# What is the vertex form of y=2x^2+5x-3?

Apr 6, 2016

$y = 2 {\left(x - \left(- \frac{5}{4}\right)\right)}^{2} + \left(- 6 \frac{1}{8}\right)$

#### Explanation:

General vertex form:
$\textcolor{w h i t e}{\text{XXX}} y = m {\left(x - \textcolor{b l u e}{a}\right)}^{2} + \textcolor{\mathmr{and} a n \ge}{b}$ with vertex at $\left(\textcolor{b l u e}{a} , \textcolor{\mathmr{and} a n \ge}{b}\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} y = 2 {x}^{2} + 5 x - 3$

Extract "spread factor" $m$
$\textcolor{w h i t e}{\text{XXX}} y = 2 \left({x}^{2} + \frac{5}{2} x\right) - 3$

Complete the square
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{red}{2} \left({x}^{2} + \frac{5}{2} x \textcolor{red}{+ {\left(\frac{5}{4}\right)}^{2}}\right) - 3 - \textcolor{red}{2 {\left(\frac{5}{4}\right)}^{2}}$

Write as a squared binomial and simplify the constant
$\textcolor{w h i t e}{\text{XXX}} y = 2 {\left(x + \frac{5}{4}\right)}^{2} - 6 \frac{1}{8}$

Re-write to match signs of standard general form:
color(white)("XXX")y=2(x-color(blue)(color(white)("")(-5/4)))^2+color(orange)(color(white)("")(-6 1/8))