The equation provided in the image is:

$\sqrt{x - 8} + \sqrt{x} = 2$

Let's solve this step by step.

Step 1: Isolate one square root term

We'll isolate one of the square root terms on one side of the equation. Let's move $\sqrt{x}$ to the other side:

$\sqrt{x - 8} = 2 - \sqrt{x}$

Step 2: Square both sides

To eliminate the square root, square both sides of the equation:

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

This simplifies to:

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

Now expand the right-hand side:

$x - 8 = 4 - 4\sqrt{x} + x$

Step 3: Simplify the equation

Cancel out the $x$ terms on both sides:

$-8 = 4 - 4\sqrt{x}$

Now subtract 4 from both sides:

$-12 = -4\sqrt{x}$

Step 4: Solve for $\sqrt{x}$

Divide both sides by $-4$:

$\sqrt{x} = 3$

Step 5: Solve for $x$

Now square both sides to get rid of the square root:

$x = 9$

Step 6: Check the solution

Substitute $x = 9$ back into the original equation:

$\sqrt{9 - 8} + \sqrt{9} = 2$

This simplifies to:

$\sqrt{1} + \sqrt{9} = 2$

$1 + 3 = 2$

Clearly, this does not hold true. Hence, there is no valid solution to this equation.

Would you like further details or explanations?

Here are five related questions for further practice:

1. How would you solve $\sqrt{x - 5} + \sqrt{x} = 3$?
2. What are the steps to solving an equation involving multiple square roots?
3. Can a square root equation have no solution? If so, why?
4. What happens if you square both sides of an equation and introduce extraneous solutions?
5. How do you check if a solution is valid after solving a square root equation?

Tip: Always check your solutions when dealing with square root equations, as squaring both sides may introduce extraneous solutions.