Solve. $LaTeX: \displaystyle \sqrt{x + 24}=\sqrt{x - 2} + 1$

Squaring both sides gives $LaTeX: \displaystyle x + 24=x + 2 \sqrt{x - 2} - 1$ . Isolating the radical gives $LaTeX: \displaystyle \frac{25}{2}=\sqrt{x - 2}$ Squaring again gives $LaTeX: \displaystyle \frac{625}{4}=x - 2$ . Solving for $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x=\frac{633}{4}$ . Checking the solution, $LaTeX: \displaystyle x = \frac{633}{4}$ , in the original equation gives $LaTeX: \displaystyle \frac{27}{2} = \frac{27}{2}$ which is true so the solution checks.