Solve the radical equation. Check all proposed solutions. \( \sqrt{8-2 x}=x \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \). (Simplify your answer. Use a comma to separate answers as needed.) B. The solution set is the empty set.

To solve the equation \( \sqrt{8 - 2x} = x \), first, square both sides to eliminate the square root: \[ 8 - 2x = x^2 \] Rearranging the equation gives: \[ x^2 + 2x - 8 = 0 \] Now, we can factor or use the quadratic formula. Factoring gives us: \[ (x + 4)(x - 2) = 0 \] So, the potential solutions are: \[ x + 4 = 0 \implies x = -4 \] \[ x - 2 = 0 \implies x = 2 \] Next, we have to check both proposed solutions in the original equation. 1. **Check \( x = -4 \)**: \[ \sqrt{8 - 2(-4)} = \sqrt{8 + 8} = \sqrt{16} = 4 \neq -4 \] So, \( x = -4 \) is not a solution. 2. **Check \( x = 2 \)**: \[ \sqrt{8 - 2(2)} = \sqrt{8 - 4} = \sqrt{4} = 2 \] So, \( x = 2 \) is a valid solution. Thus, the solution set is \( \{ 2 \} \). The correct choice is: A. The solution set is \( \{ 2 \} \).