Question

How do you factor `10x^2 - 7x - 12`?

Answer 1

Use a variant of the AC Method.

`A=10`, `B=7`, `C=12`

Look for a pair of factors of `AC=10xx12=120` whose difference is `7`. We look for the difference rather than the sum because the sign of the constant term is negative.

The pair `15, 8` works:

`15 xx 8 = 120`
`15 - 8 = 7`

Now use that pair to split the middle term, then factor by grouping...

`10x^2-7x-12 = 10x^2-15x+8x-12`

`=(10x^2-15x)+(8x-12)`

`=5x(2x-3)+4(2x-3)`

`=(5x+4)(2x-3)`

Answer 2

The method George used is the popular factoring AC Method (YouTube). To avoid the lengthy factoring by grouping, you may use the new AC Method.

`f(x) = 10x^2 - 7x - 12 = 10(x + p)(x + q).`
Convert y to `y' = x^2 - 7x - 120` = (x + p')(x + q')
Find p' and q' by composing factor pairs of (a.c = -120). a and c have different signs. Proceed: ...(-4, 30)(-5, 24)(-8, 15). This sum is 7 = -b. Then p' = 8 and q' = -15.
Then, `p = (p')/a = 8/10 = 4/5` and `q = -15/10 = -3/2.`

Factored form: `f(x) = 10(x + 4/5)(x - 3/2) = (5x + 4)(2x - 3).`

Check by developing: f(x) = 10x^2 - 15x + 8x - 12. OK.

This new AC Method is fast, systematic, no guessing, no factoring by grouping.