How do you factor 12x^2 - 5x - 3?

2 Answers
Alan P.
May 23, 2015

We start by hoping for integer coefficient terms for our factors
and note
factors of (6) = {(1,12), (2,6), (3,4)}
factors of (3) = {(1,3)}

We are looking for
(ax+-b)(cx+-d)
where (a,c) are factors of 6
and (b,d) are factors of 3
such that ad-bc = -5

With the limited pairs of factors which we have predetermined (and some juggling of plus/minus signs)
we get
12x^2-5x-3 = (3x+1)(4x-3)

Guilherme N.
May 23, 2015

You can find its roots (using Bhaskara, in this resolution) and then turn them into factors, by equaling each to zero, as follows.

(5+-sqrt(25-4(12)(-3)))/24
(5+-13)/24
x_1=3/4, which can be rewritten as the factor (4x-3)=0
x_2=-1/3, which can be rewritten as the factor (3x+1)=0

So

12x^2-5x-3=(4x-3)(3x+1)

Related questions
See all questions in Factorization of Quadratic Expressions
Impact of this question
6455 views around the world
You can reuse this answer
Creative Commons License