To solve the equation [tex]\( x^2 - 24x = -80 \)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 - 24x = -80
\][/tex]
2. Move the constant term to the left side of the equation:
[tex]\[
x^2 - 24x + 80 = 0
\][/tex]
3. To complete the square, take the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-24\)[/tex]), divide it by 2, and then square the result:
[tex]\[
\left(\frac{-24}{2}\right)^2 = (-12)^2 = 144
\][/tex]
4. Add and subtract this square within the equation:
[tex]\[
x^2 - 24x + 144 - 144 + 80 = 0
\][/tex]
Which simplifies to:
[tex]\[
(x - 12)^2 - 64 = 0
\][/tex]
5. Isolate the squared term:
[tex]\[
(x - 12)^2 = 64
\][/tex]
6. Take the square root of both sides:
[tex]\[
x - 12 = \pm \sqrt{64}
\][/tex]
Which gives us:
[tex]\[
x - 12 = \pm 8
\][/tex]
7. Solve for [tex]\(x\)[/tex] by isolating [tex]\(x\)[/tex]:
[tex]\[
x = 12 + 8 \quad \text{or} \quad x = 12 - 8
\][/tex]
[tex]\[
x = 20 \quad \text{or} \quad x = 4
\][/tex]
Therefore, the solution set of the equation [tex]\( x^2 - 24x = -80 \)[/tex] is:
[tex]\(
\{4, 20\}
\)[/tex]
So, among the given options, the correct solution set is:
[tex]\(
\{4, 20\}
\)[/tex]
