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

Answer

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

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

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

### Step 4: Find \( a^2 + \frac{1}{a^2} \)
To find \( a^2 - \frac{1}{a^2} \), we can first find \( a + \frac{1}{a} \). We know:
\[ a + \frac{1}{a} = \frac{5 \pm \sqrt{29}}{2} + \frac{2}{5 \pm \sqrt{29}} \]
To simplify \( \frac{1}{a} \), we rationalize:
\[ \frac{1}{a} = \frac{2}{5 \pm \sqrt{29}} \cdot \frac{5 \mp \sqrt{29}}{5 \mp \sqrt{29}} = \frac{2(5 \mp \sqrt{29})}{(5^2 - 29)} = \frac{2(5 \mp \sqrt{29})}{6} = \frac{5 \mp \sqrt{29}}{3} \]
Thus,
\[ a + \frac{1}{a} = \frac{5 \pm \sqrt{29}}{2} + \frac{5 \mp \sqrt{29}}{3} \]

### Step 5: Find \( a^2 + \frac{1}{a^2} \)
Using the identity:
\[ a^2 + \frac{1}{a^2} = (a + \frac{1}{a})^2 - 2 \]
We can compute \( a^2 - \frac{1}{a^2} \) directly using:
\[ a^2 - \frac{1}{a^2} = (a + \frac{1}{a})(a - \frac{1}{a}) \]

### Step 6: Calculate \( a^2 - \frac{1}{a^2} \)
Using the values obtained:
\[ a^2 - \frac{1}{a^2} = (a + \frac{1}{a})(a - \frac{1}{a}) = (5)(5) - 2 = 25 - 2 = 23 \]

### Final Result
Thus, the value of \( a^2 - \frac{1}{a^2} \) is:
\[ \boxed{23} \]