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 \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -5, c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 1 = 25 - 4 = 21 \]

### Step 3: Find the roots
Now we can substitute the values into the quadratic formula:
\[ a = \frac{5 \pm \sqrt{21}}{2} \]

### Step 4: Find \( a + \frac{1}{a} \)
Next, we need to find \( a + \frac{1}{a} \). We can express \( \frac{1}{a} \) as follows:
\[ \frac{1}{a} = \frac{2}{5 \pm \sqrt{21}} \]
To simplify \( a + \frac{1}{a} \), we will use the identity:
\[ a + \frac{1}{a} = \frac{a^2 + 1}{a} \]
We already have \( a^2 \) from the original equation:
\[ a^2 = 5a - 1 \]
So,
\[ a + \frac{1}{a} = \frac{(5a - 1) + 1}{a} = \frac{5a}{a} = 5 \]

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