If x is a negative real number such that 2x^2 + 5x - 12 = 0, what is the value of (2x - 3)… — GRE General Test

If $x$ is a negative real number such that $2x^2 + 5x - 12 = 0$ , what is the value of $(2x - 3)^2$ ?

121

Factoring

2x^2 + 5x - 12 = 0

gives

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

, which yields the solutions

x = \frac{3}{2}

and

x = -4

. Because the problem specifies that

x

is negative,

x

must equal

-4

. Substituting

-4

for

x

in the target expression yields

(2(-4) - 3)^2 = (-11)^2 = 121

Factor the given quadratic equation to find its roots.

2x^2 + 5x - 12 = (2x - 3)(x + 4) = 0

, giving solutions

x = \frac{3}{2}

and

x = -4

Factoring isolates the linear terms to solve for the values of

x

Apply the given constraint to select the appropriate root.

Since

x

is specified as a negative real number,

x = -4

The root

x = \frac{3}{2}

is positive and violates the problem constraint.

Substitute the valid root into the target expression

(2x - 3)^2

(2(-4) - 3)^2 = (-8 - 3)^2 = (-11)^2 = 121

Evaluating the expression with

x = -4

gives the final requested numerical value.

Solving quadratic equations by factoring and evaluating algebraic expressions under given sign constraints.

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