Question

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

Answer 1

`3x(2x + 5)^2`

Explanation:

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