Question

How do you factor `4x^2-12x+9`?

Answer 1

`(2x-3)(2x-3)` or `(2x-3)^2`

Explanation:

Use the rainbow method.
Multiply the two outer coefficients.
`4 * 9 = 36`
Find two numbers, that when multiplied equal 36, and when added equal -12.
-6 and -6.
`-6 + -6 = 12`
`-6 * -6 = 36`

Rewrite the equation with the x value replaced by the two new values.
`4x^2 -6x - 6x + 9`
Seperate the equation into two parts.
`4x^2 -6x` and `-6x +9`
Find the GCF of the two parts.
`2x(2x - 3)` and `-3(2x-3)`

Take the GCF as your first factor, and the two remaining values as the second.
2x-3 and 2x-3

Answer 2

`4x^2 - 12x+ 9 = 4(x-3/2)^2`

Explanation:

First, factor x^2

`4x^2 - 12x + 9 = 4(x^2 - 3x + 9/4)`

Then, identify the y element in the form `(x+y)^2 = x^2 + 2xy + y^2`

`4(x^2 - 3x + 9/4) = 4(x^2 -2*3/2 x + (3/2)^2)`

Last, factor the perfect square trinomial

`4(x^2 -2*3/2 x + (3/2)^2) = 4(x-3/2)^2`