If x is a real number such that x^3 - 3x^2 = 4x - 12, what is the sum of all possible valu… — GMAT

If $x$ is a real number such that $x^3 - 3x^2 = 4x - 12$ , what is the sum of all possible values of $x$ ?

The sum of all possible values of

x

is 3.

Moving all terms to one side gives

x^3 - 3x^2 - 4x + 12 = 0

. Grouping the terms yields

x^2(x - 3) - 4(x - 3) = 0

, which factors completely into

(x - 2)(x + 2)(x - 3) = 0

. The real solutions are

x = 2

x = -2

, and

x = 3

. Adding these values together yields

2 + (-2) + 3 = 3

Rearrange the equation to set it equal to zero.

x^3 - 3x^2 - 4x + 12 = 0

Grouping terms requires having all terms on one side of the equation.

Factor by grouping terms in pairs.

x^2(x - 3) - 4(x - 3) = 0 \Rightarrow (x^2 - 4)(x - 3) = 0

Extracting the common factor

(x - 3)

allows reducing the cubic polynomial into a linear factor and a quadratic factor.

Factor the difference of squares and set each factor to zero to find all real roots.

(x - 2)(x + 2)(x - 3) = 0 \Rightarrow x = 2, x = -2, x = 3

By the zero-product property, the expression equals zero when any factor is equal to zero.

Calculate the sum of all distinct real solutions.

2 + (-2) + 3 = 3

The question asks for the sum of all possible real values of

x

Polynomial Factoring by Grouping and Zero-Product Property

Estimated Time:1m 30s

Question