If `alpha, beta` are the roots of `x^(2) - x + 3 = 0`, then `alpha^(4) +beta^(4)` =

Options

A. 10
B. 7
C. -10
D. 8

Answer

To solve for \( \alpha^4 + \beta^4 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - x + 3 = 0 \), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is:
\[ x^2 - x + 3 = 0 \]
Here, we can identify:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 3 \)

### Step 2: Find \( \alpha + \beta \) and \( \alpha \beta \)
Using Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-1}{1} = 1 \]
\[ \alpha \beta = \frac{c}{a} = \frac{3}{1} = 3 \]

### Step 3: Find \( \alpha^2 + \beta^2 \)
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (1)^2 - 2(3) = 1 - 6 = -5 \]

### Step 4: Find \( \alpha^4 + \beta^4 \)
We can use the identity:
\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha^2 \beta^2) \]
First, we need to find \( \alpha^2 \beta^2 \):
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = (3)^2 = 9 \]
Now substituting into the equation:
\[ \alpha^4 + \beta^4 = (-5)^2 - 2(9) = 25 - 18 = 7 \]

### Final Answer
Thus, the value of \( \alpha^4 + \beta^4 \) is:
\[ \boxed{7} \]