If `alpha and beta` are the root of the equation `x^(2) - 2x + 4 = 0`. Then which of the following are the roots of the equation `x^(2) - x +1 = 0`?

Options

A. `alpha^(7) and beta^(13)`
B. `alpha^(13) and beta^(7)`
C. `alpha^(20) and beta^(20)`
D. None of these

Answer

To solve the problem, we will first find the roots of the given quadratic equations step by step.

### Step 1: Identify the first equation and its coefficients
The first equation given is:
\[ x^2 - 2x + 4 = 0 \]
Here, we have:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 4 \)

### Step 2: Use the quadratic formula to find the roots
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]
\[ x = \frac{2 \pm \sqrt{4 - 16}}{2} \]
\[ x = \frac{2 \pm \sqrt{-12}}{2} \]
\[ x = \frac{2 \pm \sqrt{12}i}{2} \]
\[ x = 1 \pm \sqrt{3}i \]

Let:
- \( \alpha = 1 + \sqrt{3}i \)
- \( \beta = 1 - \sqrt{3}i \)

### Step 3: Identify the second equation and its coefficients
The second equation is:
\[ x^2 - x + 1 = 0 \]
Here, we have:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 1 \)

### Step 4: Use the quadratic formula to find the roots of the second equation
Using the quadratic formula again:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 - 4}}{2} \]
\[ x = \frac{1 \pm \sqrt{-3}}{2} \]
\[ x = \frac{1 \pm \sqrt{3}i}{2} \]

Let:
- \( \gamma = \frac{1 + \sqrt{3}i}{2} \)
- \( \delta = \frac{1 - \sqrt{3}i}{2} \)

### Step 5: Compare the roots of the second equation with the roots of the first equation
We can express \( \gamma \) and \( \delta \) in terms of \( \alpha \) and \( \beta \):
- \( \gamma = \frac{\alpha}{2} \)
- \( \delta = \frac{\beta}{2} \)

### Step 6: Determine the correct option
The roots of the equation \( x^2 - x + 1 = 0 \) are \( \frac{\alpha}{2} \) and \( \frac{\beta}{2} \). The options provided do not match this form, so the correct answer is:
**None of these.**