If x is a real number such that (x^2 - 4)^2 - 3(x^2 - 4) - 10 = 0, which of the following … — GRE General Test

If $x$ is a real number such that $(x^2 - 4)^2 - 3(x^2 - 4) - 10 = 0$ , which of the following could be the value of $x$ ? Select all such values.

The values that satisfy the equation are

-3

-\sqrt{2}

, and

3

The given equation is in quadratic form with respect to

(x^2 - 4)

. Letting

u = x^2 - 4

, we factor the equation as

(u - 5)(u + 2) = 0

, giving

u = 5

u = -2

. Setting

x^2 - 4 = 5

gives

x^2 = 9

, which yields

x = 3

x = -3

. Setting

x^2 - 4 = -2

gives

x^2 = 2

, which yields

x = \sqrt{2}

x = -\sqrt{2}

. Thus,

-3

-\sqrt{2}

, and

3

are all valid solutions.

Use algebraic substitution to simplify the quadratic form.

Let

u = x^2 - 4

. The equation becomes

u^2 - 3u - 10 = 0

Recognizing the structure as a quadratic equation in terms of

(x^2 - 4)

simplifies factoring.

Factor the quadratic equation in terms of

u

(u - 5)(u + 2) = 0

, giving solutions

u = 5

u = -2

Finding the roots of the substituted variable

u

leads to the possible values for

x^2 - 4

Substitute back

x^2 - 4

for

u

and solve for

x

in each case.

For

u = 5

x^2 - 4 = 5 \implies x^2 = 9 \implies x = 3

x = -3

.
For

u = -2

x^2 - 4 = -2 \implies x^2 = 2 \implies x = \sqrt{2}

x = -\sqrt{2}

Taking both positive and negative square roots provides all valid real solutions.

Quadratic Form Substitution and Factoring

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