If x is a real number that satisfies the equation x^4 - 13x^2 + 36 = 0, which of the follo… — GRE General Test

If $x$ is a real number that satisfies the equation $x^4 - 13x^2 + 36 = 0$ , which of the following values could be the value of $x$ ? Select all such values.

$-3$ Answer
$-2$ Answer
$3$ Answer
D
$4$
E
$9$

Answer

The correct values are

-3

-2

, and

3

Factoring the equation yields

(x^2 - 4)(x^2 - 9) = (x - 2)(x + 2)(x - 3)(x + 3) = 0

, giving the four distinct real solutions

x = -3, -2, 2, 3

. The options corresponding to

-3

-2

, and

3

represent valid values of

x

Step-by-Step Solution

Substitute a variable

u = x^2

to express the equation in quadratic form.

u^2 - 13u + 36 = 0

This reduces the fourth-degree polynomial into a standard quadratic equation.

Factor the quadratic equation to solve for

u

(u - 4)(u - 9) = 0 \implies u = 4 \text{ or } u = 9

Factoring determines the values of

x^2

Substitute back

x^2 = u

and solve for

x

by taking both positive and negative square roots.

x^2 = 4 \implies x = \pm 2

, and

x^2 = 9 \implies x = \pm 3

Each positive value of

u

yields two real solutions for

x

Key Concept

Solving quadratic-form equations by factoring and taking positive and negative square roots.

Question

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