Question

How do you factor `8x^2 - 4x - 24`?

Answer 1

Well, you can factor out 4 immediately:

`4(2x^2 - x - 6)`

You can work out that the further factorization will look something like:

`(2x + a)(x + b)` where `a*b = -6`

and

`(2x * b) + (a * x) = -1x`

so you know that `2a + b= -1`
and

`a*b = -6`

You know that either a or b must be negative, and the other number must be positive.

At this point, I usually resort to trial and error guessing. There are usually a couple of choices, so try one, then the other.

so what if `a = 3` and `b = -2`, does that work?

`(2x + 3)(x - 2) = 2x^2 -4x + 3x -6 = 2x^2 - x -6`

...good guess. So your factorization will be:

`4(2x + 3)(x - 2)`

Answer 2

`=4(2x+3)(x-2)`

Explanation:

First take out the common factor of `4`

`8x^2 -4x -24`

`=4(2x^2 -x-6)`

Find factors of `2 and 6` whose products differ by `1`

`" "2 and 6`
`" "darr" "darr`

`" "2" "3" "rarr 1 xx 3 = 3`
`" "1" "2" "rarr 2xx2 = ul4`
`color(white)(xxxxxxxxx.x.xxxxxx)1`

`8x^2 -4x -24`

`=4(2x^2 -x-6)`

`=4(2x+3)(x-2)`