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

Options

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

Explanation

To solve the problem, we need to analyze the quadratic equation given and find the roots, then evaluate the options provided based on these roots.

### Step-by-Step Solution:

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

2. **Calculate the roots using the quadratic formula:**
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -1\), and \(c = 1\). Plugging in these values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]
Hence, the roots are:
\[ \alpha = \frac{1 + i\sqrt{3}}{2}, \quad \beta = \frac{1 - i\sqrt{3}}{2} \]

3. **Find \(\alpha^4 - \beta^4\):**
We can use the identity:
\[ \alpha^4 - \beta^4 = (\alpha^2 - \beta^2)(\alpha^2 + \beta^2) \]
First, we need to calculate \(\alpha^2\) and \(\beta^2\):
\[ \alpha^2 = \left(\frac{1 + i\sqrt{3}}{2}\right)^2 = \frac{1 + 2i\sqrt{3} - 3}{4} = \frac{-2 + 2i\sqrt{3}}{4} = \frac{-1 + i\sqrt{3}}{2} \]
\[ \beta^2 = \left(\frac{1 - i\sqrt{3}}{2}\right)^2 = \frac{1 - 2i\sqrt{3} - 3}{4} = \frac{-2 - 2i\sqrt{3}}{4} = \frac{-1 - i\sqrt{3}}{2} \]

Now, calculate \(\alpha^2 - \beta^2\):
\[ \alpha^2 - \beta^2 = \frac{-1 + i\sqrt{3}}{2} - \frac{-1 - i\sqrt{3}}{2} = \frac{2i\sqrt{3}}{2} = i\sqrt{3} \]

Next, calculate \(\alpha^2 + \beta^2\):
\[ \alpha^2 + \beta^2 = \frac{-1 + i\sqrt{3}}{2} + \frac{-1 - i\sqrt{3}}{2} = \frac{-2}{2} = -1 \]

Now substitute back into the identity:
\[ \alpha^4 - \beta^4 = (i\sqrt{3})(-1) = -i\sqrt{3} \]

4. **Evaluate the options:**
Now we need to check which of the options is correct based on our calculations.

- **Option A:** \(\alpha^4 - \beta^4\) is not real (it is \(-i\sqrt{3}\)).
- **Option B:** \(2\alpha^6 + \beta^6 = \alpha\beta^5\) needs to be checked.
- **Option C:** \(\alpha^6 - \beta^6 = 0\) needs to be checked.

After evaluating these, we find:
- **Option C** is correct because \(\alpha^6 - \beta^6 = 0\) holds true.

### Final Answer:
The correct option is **C: \(\alpha^6 - \beta^6 = 0\)**.