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

Options

A. `alpha^(2) and beta^(13)`
B. ` alpha^(13) and beta^7`
C. ` alpha ^(20) and beta^(20)`
D. None of these

Explanation

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

### Step 1: Find the roots of the equation \(x^2 + x + 1 = 0\)

We start with the quadratic equation:

\[ x^2 + x + 1 = 0 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = 1\):

\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

Calculating the discriminant:

\[ b^2 - 4ac = 1 - 4 = -3 \]

Since the discriminant is negative, the roots are complex:

\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]

This can be simplified to:

\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]

Thus, the roots are:

\[ \alpha = \frac{-1 + i\sqrt{3}}{2}, \quad \beta = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Find the roots of the equation \(x^2 - x + 1 = 0\)

Now we need to find the roots of the equation:

\[ x^2 - x + 1 = 0 \]

Using the quadratic formula again:

\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

Calculating the discriminant:

\[ (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

Again, the discriminant is negative, so the roots are complex:

\[ x = \frac{1 \pm \sqrt{-3}}{2} \]

This can be simplified to:

\[ x = \frac{1 \pm i\sqrt{3}}{2} \]

Thus, the roots are:

\[ \gamma = \frac{1 + i\sqrt{3}}{2}, \quad \delta = \frac{1 - i\sqrt{3}}{2} \]

### Step 3: Relate the roots of the two equations

Notice that the roots \(\alpha\) and \(\beta\) can be expressed in terms of the cube roots of unity:

\[ \alpha = \omega, \quad \beta = \omega^2 \]

where \(\omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2}\) and \(\omega^2 = e^{-2\pi i / 3} = \frac{-1 - i\sqrt{3}}{2}\).

The roots of the second equation can also be expressed in terms of the cube roots of unity:

\[ \gamma = \omega^2, \quad \delta = \omega \]

### Conclusion

Thus, the roots of the equation \(x^2 - x + 1 = 0\) are \(\frac{1 + i\sqrt{3}}{2}\) and \(\frac{1 - i\sqrt{3}}{2}\), which correspond to \(\beta\) and \(\alpha\) respectively.