If ` x^2 - 3x +1=0,` then the value of ` x +1/x ` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( x^2 - 3x + 1 = 0 \) and find the value of \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ x^2 - 3x + 1 = 0 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -3, c = 1 \).

### 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: Find the roots
Now we can find the roots using the quadratic formula:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 4: Find \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). We can express \( \frac{1}{x} \) in terms of \( x \):
\[ \frac{1}{x} = \frac{2}{3 \pm \sqrt{5}} \]
To simplify \( x + \frac{1}{x} \), we can multiply \( x \) by \( \frac{1}{x} \):
\[ x + \frac{1}{x} = x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
Using the original equation \( x^2 - 3x + 1 = 0 \), we can rearrange it to find \( x^2 \):
\[ x^2 = 3x - 1 \]
Now substituting this into our expression:
\[ x + \frac{1}{x} = \frac{(3x - 1) + 1}{x} = \frac{3x}{x} = 3 \]

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