Solve. $LaTeX: \displaystyle \sqrt{x + 24}=\sqrt{x - 6} + 2$

Squaring both sides gives $LaTeX: \displaystyle x + 24=x + 4 \sqrt{x - 6} - 2$ . Isolating the radical gives $LaTeX: \displaystyle \frac{13}{2}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{169}{4}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{193}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{193}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{17}{2} = \frac{17}{2}$ which is true so the solution checks.