To solve the equation [tex]\(\sqrt{x-1} = x - 13\)[/tex], we will proceed step by step.
1. Square both sides of the equation to eliminate the square root. This gives:
[tex]\[
(\sqrt{x-1})^2 = (x-13)^2
\][/tex]
Simplifying, we get:
[tex]\[
x - 1 = (x - 13)^2
\][/tex]
2. Expand the right-hand side of the equation:
[tex]\[
x - 1 = x^2 - 26x + 169
\][/tex]
3. Rearrange the equation to bring all terms to one side, forming a quadratic equation:
[tex]\[
0 = x^2 - 26x + 169 - x + 1
\][/tex]
Combine like terms:
[tex]\[
x^2 - 27x + 170 = 0
\][/tex]
4. Solve the quadratic equation using the quadratic formula, [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -27\)[/tex], and [tex]\(c = 170\)[/tex].
Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-27)^2 - 4(1)(170) = 729 - 680 = 49
\][/tex]
Use the quadratic formula to find the roots:
[tex]\[
x = \frac{27 \pm \sqrt{49}}{2} = \frac{27 \pm 7}{2}
\][/tex]
Simplifying the two potential solutions:
For the positive root:
[tex]\[
x = \frac{27 + 7}{2} = \frac{34}{2} = 17
\][/tex]
For the negative root:
[tex]\[
x = \frac{27 - 7}{2} = \frac{20}{2} = 10
\][/tex]
5. Validate the potential solutions to ensure they satisfy the original equation [tex]\(\sqrt{x - 1} = x - 13\)[/tex].
For [tex]\(x = 17\)[/tex]:
[tex]\[
\sqrt{17 - 1} = \sqrt{16} = 4
\][/tex]
and
[tex]\[
17 - 13 = 4
\][/tex]
Both sides equal, so [tex]\(x = 17\)[/tex] is a valid solution.
For [tex]\(x = 10\)[/tex]:
[tex]\[
\sqrt{10 - 1} = \sqrt{9} = 3
\][/tex]
and
[tex]\[
10 - 13 = -3
\][/tex]
Both sides do not equal, so [tex]\(x = 10\)[/tex] is not a valid solution.
Therefore, the correct solution to the equation [tex]\(\sqrt{x-1} = x - 13\)[/tex] is:
[tex]\[
\boxed{17}
\][/tex]
