Solve. $LaTeX: \displaystyle \sqrt{x + 12}=\sqrt{x - 6} - 1$

Squaring both sides gives $LaTeX: \displaystyle x + 12=x - 2 \sqrt{x - 6} - 5$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{17}{2}=\sqrt{x - 6}$ Squaring again gives $LaTeX: \displaystyle \frac{289}{4}=x - 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{313}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{313}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{19}{2} = \frac{15}{2}$ which is false so it is an extraneous solution and there is no solution.