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

Answer

To solve the problem where \( x = \frac{1}{x - 5} \) and we need to find \( x + \frac{1}{x} \), 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 form a standard quadratic equation**:
\[ x^2 - 5x - 1 = 0 \]

4. **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 \( c = -1 \). Plugging in these values:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
Simplifying this:
\[ x = \frac{5 \pm \sqrt{25 + 4}}{2} \]
\[ x = \frac{5 \pm \sqrt{29}}{2} \]

5. **Now, calculate \( \frac{1}{x} \)**:
Since \( x = \frac{5 \pm \sqrt{29}}{2} \), we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{2}{5 \pm \sqrt{29}} \]

6. **Find \( x + \frac{1}{x} \)**:
To find \( x + \frac{1}{x} \), we can use the identity:
\[ x + \frac{1}{x} = \frac{5 \pm \sqrt{29}}{2} + \frac{2}{5 \pm \sqrt{29}} \]
To simplify, we can find a common denominator:
\[ x + \frac{1}{x} = \frac{(5 \pm \sqrt{29})^2 + 2}{(5 \pm \sqrt{29})} \]
Expanding \( (5 \pm \sqrt{29})^2 \):
\[ = 25 + 29 \pm 10\sqrt{29} + 2 = 27 \pm 10\sqrt{29} \]
Therefore:
\[ x + \frac{1}{x} = \frac{27 \pm 10\sqrt{29}}{(5 \pm \sqrt{29})} \]

7. **Final Result**:
We can conclude that:
\[ x + \frac{1}{x} = \sqrt{29} \text{ or } -\sqrt{29} \]