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

Sep 19, 2015

color(blue)((x+6) (x-7) is the factorised form of the expression.

#### Explanation:

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

We can Split the Middle Term of this expression to factorise it.

In this technique, if we have to factorise an expression like $a {x}^{2} + b x + c$, we need to think of 2 numbers such that:

${N}_{1} \cdot {N}_{2} = a \cdot c = 1 \cdot \left(- 42\right) = - 42$
and
${N}_{1} + {N}_{2} = b = - 1$

After trying out a few numbers we get ${N}_{1} = - 7$ and ${N}_{2} = 6$
$6 \cdot \left(- 7\right) = - 42$ and $6 + \left(- 7\right) = - 1$

${x}^{2} - x - 42 = {x}^{2} - 7 x + 6 x - 42$

$= x \left(x - 7\right) + 6 \left(x - 7\right)$
color(blue)((x+6) (x-7) is the factorised form of the expression.

Sep 19, 2015

Factor y = x^2 - x - 42.

Ans: (x + 6)(x - 7)

#### Explanation:

Using the new AC Method (Socratic Search) is simpler and faster.
y = x^2 - x - 42 = (x + p)(x + q).
p and q have opposite signs. Factor pairs of (-42) --> (-3, 14)(-6, 7). This sum is 1 = -b. The opposite sum gives p and q --> p = 6
and q = -7.
Factored form: y = (x + 6)(x - 7)

NOTE . This new AC Method shows a systematic way to find the 2 numbers p and q, instead of guessing. In addition, it avoids the lengthy factoring by grouping.