If x(x^2 - 7x + 12) = 2(x^2 - 7x + 12), what is the sum of all distinct real values of x t… — GMAT

If $x(x^2 - 7x + 12) = 2(x^2 - 7x + 12)$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

To find all solutions to

x(x^2 - 7x + 12) = 2(x^2 - 7x + 12)

, subtract the right side from the left to obtain

(x - 2)(x^2 - 7x + 12) = 0

. Factoring the quadratic expression gives

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

. Setting each factor to zero reveals three distinct real roots: 2, 3, and 4. The sum of these roots is

2 + 3 + 4 = 9

Rearrange the equation to set one side to zero

x(x^2 - 7x + 12) - 2(x^2 - 7x + 12) = 0

Setting the equation to zero allows the application of the zero-product property without illegally dividing by variable expressions.

Factor out the common quadratic expression

(x^2 - 7x + 12)

(x - 2)(x^2 - 7x + 12) = 0

Both terms share the common factor

(x^2 - 7x + 12)

Factor the quadratic component

(x^2 - 7x + 12)

into linear factors

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

The quadratic factors into

(x - 3)(x - 4)

because

(-3) \times (-4) = 12

and

(-3) + (-4) = -7

Apply the zero-product property to find all distinct real roots and compute their sum

Roots are

x = 2

x = 3

, and

x = 4

. Sum =

2 + 3 + 4 = 9

Setting each linear factor to zero gives the solutions. Summing these solutions yields 9.

Polynomial Factoring and the Zero-Product Property

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