Solve. $LaTeX: \displaystyle \sqrt{x + 24}=\sqrt{x - 6} - 4$

Squaring both sides gives $LaTeX: \displaystyle x + 24=x - 8 \sqrt{x - 6} + 10$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{7}{4}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{49}{16}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{145}{16}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{145}{16}$ , in the original equation gives $LaTeX: \displaystyle \frac{23}{4} = - \frac{9}{4}$ which is false so it is an extraneous solution and there is no solution.