Answer :
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]
[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]