If x is a real number satisfying the equation \sqrt{15 - 2x} = 2x - 3, what is the sum of … — GMAT

If $x$ is a real number satisfying the equation $\sqrt{15 - 2x} = 2x - 3$ , what is the sum of all real values of $x$ that satisfy this equation?

The sum of all real values of

x

satisfying the equation is

3

Squaring both sides of

\sqrt{15 - 2x} = 2x - 3

gives

15 - 2x = 4x^2 - 12x + 9

, which simplifies to

2x^2 - 5x - 3 = 0

. Factoring yields candidate roots

x = 3

and

x = -1/2

. Substituting

x = 3

into the original equation gives

\sqrt{9} = 3

, which is true. Substituting

x = -1/2

yields

\sqrt{16} = -4

, which is false because principal radicals cannot evaluate to negative values. Therefore,

x = 3

is the sole valid solution, making the sum equal to

3

Isolate the radical and state domain constraints.

The principal square root

\sqrt{15 - 2x}

must be non-negative, requiring

15 - 2x \ge 0 \implies x \le 7.5

, and

2x - 3 \ge 0 \implies x \ge 1.5

Radical expressions produce non-negative principal square roots.

Square both sides of the equation.

15 - 2x = (2x - 3)^2 \implies 15 - 2x = 4x^2 - 12x + 9

Eliminate the radical to form a polynomial equation.

Rearrange into standard quadratic form and solve for

x

4x^2 - 10x - 6 = 0 \implies 2x^2 - 5x - 3 = 0 \implies (2x + 1)(x - 3) = 0

, giving candidate solutions

x = 3

and

x = -1/2

Solve the quadratic equation using factoring.

Test candidate solutions in the original equation to eliminate extraneous roots.

For

x = 3

\sqrt{15 - 2(3)} = \sqrt{9} = 3

and

2(3) - 3 = 3

(Valid). For

x = -1/2

\sqrt{15 - 2(-1/2)} = \sqrt{16} = 4

, but

2(-1/2) - 3 = -4

(Extraneous).

Squaring an equation can introduce extraneous roots.

Solving Radical Equations and Identifying Extraneous Solutions

Question