Step-by-Step Solution

Solve the radical equation: $\sqrt{x + 5} - \sqrt{x - 3} = 2$

Step-by-Step Solution

Step 1: Transpose negative radical to the right

$\sqrt{x + 5} = 2 + \sqrt{x - 3}$

Step 2: Square both sides

$\left(\sqrt{x + 5}\right)^2 = \left(2 + \sqrt{x - 3}\right)^2$

$x + 5 = 4 + 4\sqrt{x - 3} + (x - 3)$

$x + 5 = x + 1 + 4\sqrt{x - 3}$

Step 3: Isolate and solve for $x$

Subtract $x + 1$ from both sides: $4 = 4\sqrt{x - 3}$

Divide by $4$: $\sqrt{x - 3} = 1$

Square both sides: $x - 3 = 1^2 \implies x = 4$

Step 4: Check solution

$\text{LHS} = \sqrt{4 + 5} - \sqrt{4 - 3} = \sqrt{9} - \sqrt{1} = 3 - 1 = 2 = \text{RHS} \quad \checkmark$

$\boxed{\text{Solution Set: } \{4\}}$