x + \textcolor{m a \ge n t a}{4}\right) + 4 \textcolor{b r o w n}{- 8}$
Simplifying
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{2} {\left(x + 2\right)}^{2} - 4$

Adjusting to match the sign requirements of the vertex form:
$\textcolor{w h i t e}{\text{XXX")y=color(green)2(x-color(red)(""(-2)))^2+color(blue)(} \left(- 4\right)}$

The graph below of the original equation supports this result.
graph{2x^2+8x+4 [-8.386, 2.71, -5.243, 0.304]}