# What is the vertex form of y=-1/3(x-2)(2x+5) ?

The Vertex Form is ${\left(x - - \frac{1}{4}\right)}^{2} = - \frac{3}{2} \cdot \left(y - \frac{27}{8}\right)$

#### Explanation:

We start from the given
$y = - \frac{1}{3} \left(x - 2\right) \left(2 x + 5\right)$
Expand first
$y = - \frac{1}{3} \left(2 {x}^{2} - 4 x + 5 x - 10\right)$
simplify
$y = - \frac{1}{3} \left(2 {x}^{2} + x - 10\right)$
insert a $1 = \frac{2}{2}$ to make factoring of 2 clear

$y = - \frac{1}{3} \left(2 {x}^{2} + \frac{2}{2} x - 10\right)$

now, factor out the 2

$y = - \frac{2}{3} \left({x}^{2} + \frac{x}{2} - 5\right)$
complete the square now by adding $\frac{1}{16}$ and subtracting $\frac{1}{16}$ inside the grouping symbol

$y = - \frac{2}{3} \left({x}^{2} + \frac{x}{2} + \frac{1}{16} - \frac{1}{16} - 5\right)$

the first 3 terms inside the grouping symbol is now a Perfect Square Trinomial so that the equation becomes

$y = - \frac{2}{3} \left({\left(x + \frac{1}{4}\right)}^{2} - \frac{81}{16}\right)$
Distribute the $- \frac{2}{3}$ inside the grouping symbol
$y = - \frac{2}{3} {\left(x + \frac{1}{4}\right)}^{2} - \frac{2}{3} \left(- \frac{81}{16}\right)$

$y = - \frac{2}{3} {\left(x - - \frac{1}{4}\right)}^{2} + \frac{27}{8}$
let us simplify now to the Vertex Form

$y - \frac{27}{8} = - \frac{2}{3} {\left(x - - \frac{1}{4}\right)}^{2}$

Finally

${\left(x - - \frac{1}{4}\right)}^{2} = - \frac{3}{2} \left(y - \frac{27}{8}\right)$

graph{(x--1/4)^2=-3/2(y-27/8)[-20,20,-10,10]}

God bless...I hope the explanation is useful..