Solve $LaTeX: \displaystyle x + 1 = \sqrt{7 x + 15}$ .

Squaring both sides gives $LaTeX: \displaystyle x^{2} + 2 x + 1 = 7 x + 15$ . The equation is quadratic setting it equal to zero gives $LaTeX: \displaystyle x^{2} - 5 x - 14 = 0$ . Factoring gives $LaTeX: \displaystyle (x - 7)(x + 2)=0$ so the possible solutions are $LaTeX: \displaystyle x = 7$ and $LaTeX: \displaystyle x = -2$ . Checking the solution $LaTeX: \displaystyle x = 7$ in the original equation gives $LaTeX: \displaystyle 8 = 8$ . The solution checks, so $LaTeX: \displaystyle x = 7$ is a true solution. Checking the solution $LaTeX: \displaystyle x = -2$ in the original equation gives $LaTeX: \displaystyle -1 = 1$ . The solution does no check, so $LaTeX: \displaystyle x = -2$ is an extraneous solution.