Solve $LaTeX: \displaystyle x + 3 = \sqrt{8 x + 57}$ .

Squaring both sides gives $LaTeX: \displaystyle x^{2} + 6 x + 9 = 8 x + 57$ . The equation is quadratic setting it equal to zero gives $LaTeX: \displaystyle x^{2} - 2 x - 48 = 0$ . Factoring gives $LaTeX: \displaystyle (x - 8)(x + 6)=0$ so the possible solutions are $LaTeX: \displaystyle x = 8$ and $LaTeX: \displaystyle x = -6$ . Checking the solution $LaTeX: \displaystyle x = 8$ in the original equation gives $LaTeX: \displaystyle 11 = 11$ . The solution checks, so $LaTeX: \displaystyle x = 8$ is a true solution. Checking the solution $LaTeX: \displaystyle x = -6$ in the original equation gives $LaTeX: \displaystyle -3 = 3$ . The solution does no check, so $LaTeX: \displaystyle x = -6$ is an extraneous solution.