Solve. $LaTeX: \displaystyle \sqrt{x + 32}=\sqrt{x + 8} - 1$

Squaring both sides gives $LaTeX: \displaystyle x + 32=x - 2 \sqrt{x + 8} + 9$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{23}{2}=\sqrt{x + 8}$ Squaring again gives $LaTeX: \displaystyle \frac{529}{4}=x + 8$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{497}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{497}{4}$ , 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.