Solve. $LaTeX: \displaystyle \sqrt{x + 12}=\sqrt{x - 6} - 4$

Squaring both sides gives $LaTeX: \displaystyle x + 12=x - 8 \sqrt{x - 6} + 10$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{1}{4}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{1}{16}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{97}{16}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{97}{16}$ , in the original equation gives $LaTeX: \displaystyle \frac{17}{4} = - \frac{15}{4}$ which is false so it is an extraneous solution and there is no solution.