\[ \begin{array}{|c|c|} \hline \text{List - I} & \text{List - II} \\ \hline \text{(A) } \text{Ti}^{3+} & \text{(I) } 3.87 \\ \text{(B) } \text{V}^{2+} & \text{(II) } 0.00 \\ \text{(C) } \text{Ni}^{2+} & \text{(III) } 1.73 \\ \text{(D) } \text{Sc}^{3+} & \text{(IV) } 2.84 \\ \hline \end{array} \]
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: