If `alpha` and `beta` are the roots of the equations `x^(2)-2x-1=0`, then what is the value of `alpha^(2)beta^(-2)+beta^(2)alpha^(-2)`

Options

A. `-2`
B. 0
C. 30
D. 34

Answer

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

### Step-by-step Solution:

1. **Find the roots \( \alpha \) and \( \beta \)**:
We can use the quadratic formula to find the roots of the equation \( x^2 - 2x - 1 = 0 \).
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -2, c = -1 \).
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \]
Thus, the roots are:
\[ \alpha = 1 + \sqrt{2}, \quad \beta = 1 - \sqrt{2} \]

2. **Calculate \( \alpha^2 \) and \( \beta^2 \)**:
\[ \alpha^2 = (1 + \sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \]
\[ \beta^2 = (1 - \sqrt{2})^2 = 1 - 2\sqrt{2} + 2 = 3 - 2\sqrt{2} \]

3. **Calculate \( \alpha^2 \beta^{-2} + \beta^2 \alpha^{-2} \)**:
We can rewrite the expression:
\[ \alpha^2 \beta^{-2} + \beta^2 \alpha^{-2} = \frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2} \]
Let \( z = \frac{\alpha^2}{\beta^2} \). Then:
\[ z + \frac{1}{z} = \frac{\alpha^4 + \beta^4}{\alpha^2 \beta^2} \]

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 \( \alpha^2 + \beta^2 \):
\[ \alpha + \beta = 2 \quad \text{and} \quad \alpha \beta = -1 \]
Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta = 2^2 - 2(-1) = 4 + 2 = 6 \]
Now, substituting:
\[ \alpha^4 + \beta^4 = 6^2 - 2(-1)^2 = 36 - 2 = 34 \]

5. **Find \( \alpha^2 \beta^2 \)**:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = (-1)^2 = 1 \]

6. **Final calculation**:
Now substituting back:
\[ z + \frac{1}{z} = \frac{34}{1} = 34 \]

Thus, the value of \( \alpha^2 \beta^{-2} + \beta^2 \alpha^{-2} \) is \( 34 \).

### Final Answer:
\[ \boxed{34} \]