Let `p, q` be integers and let `alpha,beta` be the roots of the equation `x^2-2x+3=0` where `alpha != beta` For `n= 0, 1, 2,.......,` Let `alpha_n=palpha^n+qbeta^n` value `alpha_9=`

Options

A. `3a_(8)-5a_(7)+3a_(6)`
B. `a_(7)-6a_(6)`
C. `a_(7)-12a_(5)+18a_(4)`
D. `2a_(8)+3a_(6)`

Explanation

To solve the problem, we need to find the value of \( \alpha_9 \) given the roots \( \alpha \) and \( \beta \) of the quadratic equation \( x^2 - 2x + 3 = 0 \).

### Step-by-Step Solution:

1. **Find the roots \( \alpha \) and \( \beta \)**:
The roots of the equation \( x^2 - 2x + 3 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -2, c = 3 \).
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 12}}{2} = \frac{2 \pm \sqrt{-8}}{2} = \frac{2 \pm 2i\sqrt{2}}{2} = 1 \pm i\sqrt{2} \]
Thus, the roots are:
\[ \alpha = 1 + i\sqrt{2}, \quad \beta = 1 - i\sqrt{2} \]

2. **Set up the recurrence relation**:
We know that:
\[ \alpha_n = p\alpha^n + q\beta^n \]
From the characteristic equation derived from the roots, we can establish the recurrence relation:
\[ \alpha_{n+2} = 2\alpha_{n+1} - 3\alpha_n \]
This is derived from the fact that both \( \alpha \) and \( \beta \) satisfy the original quadratic equation.

3. **Calculate \( \alpha_0 \) and \( \alpha_1 \)**:
We need initial conditions for the recurrence relation:
- For \( n = 0 \):
\[ \alpha_0 = p\alpha^0 + q\beta^0 = p + q \]
- For \( n = 1 \):
\[ \alpha_1 = p\alpha^1 + q\beta^1 = p\alpha + q\beta \]

4. **Use the recurrence relation to find \( \alpha_9 \)**:
We can calculate \( \alpha_2, \alpha_3, \ldots, \alpha_9 \) using the recurrence relation:
\[ \alpha_2 = 2\alpha_1 - 3\alpha_0 \]
\[ \alpha_3 = 2\alpha_2 - 3\alpha_1 \]
\[ \alpha_4 = 2\alpha_3 - 3\alpha_2 \]
\[ \alpha_5 = 2\alpha_4 - 3\alpha_3 \]
\[ \alpha_6 = 2\alpha_5 - 3\alpha_4 \]
\[ \alpha_7 = 2\alpha_6 - 3\alpha_5 \]
\[ \alpha_8 = 2\alpha_7 - 3\alpha_6 \]
\[ \alpha_9 = 2\alpha_8 - 3\alpha_7 \]

5. **Final Calculation**:
Substitute the values of \( \alpha_0 \) and \( \alpha_1 \) into the recurrence relation to compute \( \alpha_9 \).