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

Apr 25, 2018

$\left(2 x - 3\right) \left(2 x - 3\right)$ or ${\left(2 x - 3\right)}^{2}$

#### Explanation:

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

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

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

Apr 25, 2018

$4 {x}^{2} - 12 x + 9 = 4 {\left(x - \frac{3}{2}\right)}^{2}$

#### Explanation:

First, factor x^2

$4 {x}^{2} - 12 x + 9 = 4 \left({x}^{2} - 3 x + \frac{9}{4}\right)$

Then, identify the y element in the form ${\left(x + y\right)}^{2} = {x}^{2} + 2 x y + {y}^{2}$

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

Last, factor the perfect square trinomial

$4 \left({x}^{2} - 2 \cdot \frac{3}{2} x + {\left(\frac{3}{2}\right)}^{2}\right) = 4 {\left(x - \frac{3}{2}\right)}^{2}$