Question

How do you factor `3x^2+29x-44`?

Answer 1

`(x +11)(3x-4)`
Look at explanation to see the logic

Explanation:

The only possible factors of 3 are {1 , 3}

So we have `(x+?)(3x+?)`

Looking at factors of 44 that produce a difference of +29
It has to be a difference as 44 is negative. That means one is +ve and the other -ve. The larger will be +ve as 29 is +ve.

Obviously {1, 44} are no good!

We need `(3xx?)+(1xx?) =29`

`color(blue)(Try 1)`

`(1x +11)(3x-4) = 3x^2+33x-4x-44`

`=3x^2+29x-44`

Answer 2

Factor `f(x) = 3x^2 + 29x - 44`

Ans: (3x - 4)(x + 11).

Explanation:

I use the new AC Method to factor trinomials (Socratic Search).
`f(x) = 3x^2 + 29x - 44`= 3(x + p)(x + q).
Converted trinomial: `f'(x) = x^2 + 29x - 132` = (x + p')(x + q').
p' and q' have opposite signs. Factor pairs of (-132) --> (-3, 22)(-4, 33). This sum is (33 - 4 = 29 = b). Then, p' = (-4) and q' = 33.
Therefor,
`p = (p')/a = -4/3` and `q' = 33/3 = 11`.
Factored form: `f(x) = 3(x - 4/3)(x + 11) = (3x - 4)(x + 11)`

NOTE. This method avoids guessing that becomes confusing when (ac) is a large number. This method is systematic and also avoids the lengthy factoring by grouping.