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

Jan 31, 2016

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

#### Explanation:

Firstly, we expand the right hand side,

$y = {x}^{2} - 5 x + 6$

Now we complete the square and do a bit of algebraic simplification,

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

$y = {\left(x - \frac{5}{2}\right)}^{2} - \frac{25}{4} + 6$

$y = {\left(x - \frac{5}{2}\right)}^{2} - \frac{25}{4} + \frac{24}{4}$

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

Jan 31, 2016

vertex form: $y = 1 {\left(x - \frac{5}{2}\right)}^{2} + \left(- \frac{1}{4}\right)$

#### Explanation:

The general vertex form is:
$\textcolor{w h i t e}{\text{XXX}} y = m {\left(x - \textcolor{b l u e}{a}\right)}^{2} + \textcolor{c y a n}{b}$
with a vertex at $\left(\textcolor{b l u e}{a} , \textcolor{c y a n}{b}\right)$
(So that's our target).

Given
$\textcolor{w h i t e}{\text{XXX}} y = \left(x - 3\right) \left(x - 2\right)$
Expanding the right-side by multiplying:
$\textcolor{w h i t e}{\text{XXX}} y = {x}^{2} - 5 x + 6$
Complete the square
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{{x}^{2} - 5 x} \textcolor{red}{+ {\left(\frac{5}{2}\right)}^{2}} + 6 \textcolor{red}{- \frac{25}{4}}$
Re-write as a squared binomial and simplified constant
$\textcolor{w h i t e}{\text{XXX")y=(x-color(blue)(5/2))^2+color(cyan)("("-1/4")}}$
which is in the general form (assuming a default value $m = 1$)

The graph below for $y = \left(x - 2\right) \left(x - 3\right)$ helps verify that this solution is reasonable.
graph{(x-2)(x-3) [-0.45, 10.647, -2.48, 3.07]}