The characteristic polynomial for the Fibonacci recurrence $f_n = f_{n-1}+f_{n-2}$ is $$ x^2 = x +1. $$

The solutions of the characteristic equation $x^2-x-1=0$ are $$ \phi=\frac{1+\sqrt5}2,\qquad \psi=\frac{1-\sqrt5}2 $$ so the closed formula for the Fibonacci sequence must be of the form $$ f_n = u\phi^n +v\psi^n $$ for some real numbers $u,v$ . Now we use the boundary conditions of the recurrence, that is, $f_0=0, f_1=1$ , which means we have to solve the system $$ 0=u \phi^0 +v\psi^0, \qquad 1=u\phi^1 + v\psi^1 $$ The first equation simplifies to $u=-v$ and substituting into the second one gives: $$ 1=u\left(\frac{1+\sqrt5}2\right) - u\left(\frac{1-\sqrt5}2\right) = u\left(\frac{2\sqrt{5}}2\right)=u\sqrt{5}. $$

Therefore $$ u=\frac{1}{\sqrt5},\qquad v=\frac{-1}{\sqrt5} $$ and so $$ f_n = \frac{\phi^n}{\sqrt5}- \frac{\psi^n}{\sqrt5}=\frac{\phi^n-\psi^n}{\sqrt5}. $$