Solve. $LaTeX: \displaystyle \sqrt{x + 30}=\sqrt{x - 6} - 3$

Squaring both sides gives $LaTeX: \displaystyle x + 30=x - 6 \sqrt{x - 6} + 3$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{9}{2}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{81}{4}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{105}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{105}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{15}{2} = \frac{3}{2}$ which is false so it is an extraneous solution and there is no solution.