# What is the vertex form of y= (x+2) (2x+5) ?

Mar 10, 2018

$y = 2 {\left(x + \frac{9}{4}\right)}^{2} - \frac{1}{8}$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a }$
$\text{is a multiplier}$

$y = \left(x + 2\right) \left(2 x + 5\right) \leftarrow \textcolor{b l u e}{\text{expand the factors}}$

$\textcolor{w h i t e}{y} = 2 {x}^{2} + 9 x + 10$

$\text{to obtain vertex form use "color(blue)"completing the square}$

• " the coefficient of the "x^2" term must be 1"

$\Rightarrow y = 2 \left({x}^{2} + \frac{9}{2} x + 5\right)$

• " add/subtract "(1/2"coefficient of the x-term")^2" to"
${x}^{2} + \frac{9}{2} x$

$\Rightarrow y = 2 \left({x}^{2} + 2 \left(\frac{9}{4}\right) x \textcolor{red}{+ \frac{81}{16}} \textcolor{red}{- \frac{81}{16}} + 5\right)$

$\textcolor{w h i t e}{y} = 2 {\left(x + \frac{9}{4}\right)}^{2} + 2 \left(- \frac{81}{16} + 5\right)$

$\textcolor{w h i t e}{y} = 2 {\left(x + \frac{9}{4}\right)}^{2} - \frac{1}{8} \leftarrow \textcolor{red}{\text{in vertex form}}$