# How do you factor 12x²-24x+9?

Nov 3, 2016

$12 {x}^{2} - 24 x + 9 = 3 \left(2 x - 1\right) \left(2 x - 3\right)$

#### Explanation:

First separate out the common scalar factor $3$:

$12 {x}^{2} - 24 x + 9 = 3 \left(4 {x}^{2} - 8 x + 3\right)$

To factor the remaining quadratic $4 {x}^{2} - 8 x + 3$ use an AC method:

Find a pair of factors of $A C = 4 \cdot 3 = 12$ with sum $B = 8$

The pair $6 , 2$ works.

Use this pair to split the middle term and factor by grouping:

$4 {x}^{2} - 8 x + 3 = 4 {x}^{2} - 6 x - 2 x + 3$

$\textcolor{w h i t e}{4 {x}^{2} - 8 x + 3} = \left(4 {x}^{2} - 6 x\right) - \left(2 x - 3\right)$

$\textcolor{w h i t e}{4 {x}^{2} - 8 x + 3} = 2 x \left(2 x - 3\right) - 1 \left(2 x - 3\right)$

$\textcolor{w h i t e}{4 {x}^{2} - 8 x + 3} = \left(2 x - 1\right) \left(2 x - 3\right)$

Putting it all together:

$12 {x}^{2} - 24 x + 9 = 3 \left(2 x - 1\right) \left(2 x - 3\right)$