Question

solve the following equation by making an appropriate substitution.
\\((x^2 - x)^2 - 14(x^2 - x) + 24 = 0\\)

solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\(\bigcirc\\) a. the solution set is \\(\\{\\ \\}\\)
(simplify your answer. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. the solution set is the empty set.

Explanation:

Step1: Substitute $u=x^2-x$

Let $u = x^2 - x$. The equation becomes:
$$u^2 - 14u + 24 = 0$$

Step2: Factor the quadratic equation

Factor the trinomial:
$$(u - 12)(u - 2) = 0$$

Step3: Solve for $u$

Set each factor equal to 0:
$u - 12 = 0 \implies u = 12$
$u - 2 = 0 \implies u = 2$

Step4: Substitute back $u=x^2-x$

First for $u=12$:
$$x^2 - x = 12$$
Rearrange to standard quadratic:
$$x^2 - x - 12 = 0$$
Factor:
$$(x - 4)(x + 3) = 0$$
Solve for $x$:
$x - 4 = 0 \implies x = 4$
$x + 3 = 0 \implies x = -3$

Now for $u=2$:
$$x^2 - x = 2$$
Rearrange to standard quadratic:
$$x^2 - x - 2 = 0$$
Factor:
$$(x - 2)(x + 1) = 0$$
Solve for $x$:
$x - 2 = 0 \implies x = 2$
$x + 1 = 0 \implies x = -1$

Answer:

A. The solution set is {-3, -1, 2, 4}