Solve. $LaTeX: \displaystyle \sqrt{x + 14}=\sqrt{x - 4} - 2$

Squaring both sides gives $LaTeX: \displaystyle x + 14=x - 4 \sqrt{x - 4}$ . Isolating the radical gives $LaTeX: \displaystyle - \frac{7}{2}=\sqrt{x - 4}$ Squaring again gives $LaTeX: \displaystyle \frac{49}{4}=x - 4$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{65}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{65}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{11}{2} = \frac{3}{2}$ which is false so it is an extraneous solution and there is no solution.