If `alpha and beta` are the roots of the equation `x^(2) -x + 2 =0`, then what is `(alpha^(10) + beta^(10))/(alpha^(-10) +beta^(-10))` equal to ?

Options

A. 4096
B. 2048
C. 1024
D. 512

Answer

To solve the problem, we need to find the value of \((\alpha^{10} + \beta^{10}) / (\alpha^{-10} + \beta^{-10})\), where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 - x + 2 = 0\).

### Step 1: Find the roots of the quadratic equation
The roots of the quadratic equation \(x^2 - x + 2 = 0\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -1\), and \(c = 2\).

Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \]
Since the discriminant is negative, the roots are complex:
\[ x = \frac{1 \pm \sqrt{-7}}{2} = \frac{1 \pm i\sqrt{7}}{2} \]
Thus, we have:
\[ \alpha = \frac{1 + i\sqrt{7}}{2}, \quad \beta = \frac{1 - i\sqrt{7}}{2} \]

### Step 2: Calculate \(\alpha^{10} + \beta^{10}\)
To find \(\alpha^{10} + \beta^{10}\), we can use the recurrence relation for powers of roots of quadratic equations. The recurrence relation is given by:
\[ s_n = \alpha^n + \beta^n \]
where \(s_n\) satisfies the relation:
\[ s_n = s_{n-1} + 2s_{n-2} \]
with initial conditions:
\[ s_0 = 2, \quad s_1 = 1 \]
Now we can calculate \(s_n\) for \(n = 2\) to \(10\):
- \(s_2 = s_1 + 2s_0 = 1 + 2 \cdot 2 = 5\)
- \(s_3 = s_2 + 2s_1 = 5 + 2 \cdot 1 = 7\)
- \(s_4 = s_3 + 2s_2 = 7 + 2 \cdot 5 = 17\)
- \(s_5 = s_4 + 2s_3 = 17 + 2 \cdot 7 = 31\)
- \(s_6 = s_5 + 2s_4 = 31 + 2 \cdot 17 = 65\)
- \(s_7 = s_6 + 2s_5 = 65 + 2 \cdot 31 = 127\)
- \(s_8 = s_7 + 2s_6 = 127 + 2 \cdot 65 = 257\)
- \(s_9 = s_8 + 2s_7 = 257 + 2 \cdot 127 = 511\)
- \(s_{10} = s_9 + 2s_8 = 511 + 2 \cdot 257 = 1025\)

Thus, we have:
\[ \alpha^{10} + \beta^{10} = s_{10} = 1025 \]

### Step 3: Calculate \(\alpha^{-10} + \beta^{-10}\)
Using the property of roots:
\[ \alpha^{-n} + \beta^{-n} = \frac{1}{\alpha^n + \beta^n} \]
Thus,
\[ \alpha^{-10} + \beta^{-10} = \frac{1}{\alpha^{10} + \beta^{10}} = \frac{1}{1025} \]

### Step 4: Calculate the final expression
Now we can substitute these values into the expression:
\[ \frac{\alpha^{10} + \beta^{10}}{\alpha^{-10} + \beta^{-10}} = \frac{1025}{\frac{1}{1025}} = 1025 \cdot 1025 = 1025^2 \]
Calculating \(1025^2\):
\[ 1025^2 = 1050625 \]

Thus, the final answer is:
\[ \frac{\alpha^{10} + \beta^{10}}{\alpha^{-10} + \beta^{-10}} = 1050625 \]