Solve. $LaTeX: \displaystyle \sqrt{x + 22}=\sqrt{x - 4} - 4$

Squaring both sides gives $LaTeX: \displaystyle x + 22=x - 8 \sqrt{x - 4} + 12$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{5}{4}=\sqrt{x - 4}$ Squaring again gives $LaTeX: \displaystyle \frac{25}{16}=x - 4$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{89}{16}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{89}{16}$ , in the original equation gives $LaTeX: \displaystyle \frac{21}{4} = - \frac{11}{4}$ which is false so it is an extraneous solution and there is no solution.