Solve. $LaTeX: \displaystyle \sqrt{x + 22}=\sqrt{x - 2} - 1$

Squaring both sides gives $LaTeX: \displaystyle x + 22=x - 2 \sqrt{x - 2} - 1$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{23}{2}=\sqrt{x - 2}$ Squaring again gives $LaTeX: \displaystyle \frac{529}{4}=x - 2$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{537}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{537}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{25}{2} = \frac{21}{2}$ which is false so it is an extraneous solution and there is no solution.