Solve. $LaTeX: \displaystyle \sqrt{x + 24}=\sqrt{x - 8} - 3$

Squaring both sides gives $LaTeX: \displaystyle x + 24=x - 6 \sqrt{x - 8} + 1$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{23}{6}=\sqrt{x - 8}$ Squaring again gives $LaTeX: \displaystyle \frac{529}{36}=x - 8$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{817}{36}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{817}{36}$ , in the original equation gives $LaTeX: \displaystyle \frac{41}{6} = \frac{5}{6}$ which is false so it is an extraneous solution and there is no solution.