Solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

1. Problem Description

Solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

2. Solution Steps

We want to solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

First, isolate one of the square roots.

$\sqrt{x+8} = 7 - \sqrt{x+1}$

Square both sides of the equation:

$(\sqrt{x+8})^2 = (7 - \sqrt{x+1})^2$

$x+8 = 49 - 14\sqrt{x+1} + (x+1)$

$x+8 = 50 + x - 14\sqrt{x+1}$

Subtract $x$ from both sides:

$8 = 50 - 14\sqrt{x+1}$

Subtract 50 from both sides:

$-42 = -14\sqrt{x+1}$

Divide both sides by -14:

$3 = \sqrt{x+1}$

Square both sides of the equation:

$3^2 = (\sqrt{x+1})^2$

$9 = x+1$

Subtract 1 from both sides:

$x = 8$

Now, we check if $x=8$ is a solution to the original equation.

$\sqrt{8+8} + \sqrt{8+1} = \sqrt{16} + \sqrt{9} = 4+3 = 7$

Thus, $x=8$ is a solution.

3. Final Answer

$x=8$