If `a^(2)- 5a-1= 0 and a ne 0`, find : `a - (1)/(a)`

Answer

To solve the equation \( a^2 - 5a - 1 = 0 \) and find \( a - \frac{1}{a} \), we will follow these steps:

### Step 1: Solve the quadratic equation
We start with the equation:
\[ a^2 - 5a - 1 = 0 \]
We can use the quadratic formula to find the values of \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -5 \), and \( c = -1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot (-1) = 25 + 4 = 29 \]

### Step 3: Substitute values into the quadratic formula
Now we substitute the values into the quadratic formula:
\[ a = \frac{5 \pm \sqrt{29}}{2} \]

### Step 4: Find \( a - \frac{1}{a} \)
Next, we need to find \( a - \frac{1}{a} \). We can express \( \frac{1}{a} \) as follows:
\[ \frac{1}{a} = \frac{2}{5 \pm \sqrt{29}} \]
To find \( a - \frac{1}{a} \), we can rewrite it as:
\[ a - \frac{1}{a} = a - \frac{2}{5 \pm \sqrt{29}} \]

### Step 5: Simplify \( a - \frac{1}{a} \)
Using the value of \( a \):
\[ a - \frac{1}{a} = \frac{5 \pm \sqrt{29}}{2} - \frac{2}{\frac{5 \pm \sqrt{29}}{2}} \]
This can be simplified further, but we can also use the fact that:
\[ a - \frac{1}{a} = a - \frac{2}{5 \pm \sqrt{29}} = \frac{(5 \pm \sqrt{29})^2 - 2 \cdot 2}{(5 \pm \sqrt{29})} \]

### Step 6: Final calculation
Calculating this will yield:
\[ a - \frac{1}{a} = 5 \]

### Conclusion
Thus, the value of \( a - \frac{1}{a} \) is:
\[ \boxed{5} \]