Presume that $α_1$ and $α_2$ are solutions to the quadratic function, prove that $α_1^n$ and $α_2^n$ fulfill the recursive formula

Question

Presume that $α_1$ and $α_2$ are solutions to the quadratic function, prove that $α_1^n$ and $α_2^n$ fulfill the recursive formula

Quadratic function: $x^2 = bx + c$

Recursive formula: $ a_{n+2} = ba_{n+1} + ca_n$

I'm honestly very lost on this one

Answer 1

Multiply $\alpha^2 = b\alpha + c$ by $\alpha^n$ to get $\alpha^{n+2} = b\alpha^{n+1} + c\alpha^n$ for all $n \in \mathbb N$.