Question

What is the factored form of `2x^3+4x^2-x`?

Answer 1

2x(x + 2 - sqrt6)(x + 2 + sqrt6)

Explanation:

f(x) = xy = x(2x^2 + 4x - 1) = ax(x - x1)(x - x2)
x1 and x2 are the 2 real roots of y.
Find these 2 real roots by the improved quadratic formula (Socratic Search)
`D = d^2 = b^2 - 4ac = 16 + 8 = 24 --> d = +- 2sqrt6`
There are 2 real roots:
`x1 = -b/(2a) +- d/(2a) = - 2 +- 2sqrt6/2`
`x1 = - 2 + sqrt6`
`x2 = - 2 - sqrt6`

Factored form:
`y = 2x(x - x1)(x - x2) = 2x(x + 2 - sqrt6)(x + 2 + sqrt6)`