# How do you factor 12x^3 - 60x^2 + 75x = 0?

Apr 29, 2016

$3 x {\left(2 x + 5\right)}^{2}$

#### Explanation:

f(x) = x(12x^2 - 60x + 75)
Factor the trinomial y in parentheses by the new AC Method (Socratic Search).
$y = 12 {x}^{2} - 60 x + 75 =$12(x + p)(x + q)
Converted trinomial $y ' = {x}^{2} - 60 x + 900 =$ (x + p')(x + q').
p'and q' have same sign because ac > 0.
$y ' = {x}^{2} - 60 x + 900 = {\left(x - 30\right)}^{2}$. Then p' = q' = 30 (double roots)
Back to y, $p = q = p \frac{'}{a} = q \frac{'}{a} = \frac{30}{12} = \frac{5}{2}$.
$f \left(x\right) = 12 x {\left(x + \frac{5}{2}\right)}^{2} = 3 x {\left(2 x + 5\right)}^{2}$