square root (x) - square root of (x-5) = 1 

We are given the equation:

\[
\sqrt{x} - \sqrt{x-5} = 1
\]

### Steps to solve:

1. **Isolate one of the square roots**:
   \[
   \sqrt{x} = 1 + \sqrt{x-5}
   \]

2. **Square both sides to eliminate the square roots**:
   \[
   (\sqrt{x})^2 = (1 + \sqrt{x-5})^2
   \]
   This simplifies to:
   \[
   x = (1)^2 + 2(1)(\sqrt{x-5}) + (\sqrt{x-5})^2
   \]
   \[
   x = 1 + 2\sqrt{x-5} + (x - 5)
   \]
   Simplify further:
   \[
   x = x - 4 + 2\sqrt{x-5}
   \]

3. **Isolate the square root term**:
   \[
   2\sqrt{x-5} = 4
   \]

4. **Divide by 2**:
   \[
   \sqrt{x-5} = 2
   \]

5. **Square both sides again**:
   \[
   x - 5 = 4
   \]

6. **Solve for \(x\)**:
   \[
   x = 9
   \]

### Verify the solution:
Substitute \(x = 9\) back into the original equation:
\[
\sqrt{9} - \sqrt{9-5} = 3 - 2 = 1
\]
This satisfies the equation.

Thus, the solution is:
\[
x = 9
\]

Would you like more details on any of these steps?

### Related Questions:
1. What happens if you square both sides of an equation incorrectly?
2. How do you handle more complex radicals in equations?
3. What are common mistakes to avoid when solving radical equations?
4. Can radical equations have extraneous solutions?
5. How can squaring both sides affect the number of solutions?

### Tip:
Always check for extraneous solutions when solving equations involving square roots, as squaring both sides can introduce invalid answers.

square root (x) - square root of (x-5) = 1

Learn how to solve the radical equation √x - √(x-5) = 1 with step-by-step guidance. This problem involves isolating square root terms and squaring both sides, followed by solving for x and verifying the solution. Suitable for students in Grades 9-11.

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