To solve the equation [tex]\((x - 7)^2 = 36\)[/tex], let’s follow these steps:
1. Recognize that this equation is a quadratic equation in the form [tex]\((x - a)^2 = b^2\)[/tex], where [tex]\(a = 7\)[/tex] and [tex]\(b^2 = 36\)[/tex]. Therefore, [tex]\(b = 6\)[/tex] or [tex]\(b = -6\)[/tex].
2. Solve the equation by taking the square root of both sides:
[tex]\[
\sqrt{(x - 7)^2} = \sqrt{36}
\][/tex]
This gives us:
[tex]\[
|x - 7| = 6
\][/tex]
3. The absolute value equation [tex]\(|x - 7| = 6\)[/tex] leads to two possible cases:
- Case 1: [tex]\(x - 7 = 6\)[/tex]
- Case 2: [tex]\(x - 7 = -6\)[/tex]
4. Solve each case separately:
- For Case 1: [tex]\(x - 7 = 6\)[/tex]
[tex]\[
x = 6 + 7
\][/tex]
[tex]\[
x = 13
\][/tex]
- For Case 2: [tex]\(x - 7 = -6\)[/tex]
[tex]\[
x = -6 + 7
\][/tex]
[tex]\[
x = 1
\][/tex]
So, the solutions to the equation [tex]\((x - 7)^2 = 36\)[/tex] are [tex]\(x = 13\)[/tex] and [tex]\(x = 1\)[/tex].
Now, let's check these against the given options:
- [tex]\(x = 13\)[/tex]: This is one of our solutions.
- [tex]\(x = 1\)[/tex]: This is also one of our solutions.
- [tex]\(x = -29\)[/tex]: This is not a solution.
- [tex]\(x = 42\)[/tex]: This is not a solution.
Therefore, the correct values of [tex]\(x\)[/tex] from the given options are:
[tex]\[
\boxed{13 \text{ and } 1}
\][/tex]
