If `a^(2)- 3a + 1= 0 and a ne 0`, find : `a+ (1)/(a)`

Answer

To solve the equation \( a^2 - 3a + 1 = 0 \) and find \( a + \frac{1}{a} \), 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 to find the values of \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3 \), and \( c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]

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

### Step 4: Find \( a + \frac{1}{a} \)
Next, we need to find \( a + \frac{1}{a} \). We can express \( \frac{1}{a} \) in terms of \( a \):
\[ \frac{1}{a} = \frac{2}{3 \pm \sqrt{5}} \]
To simplify \( a + \frac{1}{a} \):
\[ a + \frac{1}{a} = a + \frac{2}{3 \pm \sqrt{5}} \]

### Step 5: Rationalize the denominator
To simplify \( \frac{2}{3 \pm \sqrt{5}} \), we can multiply the numerator and denominator by the conjugate:
\[ \frac{2(3 \mp \sqrt{5})}{(3 \pm \sqrt{5})(3 \mp \sqrt{5})} = \frac{2(3 \mp \sqrt{5})}{9 - 5} = \frac{2(3 \mp \sqrt{5})}{4} = \frac{3 \mp \sqrt{5}}{2} \]

### Step 6: Combine the results
Now we can combine the results:
\[ a + \frac{1}{a} = \frac{3 \pm \sqrt{5}}{2} + \frac{3 \mp \sqrt{5}}{2} \]
This simplifies to:
\[ a + \frac{1}{a} = \frac{(3 + 3) + (\sqrt{5} - \sqrt{5})}{2} = \frac{6}{2} = 3 \]

### Final Answer
Thus, the value of \( a + \frac{1}{a} \) is:
\[ \boxed{3} \]