Characteristic Roots.
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*}
If \(r_1\) and \(r_2\) are two distinct roots of the characteristic polynomial (i.e., solutions to the characteristic equation), then the solution to the recurrence relation is
\begin{equation*}
a_n = ar_1^n + br_2^n\text{,}
\end{equation*}
where \(a\) and \(b\) are constants determined by the initial conditions.