1. **State the problem:** Solve the equation $x + 1 = \sqrt{7x + 15}$. 2. **Understand the equation:** The left side is linear, and the right side is a square root function. To solve, we will isolate and then square both sides to eliminate the square root. 3. **Square both sides:** $$ (x + 1)^2 = (\sqrt{7x + 15})^2 $$ $$ (x + 1)^2 = 7x + 15 $$ 4. **Expand the left side:** $$ x^2 + 2x + 1 = 7x + 15 $$ 5. **Bring all terms to one side:** $$ x^2 + 2x + 1 - 7x - 15 = 0 $$ $$ x^2 - 5x - 14 = 0 $$ 6. **Factor the quadratic:** $$ (x - 7)(x + 2) = 0 $$ 7. **Solve for $x$:** $$ x - 7 = 0 \Rightarrow x = 7 $$ $$ x + 2 = 0 \Rightarrow x = -2 $$ 8. **Check for extraneous solutions:** Substitute back into original equation. - For $x=7$: $$ 7 + 1 = 8 $$ $$ \sqrt{7(7) + 15} = \sqrt{49 + 15} = \sqrt{64} = 8 $$ Valid. - For $x=-2$: $$ -2 + 1 = -1 $$ $$ \sqrt{7(-2) + 15} = \sqrt{-14 + 15} = \sqrt{1} = 1 $$ Not valid since $-1 \neq 1$. **Final answer:** $$ x = 7 $$