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 the value of \( a - \frac{1}{a} \), we can 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 \( b^2 - 4ac \):
\[ (-5)^2 - 4 \cdot 1 \cdot (-1) = 25 + 4 = 29 \]

### Step 3: Substitute into the quadratic formula
Now we substitute back 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:
\[ \frac{1}{a} = \frac{2}{5 \pm \sqrt{29}} \]
Now, we rationalize the denominator:
\[ \frac{1}{a} = \frac{2(5 \mp \sqrt{29})}{(5 + \sqrt{29})(5 - \sqrt{29})} = \frac{2(5 \mp \sqrt{29})}{25 - 29} = \frac{2(5 \mp \sqrt{29})}{-4} = -\frac{1}{2}(5 \mp \sqrt{29}) \]

### Step 5: Combine the results
Now, we can find \( a - \frac{1}{a} \):
\[ a - \frac{1}{a} = \left(\frac{5 \pm \sqrt{29}}{2}\right) - \left(-\frac{1}{2}(5 \mp \sqrt{29})\right) \]
This simplifies to:
\[ a - \frac{1}{a} = \frac{5 \pm \sqrt{29}}{2} + \frac{5 \mp \sqrt{29}}{2} = \frac{10}{2} = 5 \]

Thus, the final answer is:
\[ \boxed{5} \]