Let's solve the equation step-by-step to find all the possible solutions for:
[tex]\[
(5 - x)^{\frac{1}{2}} - 1 = x
\][/tex]
1. Rewrite the equation:
Isolate the square root term:
[tex]\[
(5 - x)^{\frac{1}{2}} = x + 1
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(5 - x) = (x + 1)^2
\][/tex]
Expanding the right side:
[tex]\[
5 - x = x^2 + 2x + 1
\][/tex]
3. Rearrange the equation to standard quadratic form:
Move all terms to one side:
[tex]\[
x^2 + 3x - 4 = 0
\][/tex]
4. Solve the quadratic equation:
To factorize [tex]\( x^2 + 3x - 4 \)[/tex]:
Find two numbers that multiply to [tex]\(-4\)[/tex] and add to [tex]\(3\)[/tex]. These numbers are [tex]\(4\)[/tex] and [tex]\(-1\)[/tex].
Hence:
[tex]\[
(x + 4)(x - 1) = 0
\][/tex]
So, we set each factor to zero:
[tex]\[
x + 4 = 0 \quad \text{or} \quad x - 1 = 0
\][/tex]
Solving these, we get:
[tex]\[
x = -4 \quad \text{or} \quad x = 1
\][/tex]
5. Check for extraneous solutions:
Substitute [tex]\( x = -4 \)[/tex] back into the original equation to check:
[tex]\[
(5 - (-4))^{\frac{1}{2}} - 1 = -4
\][/tex]
[tex]\[
(5 + 4)^{\frac{1}{2}} - 1 = -4
\][/tex]
[tex]\[
3 - 1 \neq -4
\][/tex]
Therefore, [tex]\( x = -4 \)[/tex] is not a solution.
Substitute [tex]\( x = 1 \)[/tex] back into the original equation to check:
[tex]\[
(5 - 1)^{\frac{1}{2}} - 1 = 1
\][/tex]
[tex]\[
2 - 1 = 1
\][/tex]
[tex]\[
1 = 1
\][/tex]
Therefore, [tex]\( x = 1 \)[/tex] is a solution.
Based on these checks, the correct solution is:
[tex]\[
\boxed{1}
\][/tex]
