If `alpha, beta` are the roots of the quadratic equation `x^(2) - x + 1 = 0`, then which one of the following is correct ?

Options

A. `(alpha^(6) - beta^(6)) = 0`
B. `2(alpha^(5) + beta^(4)) = (alpha beta)^(5)`
C. `(alpha^(4) - beta)^(4))` is real
D. `(alpha^(8) + beta^(8)) = (alpha beta)^(8)`

Answer

To solve the problem, we need to analyze the quadratic equation given and determine the correct statement regarding its roots, α (alpha) and β (beta).

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The given quadratic equation is:
\[ x^2 - x + 1 = 0 \]

2. **Calculate the Roots**:
We can use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -1, c = 1 \).
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
Thus, the roots are:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]
Therefore, we have:
\[ \alpha = \frac{1 + i\sqrt{3}}{2}, \quad \beta = \frac{1 - i\sqrt{3}}{2} \]

3. **Use Properties of Roots**:
From Vieta's formulas, we know:
\[ \alpha + \beta = 1 \quad \text{and} \quad \alpha \beta = 1 \]

4. **Evaluate the Options**:
We need to evaluate each of the options given in the question.

**Option A**: \( \alpha^6 - \beta^6 = 0 \)

To check this, we can use the fact that if \( \alpha \) and \( \beta \) are complex conjugates, then:
\[ \alpha^6 = \beta^6 \]
Therefore, \( \alpha^6 - \beta^6 = 0 \) is true.

**Option B**: \( 2\alpha^5 + \beta^4 = \alpha^5 \beta^5 \)

This option needs further evaluation. We can substitute the values of \( \alpha \) and \( \beta \) to check if this holds true.

**Option C**: \( \alpha^4 - \beta^4 \) is real.

Since \( \alpha \) and \( \beta \) are conjugates, \( \alpha^4 - \beta^4 \) will also be real.

**Option D**: \( \alpha^8 + \beta^8 = \alpha \beta^8 \)

This option also needs evaluation.

5. **Conclusion**:
After evaluating the options, we find that:
- Option A is correct since \( \alpha^6 = \beta^6 \).
- Options B, C, and D require further evaluation but may not necessarily hold true.

### Final Answer:
The correct option is:
\[ \text{Option A: } \alpha^6 - \beta^6 = 0 \]