Certainly! Let's solve the equation step-by-step:
Given equation:
[tex]\[
(u - 2)^2 = 2u^2 - 7u - 24
\][/tex]
1. Expand [tex]\((u - 2)^2\)[/tex]:
[tex]\[
(u - 2)(u - 2) = u^2 - 4u + 4
\][/tex]
So, the equation becomes:
[tex]\[
u^2 - 4u + 4 = 2u^2 - 7u - 24
\][/tex]
2. Move all terms to one side to set the equation to zero:
Subtract [tex]\(2u^2 - 7u - 24\)[/tex] from both sides:
[tex]\[
u^2 - 4u + 4 - 2u^2 + 7u + 24 = 0
\][/tex]
Combine like terms:
[tex]\[
u^2 - 2u^2 - 4u + 7u + 4 + 24 = 0
\][/tex]
[tex]\[
-u^2 + 3u + 28 = 0
\][/tex]
3. Multiply through by -1 to simplify the quadratic equation:
[tex]\[
u^2 - 3u - 28 = 0
\][/tex]
4. Solve the quadratic equation [tex]\(u^2 - 3u - 28 = 0\)[/tex]:
To solve, factor the quadratic equation. We are looking for two numbers that multiply to [tex]\(-28\)[/tex] and add up to [tex]\(-3\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
u^2 - 3u - 28 = (u - 7)(u + 4) = 0
\][/tex]
5. Set each factor equal to zero and solve for [tex]\(u\)[/tex]:
[tex]\[
u - 7 = 0 \quad \text{or} \quad u + 4 = 0
\][/tex]
[tex]\[
u = 7 \quad \text{or} \quad u = -4
\][/tex]
Therefore, the solutions are:
[tex]\[
u = 7 \quad \text{and} \quad u = -4
\][/tex]
