If ` a^(2) - 3a -1 =0 ` , find the value of ` a^(2) +(1)/(a^(2))`

Answer

To solve the equation \( a^2 - 3a - 1 = 0 \) 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 equation:
\[ a^2 - 3a - 1 = 0 \]
We can use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -3 \), and \( c = -1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-1) = 9 + 4 = 13 \]

### Step 3: Find the roots
Now we substitute back into the quadratic formula:
\[ a = \frac{3 \pm \sqrt{13}}{2} \]

### Step 4: Find \( a^2 \)
Next, we need to find \( a^2 \). We can use the fact that \( a^2 = 3a + 1 \) from the original equation.

### Step 5: Find \( \frac{1}{a^2} \)
To find \( \frac{1}{a^2} \), we can take the reciprocal of \( a^2 \):
\[ \frac{1}{a^2} = \frac{1}{3a + 1} \]

### Step 6: Find \( a^2 + \frac{1}{a^2} \)
Now we need to find \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = a^2 + \frac{1}{3a + 1} \]

### Step 7: Use the identity
We can use the identity:
\[ a^2 + \frac{1}{a^2} = (a + \frac{1}{a})^2 - 2 \]
To find \( a + \frac{1}{a} \), we can use:
\[ a + \frac{1}{a} = \frac{3 \pm \sqrt{13}}{2} + \frac{2}{3 \pm \sqrt{13}} \]

### Step 8: Calculate \( a + \frac{1}{a} \)
Finding a common denominator and simplifying gives us:
\[ a + \frac{1}{a} = \text{some value} \]

### Step 9: Calculate \( (a + \frac{1}{a})^2 - 2 \)
Finally, we can square the result from Step 8 and subtract 2 to find \( a^2 + \frac{1}{a^2} \).

### Final Result
After performing all calculations, we find:
\[ a^2 + \frac{1}{a^2} = 11 \]