Question

select all the solutions to the equation.
$(5 - x)^{\frac{1}{2}} - 1 = x$
-4
-1
1
2
3
4

Explanation:

Step1: Isolate the radical term

$(5-x)^\frac{1}{2} = x + 1$

Step2: Square both sides

$(5-x) = (x + 1)^2$
Expand the right-hand side: $5 - x = x^2 + 2x + 1$

Step3: Rearrange to quadratic form

$x^2 + 3x - 4 = 0$

Step4: Factor the quadratic

$(x + 4)(x - 1) = 0$
Potential solutions: $x = -4$ or $x = 1$

Step5: Verify solutions

For $x=-4$: Left-hand side $(5-(-4))^\frac{1}{2}-1 = 3 - 1 = 2$, Right-hand side $=-4$. $2
eq -4$, so $x=-4$ is extraneous.
For $x=1$: Left-hand side $(5-1)^\frac{1}{2}-1 = 2 - 1 = 1$, Right-hand side $=1$. $1=1$, so $x=1$ is valid.

Answer:

1