Mathematics - Quadratic Equation Question with Solution | TestHub

$\begin{array}{rl}& {\mathrm{x}}^2-\sqrt{2\mathrm{x}}-\sqrt{3}=0⟨{}_{\beta }^{\alpha }\\ & {\alpha }^{\mathrm{n}+2}-\sqrt{2}{\alpha }^{\mathrm{n}+1}-\sqrt{3}{\alpha }^{\mathrm{n}}=0\\ & \text{ and }{\beta }^{\mathrm{n}+2}-\sqrt{2}{\beta }^{\mathrm{n}+1}-\sqrt{3}{\beta }^{\mathrm{n}}=0\end{array}$
Subtracting $\begin{array}{rl}& \left({\alpha }^{\mathrm{n}+2}-{\beta }^{\mathrm{n}+2}\right)-\sqrt{2}\left({\alpha }^{\mathrm{n}+1}-{\beta }^{\mathrm{n}+1}\right)-\sqrt{3}\left({\alpha }^{\mathrm{n}}-{\beta }^{\mathrm{n}}\right)=0\\ & \Rightarrow {\mathrm{P}}_{\mathrm{n}+2}-\sqrt{2}{\mathrm{P}}_{\mathrm{n}+1}-\sqrt{3}{\mathrm{P}}_{\mathrm{n}}=0\end{array}$ $\begin{array}{rl}& \text{ Put }\mathrm{n}=10\\ & {\mathrm{P}}_{12}-\sqrt{2}{\mathrm{P}}_{11}-\sqrt{3}{\mathrm{P}}_{10}=0\\ & \mathrm{n}=9\\ & {\mathrm{P}}_{11}-\sqrt{2}{\mathrm{P}}_{10}-\sqrt{3}{\mathrm{P}}_9=0\\ & 11\left(\sqrt{3}\cdot {\mathrm{P}}_{10}+\sqrt{2}{\mathrm{P}}_{11}-{\mathrm{P}}_{11}\right)-10\left(\sqrt{2}{\mathrm{P}}_{10}-{\mathrm{P}}_{11}\right)\\ & =0-10\left(-\sqrt{3}{\mathrm{P}}_9\right)=10\sqrt{3}{\mathrm{P}}_9\end{array}$