Solve. $LaTeX: \displaystyle \sqrt{x + 14}=\sqrt{x - 4} + 1$

Squaring both sides gives $LaTeX: \displaystyle x + 14=x + 2 \sqrt{x - 4} - 3$ . Isolating the radical gives $LaTeX: \displaystyle \frac{17}{2}=\sqrt{x - 4}$ Squaring again gives $LaTeX: \displaystyle \frac{289}{4}=x - 4$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{305}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{305}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{19}{2} = \frac{19}{2}$ which is true so the solution checks.