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

2 Answers
Jan 31, 2016

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.

Jan 31, 2016

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]}