What are some basic methods to solve a recursive function of the shape $f_n = \alpha f_{n-1} + \beta f_{n-2}$ where $\alpha, \beta \in\mathbb{R}$

Question

I want to understand a little bit more about determining the determinant of tridiagonal matrices. These determinants are (mostly) of the form $$f_n = \alpha f_{n-1} + \beta f_{n-2}$$ where $\alpha, \beta \in\mathbb{R}$.

I have a hard time finding literature on how to solve this. Anyone that can help me out?