Question

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

Answer 1

`y = (x - 5/2)^2 - 1/4`.

Explanation:

Firstly, we expand the right hand side,

`y = x^2 - 5x + 6`

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

`y = x^2 - 5x + (5/2)^2 - (5/2)^2 + 6`

`y = (x - 5/2)^2 - 25/4 + 6`

`y = (x - 5/2)^2 - 25/4 + 24/4`

`y = (x - 5/2)^2 - 1/4`.

Answer 2

vertex form: `y=1(x-5/2)^2+(-1/4)`

Explanation:

The general vertex form is:
`color(white)("XXX")y=m(x-color(blue)(a))^2+color(cyan)(b)`
with a vertex at `(color(blue)(a),color(cyan)(b))`
(So that's our target).

Given
`color(white)("XXX")y=(x-3)(x-2)`
Expanding the right-side by multiplying:
`color(white)("XXX")y=x^2-5x+6`
Complete the square
`color(white)("XXX")y=color(green)(x^2-5x)color(red)(+(5/2)^2)+6color(red)(-25/4)`
Re-write as a squared binomial and simplified constant
`color(white)("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=(x-2)(x-3)` helps verify that this solution is reasonable.
graph{(x-2)(x-3) [-0.45, 10.647, -2.48, 3.07]}