What is the sum of all distinct real values of x that satisfy the equation x(x^2 - 5x + 6)… — GMAT

What is the sum of all distinct real values of $x$ that satisfy the equation $x(x^2 - 5x + 6) = 2(x^2 - 5x + 6)$ ?

The sum of all distinct real solutions is 5.

To solve the equation

x(x^2 - 5x + 6) = 2(x^2 - 5x + 6)

, move all terms to the left side to get

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

. Factoring out

(x^2 - 5x + 6)

gives

(x - 2)(x^2 - 5x + 6) = 0

. Factoring the quadratic part yields

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

, or

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

. The real roots are

x = 2

and

x = 3

. Taking the sum of these distinct real roots gives

2 + 3 = 5

Move all terms to one side of the equation to avoid dividing by a variable expression.

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

Dividing by

(x^2 - 5x + 6)

directly would lose valid roots where

x^2 - 5x + 6 = 0

Factor out the common quadratic expression

(x^2 - 5x + 6)

(x - 2)(x^2 - 5x + 6) = 0

Applying the distributive property isolates the linear factor

(x - 2)

Factor the quadratic term

x^2 - 5x + 6

into linear factors.

(x - 2)(x - 2)(x - 3) = 0 \implies (x - 2)^2(x - 3) = 0

The quadratic factors into

(x - 2)(x - 3)

because

(-2) + (-3) = -5

and

(-2) \times (-3) = 6

Identify all distinct real solutions and compute their sum.

Real solutions are

x = 2

and

x = 3

. Distinct sum:

2 + 3 = 5

The root

x = 2

has multiplicity 2, but only distinct values must be summed.

Polynomial Factoring and Variable Cancellation Rules

Estimated Time:1m 30s

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