The water-gas shift reaction is a key industrial process for hydrogen production. It converts CO and H2O into CO2 and H2
The water-gas shift reaction is:
\[ CO + H_2O \rightarrow CO_2 + H_2 \]
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: