To determine the solutions to the equation [tex]\((x-4)^2 = 81\)[/tex], follow these steps:
1. Rewrite the equation in standard form:
[tex]\[
(x - 4)^2 = 81
\][/tex]
2. Take the square root of both sides:
When we take the square root of both sides of the equation, we must remember that the square root of a number can be both positive and negative. Therefore:
[tex]\[
\sqrt{(x-4)^2} = \pm \sqrt{81}
\][/tex]
Simplifying, we have:
[tex]\[
x - 4 = \pm 9
\][/tex]
3. Solve the equation for [tex]\(x\)[/tex] by considering both the positive and negative cases:
- For the positive case:
[tex]\[
x - 4 = 9
\][/tex]
Add 4 to both sides:
[tex]\[
x = 9 + 4
\][/tex]
[tex]\[
x = 13
\][/tex]
- For the negative case:
[tex]\[
x - 4 = -9
\][/tex]
Add 4 to both sides:
[tex]\[
x = -9 + 4
\][/tex]
[tex]\[
x = -5
\][/tex]
4. List the solutions:
The solutions to the equation [tex]\((x-4)^2 = 81\)[/tex] are:
[tex]\[
x = -5 \quad \text{and} \quad x = 13
\][/tex]
Therefore, the correct answer is [tex]\(x = -5\)[/tex] and [tex]\(x = 13\)[/tex]. Thus, the correct choice from the provided options is:
[tex]\[
x = -5 \text{ and } 13
\][/tex]
