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

Squaring both sides gives $LaTeX: \displaystyle x + 12=x - 8 \sqrt{x + 6} + 22$ . Isolating the radical gives $LaTeX: \displaystyle \frac{5}{4}=\sqrt{x + 6}$ Squaring again gives $LaTeX: \displaystyle \frac{25}{16}=x + 6$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=- \frac{71}{16}$ . Checking the solution, $LaTeX: \displaystyle x = - \frac{71}{16}$ , in the original equation gives $LaTeX: \displaystyle \frac{11}{4} = - \frac{11}{4}$ which is false so it is an extraneous solution and there is no solution.