# How do you factor the expression x^2-x-42?

Mar 22, 2018

$\left(x + 6\right) \left(x - 7\right)$

#### Explanation:

${x}^{2} - x - 42$

$\implies {x}^{2} - 7 x + 6 x - 42$

$\implies x \left(x - 7\right) + 6 \left(x - 7\right)$

$\implies \left(x + 6\right) \left(x - 7\right)$

Mar 22, 2018

$\left(x + 6\right) \left(x - 7\right)$

#### Explanation:

The coefficient of ${x}^{2}$ is $\boldsymbol{1}$, so the factors will be of this form:

$\left(x + a\right) \left(x + b\right)$

We are looking for an $\boldsymbol{a}$ and a $\boldsymbol{b}$ such that:

$a \times b = - 42$

$a + b = - 1$

Possible factors for $a \times b = - 42$ are:

$- 1 \times 42$

$1 \times 42$

$2 \times - 21$

$- 2 \times 21$

$6 \times - 7$

$- 6 \times 7$

$3 \times - 14$

$- 3 \times 14$

The sum has to equal -1.

We can see that the only factors with a sum of $- 1$ is $6 \mathmr{and} - 7$.

$6 + \left(- 7\right) = - 1$

These are the required values:

$\left(x + 6\right) \left(x - 7\right)$

Expanding:

$\left(x + 6\right) \left(x - 7\right) = {x}^{2} - x - 42$

As required.

With practice you will quickly find the required values without having to list them all.