If x^2(2x - 3) = 9(2x - 3), what is the sum of all distinct real solutions for x? — GMAT

If $x^2(2x - 3) = 9(2x - 3)$ , what is the sum of all distinct real solutions for $x$ ?

The sum of all distinct real solutions for

x

1.5

To find all solutions without losing roots, set the equation to zero:

x^2(2x - 3) - 9(2x - 3) = 0

. Factoring out

(2x - 3)

gives

(x^2 - 9)(2x - 3) = 0

, which fully factors into

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

. The distinct real roots are

3

-3

, and

1.5

. Adding these values together yields

3 + (-3) + 1.5 = 1.5

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

x^2(2x - 3) - 9(2x - 3) = 0

Dividing by an expression containing variables can eliminate valid roots.

Factor out the common term

(2x - 3)

(x^2 - 9)(2x - 3) = 0

Factoring allows application of the zero-product property.

Factor the quadratic term

x^2 - 9

as a difference of squares.

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

a^2 - b^2 = (a - b)(a + b)

Set each factor to zero to find all real solutions.

x - 3 = 0 \implies x = 3

x + 3 = 0 \implies x = -3

, and

2x - 3 = 0 \implies x = 1.5

If the product of factors is zero, at least one factor must equal zero.

Sum all distinct real solutions.

3 + (-3) + 1.5 = 1.5

The question asks for the sum of all distinct real solutions.

Factoring Polynomial Equations and Avoiding Variable Cancellation

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