To solve the equation [tex]\( x - 2 = \sqrt{x - 2} + 12 \)[/tex], we will proceed through the following steps:
1. Isolate the square root term:
[tex]\[
x - 2 - 12 = \sqrt{x - 2}
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
x - 14 = \sqrt{x - 2}
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(x - 14)^2 = (\sqrt{x - 2})^2
\][/tex]
Simplifying both sides, we get:
[tex]\[
(x - 14)^2 = x - 2
\][/tex]
3. Expand the left-hand side:
[tex]\[
x^2 - 28x + 196 = x - 2
\][/tex]
4. Move all terms to one side to form a quadratic equation:
[tex]\[
x^2 - 28x + 196 - x + 2 = 0
\][/tex]
Simplifying the equation:
[tex]\[
x^2 - 29x + 198 = 0
\][/tex]
5. Solve the quadratic equation:
We use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -29 \)[/tex], and [tex]\( c = 198 \)[/tex].
Computing the discriminant:
[tex]\[
b^2 - 4ac = (-29)^2 - 4 \cdot 1 \cdot 198 = 841 - 792 = 49
\][/tex]
Thus:
[tex]\[
x = \frac{29 \pm \sqrt{49}}{2 \cdot 1} = \frac{29 \pm 7}{2}
\][/tex]
This gives us two potential solutions:
[tex]\[
x = \frac{29 + 7}{2} = \frac{36}{2} = 18
\][/tex]
[tex]\[
x = \frac{29 - 7}{2} = \frac{22}{2} = 11
\][/tex]
6. Check each potential solution:
- For [tex]\( x = 18 \)[/tex]:
[tex]\[
18 - 2 = \sqrt{18 - 2} + 12
\][/tex]
Simplifying,
[tex]\[
16 = \sqrt{16} + 12
\][/tex]
[tex]\[
16 = 4 + 12
\][/tex]
[tex]\[
16 = 16 \quad (\text{True})
\][/tex]
So, [tex]\( x = 18 \)[/tex] is a valid solution.
- For [tex]\( x = 11 \)[/tex]:
[tex]\[
11 - 2 = \sqrt{11 - 2} + 12
\][/tex]
Simplifying,
[tex]\[
9 = \sqrt{9} + 12
\][/tex]
[tex]\[
9 = 3 + 12
\][/tex]
[tex]\[
9 \neq 15 \quad (\text{False})
\][/tex]
So, [tex]\( x = 11 \)[/tex] is not a valid solution.
Therefore, the solution to the equation [tex]\( x - 2 = \sqrt{x - 2} + 12 \)[/tex] is:
[tex]\[
\boxed{18}
\][/tex]
