Question

What are the solutions to the equation?
x3 – 6x2 – 9x + 54 = 0

Answer 1

x = -3 or x = 3 or x = 6

Explanation:

x3 – 6x2 – 9x + 54 = 0

There is no common factor to factor out. There are 4 terms. We try factoring by grouping. Factor a common factor out of the first two terms. Factor a common factor out of the last two terms.

x^2(x - 6) - 9(x - 6) = 0

x - 6 is a common factor, so we factor it out.

(x^2 - 9)(x - 6) = 0

x^2 - 9 is the difference of 2 squares, so we factor it.

(x + 3)(x - 3)(x - 6) = 0

x + 3 = 0 or x - 3 = 0 or x - 6 = 0

x = -3 or x = 3 or x = 6

Answer 2

x=6 x=3 x=-3

Explanation:

x^3 – 6x^2 – 9x + 54 = 0

Factor by grouping

x^3 – 6x^2 – 9x + 54 = 0

x^2(x-6) -9(x-6 ) =0

Factor out x-6

(x-6)(x^2 -9) =0

Notice x^2 -9 is the difference of squares

(x-6)(x-3)(x+3) = 0

Using the zero product property

x-6 =0 x-3 =0 x+3 =0

x=6 x=3 x=-3