To solve the equation [tex]\(\sqrt{x+4} = x - 2\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here are the steps to solve this equation:
1. Isolate the square root:
[tex]\[
\sqrt{x+4} = x - 2
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x+4})^2 = (x - 2)^2
\][/tex]
This simplifies to:
[tex]\[
x + 4 = (x - 2)^2
\][/tex]
3. Expand the right side:
[tex]\[
x + 4 = x^2 - 4x + 4
\][/tex]
4. Move all terms to one side to set the equation to zero:
[tex]\[
x + 4 - x^2 + 4x - 4 = 0
\][/tex]
Simplify the equation:
[tex]\[
-x^2 + 5x = 0
\][/tex]
5. Factor the quadratic equation:
[tex]\[
-x(x - 5) = 0
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
The factored equation gives us:
[tex]\[
x = 0 \quad \text{or} \quad x = 5
\][/tex]
7. Check the solutions in the original equation to ensure they do not produce extraneous results.
- For [tex]\(x = 0\)[/tex]:
[tex]\[
\sqrt{0 + 4} = 0 - 2 \quad \Rightarrow \quad 2 = -2 \quad \text{(False)}
\][/tex]
So, [tex]\(x = 0\)[/tex] is not a valid solution.
- For [tex]\(x = 5\)[/tex]:
[tex]\[
\sqrt{5 + 4} = 5 - 2 \quad \Rightarrow \quad \sqrt{9} = 3 \quad \Rightarrow \quad 3 = 3 \quad \text{(True)}
\][/tex]
So, [tex]\(x = 5\)[/tex] is a valid solution.
Therefore, the solution to the equation [tex]\(\sqrt{x+4} = x - 2\)[/tex] is:
[tex]\[
\boxed{5}
\][/tex]
