# How do you factor 6x^2 +5xy -21y^2?

May 30, 2016

(2x - 3y)(3x + 7y)

#### Explanation:

Use the new AC Method (Socratic Search)
Consider x as a variable, y as a constant, and factor f(x).
$f \left(x\right) = 6 {x}^{2} + 5 y x - 21 {y}^{2} =$ 6(x + p)(x + q)
Converted trinomial $f ' \left(x\right) = {x}^{2} + 5 y x - 126 {y}^{2} =$ (x + p')(x + q').
p' and q' have opposite signs because ac < 0.
Factor pairs of $\left(a c = 126 {y}^{2}\right)$ -->...(-6y, 21y)(-9y, 14y). This last sum is (5y = b). Then p' = -9y and q' = 14y.
Back to f(x) --> $p = \frac{p '}{a} = \frac{- 9 y}{6} = - \frac{3 y}{2}$, and
$q = \frac{q '}{a} = \frac{14 y}{6} = \frac{7 y}{3}$
Factored form : $f \left(x\right) = 6 \left(x - \frac{3 y}{2}\right) \left(x + \frac{7 y}{3}\right) = \left(2 x - 3 y\right) \left(3 x + 7 y\right)$