Solve. $LaTeX: \displaystyle \sqrt{x + 22}=\sqrt{x - 4} - 2$

Squaring both sides gives $LaTeX: \displaystyle x + 22=x - 4 \sqrt{x - 4}$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{11}{2}=\sqrt{x - 4}$ Squaring again gives $LaTeX: \displaystyle \frac{121}{4}=x - 4$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{137}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{137}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{15}{2} = \frac{7}{2}$ which is false so it is an extraneous solution and there is no solution.