Question

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

Answer 1

(4x + 1)(3x - 2)

Explanation:

Use the new AC Method (Socratic Search)
`y = 12x^2 - 5x - 2 =` 12(x + p)(x + q)
Converted trinomial: `y' = x^2 - 5x - 24 =` (x + p')(x + q').
p' and q' have opposite signs (because ac < 0). Factor pairs of (ac = -24) --> (-2, 12)(2, -12) (-3, 8)(3, -8). This sum is - 5 = b. Consequently, p' = 3 and q' = -8.
Back to original trinomial y, `p = (p')/a = 3/12 = 1/4`, and
`q = (q')/a = -8/12 = -2/3`

Factored form: y = 12(x + 1/4)(x - 2/3) = (4x + 1)(3x - 2)