To solve the equation [tex]\(\sqrt{x + 2} = x - 4\)[/tex], we can follow these steps:
1. Isolate the square root: The square root is already isolated on the left side of the equation.
[tex]\[
\sqrt{x + 2} = x - 4
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x + 2})^2 = (x - 4)^2
\][/tex]
This simplifies to:
[tex]\[
x + 2 = (x - 4)^2
\][/tex]
3. Expand the right side: We need to expand [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[
x + 2 = x^2 - 8x + 16
\][/tex]
4. Rearrange the equation to form a standard quadratic equation. Move all terms to one side:
[tex]\[
0 = x^2 - 8x + 16 - x - 2
\][/tex]
Combine like terms:
[tex]\[
0 = x^2 - 9x + 14
\][/tex]
5. Solve the quadratic equation [tex]\( x^2 - 9x + 14 = 0 \)[/tex]. We can factorize it if possible:
[tex]\[
(x - 7)(x - 2) = 0
\][/tex]
Setting each factor equal to zero gives:
[tex]\[
x - 7 = 0 \quad \text{or} \quad x - 2 = 0
\][/tex]
This means:
[tex]\[
x = 7 \quad \text{or} \quad x = 2
\][/tex]
6. Verify the solutions: We need to check both potential solutions in the original equation to confirm they hold true.
For [tex]\( x = 7 \)[/tex]:
[tex]\[
\sqrt{7 + 2} = 7 - 4
\][/tex]
[tex]\[
\sqrt{9} = 3 \quad \Rightarrow \quad 3 = 3 \quad \text{(True)}
\][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[
\sqrt{2 + 2} = 2 - 4
\][/tex]
[tex]\[
\sqrt{4} = -2 \quad \Rightarrow \quad 2 = -2 \quad \text{(False)}
\][/tex]
Thus, [tex]\( x = 2 \)[/tex] is not a valid solution since it does not satisfy the original equation.
7. Conclusion: The only solution that satisfies the equation is:
[tex]\[
x = 7
\][/tex]
Therefore, the correct answer is:
C. [tex]\( x = 7 \)[/tex]
