To solve the equation \(\sqrt{x} - \sqrt{x-2} = 1\), follow these steps:

1. **Isolate one of the square root terms:**

   \[
   \sqrt{x} = 1 + \sqrt{x-2}
   \]

2. **Square both sides to eliminate the square root on the left:**

   \[
   (\sqrt{x})^2 = (1 + \sqrt{x-2})^2
   \]

3. **Expand the right-hand side:**

   \[
   x = 1 + 2\sqrt{x-2} + (x-2)
   \]

   Simplify the equation:

   \[
   x = x - 1 + 2\sqrt{x-2}
   \]

4. **Subtract \(x\) from both sides:**

   \[
   0 = -1 + 2\sqrt{x-2}
   \]

5. **Isolate the remaining square root term:**

   \[
   1 = 2\sqrt{x-2}
   \]

6. **Divide both sides by 2:**

   \[
   \frac{1}{2} = \sqrt{x-2}
   \]

7. **Square both sides again to eliminate the square root:**

   \[
   \left(\frac{1}{2}\right)^2 = x - 2
   \]

   \[
   \frac{1}{4} = x - 2
   \]

8. **Solve for \(x\):**

   \[
   x = 2 + \frac{1}{4}
   \]

   \[
   x = \frac{8}{4} + \frac{1}{4}
   \]

   \[
   x = \frac{9}{4}
   \]

The solution to the equation \(\sqrt{x} - \sqrt{x-2} = 1\) is:

\[
x = \frac{9}{4}
\]

To verify, substitute \(x = \frac{9}{4}\) back into the original equation:

\[
\sqrt{\frac{9}{4}} - \sqrt{\frac{9}{4} - 2} = \frac{3}{2} - \sqrt{\frac{1}{4}} = \frac{3}{2} - \frac{1}{2} = 1
\]

The solution \(x = \frac{9}{4}\) satisfies the original equation.

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve equations involving multiple square roots?
2. What are the properties of square roots used in solving equations?
3. Can you solve \(\sqrt{x+3} - \sqrt{x-1} = 2\)?
4. How do you verify solutions to square root equations?
5. What are common mistakes to avoid when solving square root equations?
6. How can you graph the function \(y = \sqrt{x} - \sqrt{x-2}\)?
7. What is the domain of the function \(y = \sqrt{x} - \sqrt{x-2}\)?
8. Can you solve \(\sqrt{2x+1} - \sqrt{x} = 1\)?

**Tip:** Always verify your solutions by substituting them back into the original equation to check if they satisfy the equation.

Learn how to solve the square root equation sqrt(x) - sqrt(x-2) = 1 with a detailed step-by-step solution. This problem covers the properties of square roots and equations involving them. Suitable for students in Grades 10-12.

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