Solve. $LaTeX: \displaystyle \sqrt{x + 16}=\sqrt{x - 6} - 3$

Squaring both sides gives $LaTeX: \displaystyle x + 16=x - 6 \sqrt{x - 6} + 3$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{13}{6}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{169}{36}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{385}{36}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{385}{36}$ , in the original equation gives $LaTeX: \displaystyle \frac{31}{6} = - \frac{5}{6}$ which is false so it is an extraneous solution and there is no solution.