Solve. $LaTeX: \displaystyle \sqrt{x + 14}=\sqrt{x - 8} - 4$

Squaring both sides gives $LaTeX: \displaystyle x + 14=x - 8 \sqrt{x - 8} + 8$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{3}{4}=\sqrt{x - 8}$ Squaring again gives $LaTeX: \displaystyle \frac{9}{16}=x - 8$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{137}{16}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{137}{16}$ , in the original equation gives $LaTeX: \displaystyle \frac{19}{4} = - \frac{13}{4}$ which is false so it is an extraneous solution and there is no solution.