If x `= (1)/(x- 5) , ` find:
` x- (1)/(x)`

Answer

To solve the equation \( x = \frac{1}{x - 5} \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x = \frac{1}{x - 5} \]

2. **Cross-multiply to eliminate the fraction:**
\[ x(x - 5) = 1 \]
This simplifies to:
\[ x^2 - 5x = 1 \]

3. **Rearrange the equation to set it to zero:**
\[ x^2 - 5x - 1 = 0 \]

4. **Now, we will use the quadratic formula to solve for \( x \):**
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -5 \), and \( c = -1 \).

5. **Calculate the discriminant:**
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot (-1) = 25 + 4 = 29 \]

6. **Substituting the values into the quadratic formula:**
\[ x = \frac{-(-5) \pm \sqrt{29}}{2 \cdot 1} = \frac{5 \pm \sqrt{29}}{2} \]

7. **Now we need to find \( x - \frac{1}{x} \):**
First, we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{2}{5 \pm \sqrt{29}} \]

8. **To simplify \( x - \frac{1}{x} \):**
\[ x - \frac{1}{x} = \frac{5 \pm \sqrt{29}}{2} - \frac{2}{5 \pm \sqrt{29}} \]

9. **Finding a common denominator:**
The common denominator is \( 2(5 \pm \sqrt{29}) \):
\[ x - \frac{1}{x} = \frac{(5 \pm \sqrt{29})^2 - 4}{2(5 \pm \sqrt{29})} \]

10. **Calculating the numerator:**
\[ (5 \pm \sqrt{29})^2 - 4 = 25 + 29 \pm 10\sqrt{29} - 4 = 50 \pm 10\sqrt{29} \]

11. **Thus, we have:**
\[ x - \frac{1}{x} = \frac{50 \pm 10\sqrt{29}}{2(5 \pm \sqrt{29})} \]

12. **Final simplification gives us:**
\[ x - \frac{1}{x} = 5 \pm \sqrt{29} \]

### Conclusion:
The value of \( x - \frac{1}{x} \) can be expressed as \( 5 \pm \sqrt{29} \).