What is the sum of all distinct real solutions to the equation (x^2 - 4)^2 = 9(x - 2)^2? — GMAT

What is the sum of all distinct real solutions to the equation $(x^2 - 4)^2 = 9(x - 2)^2$ ?

The sum of all distinct real solutions is

-2

Factoring

(x^2 - 4)^2 = 9(x - 2)^2

gives

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

. Factoring out

(x - 2)^2

yields

(x - 2)^2[(x + 2)^2 - 9] = 0

, which simplifies to

(x - 2)^2(x - 1)(x + 5) = 0

. The distinct real solutions are

x = 2

x = 1

, and

x = -5

. Adding these values together gives

2 + 1 + (-5) = -2

Rewrite

x^2 - 4

(x - 2)(x + 2)

and substitute into the left side

[(x - 2)(x + 2)]^2 = (x - 2)^2(x + 2)^2

To identify common factors on both sides of the equation

Rearrange the equation and factor out

(x - 2)^2

(x - 2)^2(x + 2)^2 - 9(x - 2)^2 = 0 \implies (x - 2)^2[(x + 2)^2 - 9] = 0

Factoring out common terms avoids losing roots by division

Factor the inner quadratic expression using difference of squares

(x + 2)^2 - 9 = (x + 2 - 3)(x + 2 + 3) = (x - 1)(x + 5)

To break down the remaining quadratic into linear factors

Identify all distinct real roots and sum them

Roots are

x = 2

x = 1

, and

x = -5

. Sum:

2 + 1 + (-5) = -2

To find the sum of all unique real solutions

Polynomial Factoring and Root Preservation

Estimated Time:2m 0s

Question