I have seen this method used in discrete maths where we treat the following equation:

an = an-1 + an-2

as:

x2 = x + 1

then solve the equation and let’s say we got roots r1 and r2 then the recurrence relationship can be written as:

an=c1*(r1)n + c2*(r2)n

what do we do to expand that recurrence relation so that this patten emerge?