# What is the vertex form of 3y=(2x − 3)(x - 3) ?

Nov 23, 2017

$y = \frac{2}{3} {\left(x - \frac{9}{4}\right)}^{2} - \frac{3}{8}$

#### Explanation:

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

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

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

$\text{to express "3y=(2x-3)(x-3)" in this form}$

$\Rightarrow 3 y = 2 {x}^{2} - 9 x + 9$

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

$\Rightarrow 3 y = 2 \left({x}^{2} - \frac{9}{2} x + \frac{9}{2}\right)$

• " add/subtract "(1/2"coefficient of x-term")^2

$\text{to } {x}^{2} - \frac{9}{2} x$

$3 y = 2 \left({x}^{2} + 2 \left(- \frac{9}{4}\right) x \textcolor{red}{+ \frac{81}{16}} \textcolor{red}{- \frac{81}{16}} + \frac{9}{2}\right)$

$\textcolor{w h i t e}{3 y} = 2 {\left(x - \frac{9}{4}\right)}^{2} - \frac{9}{8} \leftarrow \textcolor{b l u e}{\text{divide by 3}}$

$\Rightarrow y = \frac{2}{3} {\left(x - \frac{9}{4}\right)}^{2} - \frac{3}{8} \leftarrow \textcolor{red}{\text{in vertex form}}$