If ` alpha , beta ` are the roots of ` x^2 +x+1=0` then ` alpha beta + beta alpha =`

Options

A. -1
B. 1
C. 2
D. i

Answer

To solve the problem, we need to find the value of \( \alpha \beta + \beta \alpha \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + x + 1 = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given equation is \( x^2 + x + 1 = 0 \). We can compare this with the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
- Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \).

2. **Use Vieta's formulas**:
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Applying these formulas:
- \( \alpha + \beta = -\frac{1}{1} = -1 \)
- \( \alpha \beta = \frac{1}{1} = 1 \)

3. **Calculate \( \alpha \beta + \beta \alpha \)**:
The expression \( \alpha \beta + \beta \alpha \) can be simplified as follows:
\[ \alpha \beta + \beta \alpha = 2 \alpha \beta \]
Since we have already found \( \alpha \beta = 1 \):
\[ \alpha \beta + \beta \alpha = 2 \cdot 1 = 2 \]

4. **Final answer**:
Therefore, the value of \( \alpha \beta + \beta \alpha \) is \( 2 \).

### Conclusion:
The answer is \( 2 \).