Solve. $LaTeX: \displaystyle \sqrt{x + 16}=\sqrt{x - 4} - 4$

Squaring both sides gives $LaTeX: \displaystyle x + 16=x - 8 \sqrt{x - 4} + 12$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{1}{2}=\sqrt{x - 4}$ Squaring again gives $LaTeX: \displaystyle \frac{1}{4}=x - 4$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{17}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{17}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{9}{2} = - \frac{7}{2}$ which is false so it is an extraneous solution and there is no solution.