# How do you factor x^2 - 7x + 12?

Jul 23, 2016

$\left(x - 3\right) \left(x - 4\right)$

#### Explanation:

We are looking for factors of 12 which ADD (because of + 12) up to 7.

The signs will be the same (because of + 12), they are both negative (because of -7).

The factors which satisfy both conditions are $- 3 \mathmr{and} - 4.$

$- 3 \times - 4 = + 12 , \mathmr{and} - 3 + - 4 = - 7$

$\left(x - 3\right) \left(x - 4\right)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let's look at the possible combinations of factors of $12$

$\underline{\text{subtract"color(white)(..................)12color(white)(..................)"add}}$
$\downarrow \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \downarrow \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} \downarrow$

$11 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots . .} 1 \times 12 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots} 13$
$4 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 2 \times 6 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} 8$
$1 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \textcolor{red}{3 \times 4} \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots} \textcolor{red}{7} \leftarrow \text{ } 3 \mathmr{and} 4$ are the
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .}$factors we need.

In order to factorise easily you have to know the multiplication tables well. Without that it is a long slog with lots of trial and error!
Learn the tables!