Question

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

Answer 1

`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 `ax^2 + bx + c`, we need to think of 2 numbers such that:

`N_1*N_2 = a*c = 1*(-42) = -42`
and
`N_1 +N_2 = b = -1`

After trying out a few numbers we get `N_1 = -7` and `N_2 =6`
`6*(-7) = -42` and `6+(-7)= -1`

`x^2 - x-42 = x^2 - 7x +6x-42`

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

Answer 2

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.