Given a recurrence relation \(a_n + \alpha a_{n-1} + \beta a_{n-2} = 0\text{,}\) the characteristic polynomial is

\begin{equation*} x^2 + \alpha x + \beta \end{equation*}

giving the characteristic equation:

\begin{equation*} x^2 + \alpha x + \beta = 0\text{.} \end{equation*}