Questions: Solve the radical equation. Check all proposed solutions. √x + 8 = x + 2

Solution Steps

Start with the original equation: \[ \sqrt{x} + 8 = x + 2 \] Isolate the radical by moving all other terms to the right side: \[ \sqrt{x} = x + 2 - 8 \] This simplifies to: \[ \sqrt{x} = x - 6 \]

Square both sides to eliminate the radical: \[ (\sqrt{x})^2 = (x - 6)^2 \] This results in: \[ x = x^2 - 12x + 36 \]

Rearrange the equation to set it to zero: \[ 0 = x^2 - 12x + 36 - x \] This simplifies to: \[ 0 = x^2 - 13x + 36 \]

Factor the quadratic equation: \[ 0 = (x - 9)(x - 4) \] Setting each factor to zero gives the potential solutions: \[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

Substitute \( x = 9 \) back into the original equation: \[ \sqrt{9} + 8 = 9 + 2 \] This simplifies to: \[ 3 + 8 = 11 \quad \Rightarrow \quad 11 = 11 \quad \text{(valid)} \]

Now check \( x = 4 \): \[ \sqrt{4} + 8 = 4 + 2 \] This simplifies to: \[ 2 + 8 = 6 \quad \Rightarrow \quad 10 \neq 6 \quad \text{(not valid)} \]

The only valid solution is: \[ \{9\} \]

Final Answer