For a soap bubble, there are two liquid-air surfaces, so the excess pressure \( \Delta P \) inside the bubble is given by:
\[\Delta P = 2 \left( \frac{2S}{R} \right) = \frac{4S}{R}\]
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: