Question

solve the equation by making an appropriate substitution.
$x^4 - 13x^2 + 36 = 0$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is { } (type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. the solution set is ∅.

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the equation becomes:
$$u^2 - 13u + 36 = 0$$

Step2: Factor the quadratic equation

Find two numbers that multiply to 36 and add to -13:
$$(u - 4)(u - 9) = 0$$

Step3: Solve for $u$

Set each factor equal to 0:
$u - 4 = 0 \implies u = 4$
$u - 9 = 0 \implies u = 9$

Step4: Substitute back $u=x^2$

For $u=4$: $x^2 = 4 \implies x = \pm 2$
For $u=9$: $x^2 = 9 \implies x = \pm 3$

Answer:

A. The solution set is $\{-3, -2, 2, 3\}$