Let's solve the given equation step-by-step:
The equation we need to solve is:
[tex]\[
\sqrt{x} + 6 = x
\][/tex]
Step 1: Isolate the square root term.
[tex]\[
\sqrt{x} = x - 6
\][/tex]
Step 2: Square both sides to eliminate the square root.
[tex]\[
(\sqrt{x})^2 = (x - 6)^2
\][/tex]
[tex]\[
x = (x - 6)^2
\][/tex]
Step 3: Expand the right-hand side.
[tex]\[
x = x^2 - 12x + 36
\][/tex]
Step 4: Rearrange the equation to standard quadratic form.
[tex]\[
0 = x^2 - 12x + 36 - x
\][/tex]
[tex]\[
0 = x^2 - 13x + 36
\][/tex]
Step 5: Solve the quadratic equation [tex]\(x^2 - 13x + 36 = 0\)[/tex]. We can use the quadratic formula, [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -13\)[/tex], and [tex]\(c = 36\)[/tex].
[tex]\[
x = \frac{13 \pm \sqrt{169 - 144}}{2}
\][/tex]
[tex]\[
x = \frac{13 \pm \sqrt{25}}{2}
\][/tex]
[tex]\[
x = \frac{13 \pm 5}{2}
\][/tex]
Step 6: Find the two possible solutions.
[tex]\[
x = \frac{13 + 5}{2} = 9
\][/tex]
[tex]\[
x = \frac{13 - 5}{2} = 4
\][/tex]
Step 7: Verify the solutions in the original equation to check for validity.
Check [tex]\(x = 9\)[/tex]:
[tex]\[
\sqrt{9} + 6 = 9
\][/tex]
[tex]\[
3 + 6 = 9
\][/tex]
This is correct.
Check [tex]\(x = 4\)[/tex]:
[tex]\[
\sqrt{4} + 6 = 4
\][/tex]
[tex]\[
2 + 6 = 8
\][/tex]
This is not correct.
Thus, the only valid solution is [tex]\(x = 9\)[/tex].
Therefore, the correct answer is:
B. 9
