https://file.aiquickdraw.com/mathbot/file/17270385208015jg6s2lv.jpg

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.

Solve the equation √(x - 8) + √x = 2.

Learn how to solve the square root equation √(x - 8) + √x = 2. This problem covers important algebraic concepts like isolating square roots, squaring both sides, and checking for extraneous solutions. Suitable for students in Grades 9-11.

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