Fibonacci and the algebraic expression $x^2-x-1$ $$\left(  \left( 1/2\,{\frac {-\beta+ \sqrt{{\beta}^{2}-4\,\delta\,
\alpha}}{\alpha}} \right) ^{i}- \left( -1/2\,{\frac {\beta+ \sqrt{{
\beta}^{2}-4\,\delta\,\alpha}}{\alpha}} \right) ^{i} \right)  \left( 
 \sqrt{{\beta}^{2}-4\,\delta\,\alpha} \right) ^{-1}$$
What exactly is the connection between the above formula and
$$x^2-x-c$$
c being a constant.
I know about linear recursion, but I don't understand why the Fibonacci Numbers are associated with $$x^2-x-1$$
I know about
$${\frac {\sqrt {5}+1}{{2}}}$$
But I don't see the connection. If I were to represent these terms algebraically it seems I would do something like this:
$${\alpha}^{-1}$$
$$-{\frac {\beta}{{\alpha}^{2}}}$$
$$-{\frac {-{\beta}^{2}+\delta\,\alpha}{{\alpha}^{3}}}$$
$${\frac {\beta\, \left( -{\beta}^{2}+2\,\delta\,\alpha \right) }{{
\alpha}^{4}}}$$
$${\frac {{\beta}^{4}-3\,{\beta}^{2}\delta\,\alpha+{\delta}^{2}{\alpha}^
{2}}{{\alpha}^{5}}}$$
$$-{\frac {\beta\, \left( {\beta}^{4}-4\,{\beta}^{2}\delta\,\alpha+3\,{
\delta}^{2}{\alpha}^{2} \right) }{{\alpha}^{6}}}
$$
There also seems to be a relationship with factorials when they are used as arrays in matrices.

Let $F_0=0$, $F_1=1$, $\forall n \in \Bbb N, F_{n+2}=F_{n+1}+F_n$
It's the Fibonacci sequence.
Now let $\forall n \in \Bbb N,V_n=\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}$
Let $M=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix}$
Then $MV_n=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix}\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}=\begin{pmatrix}
F_{n+1}\\F_n+F_{n+1}\end{pmatrix}$
By induction, $\forall n \in \Bbb N, V_n=M^nV_0$
So the only thing we really need to compute is $M^n$. In order to do this, we try do diagonalize it.
Its characteristic polynomial is $\chi_M(\lambda)=\det(M-\lambda I_2)=\det \begin{pmatrix}-\lambda & 1\\ 1 & 1-\lambda\end{pmatrix}=(-\lambda)(1-\lambda)-(1)(1)=\lambda^2-\lambda-1$
Let $\varphi=\cfrac{1+\sqrt{5}}{2}$ and $\psi=\cfrac{1-\sqrt{5}}{2}$ be the two roots of that polynomial.
Since $M\in M_2(\Bbb R)$, we can find $P\in GL_2(\Bbb R)$ so that $M=PDP^{-1}$ where $D=\begin{pmatrix}\varphi & 0\\ 0 & \psi\end{pmatrix}$
$D$ is the matrix of the canonical endomophism associated to $M$ in a basis $e=(e_1,e_2)$ where $Me_1=\varphi e_1$ and $Me_2=\psi e_2$
We know that $e_1\in Ker(M-\varphi I_2) = Ker\left(\begin{pmatrix}-\varphi & 1\\ 1 & 1-\varphi\end{pmatrix}\right)=Vect\left(\begin{pmatrix}\cfrac{1}{\varphi}\\1\end{pmatrix}\right)$ because $\cfrac{1}{\varphi}\begin{pmatrix}-\varphi\\ 1 &\end{pmatrix}+1\begin{pmatrix} 1\\ 1-\varphi\end{pmatrix}=\begin{pmatrix} 0\\0\end{pmatrix}$ so we can take $e_1=\varphi\begin{pmatrix}\cfrac{1}{\varphi}\\ 1\end{pmatrix}=\begin{pmatrix}1\\ \varphi\end{pmatrix}$
Similarly, $e_2\in Ker(M-\psi I_2) = Ker\left(\begin{pmatrix}-\psi & 1\\ 1 & 1-\psi\end{pmatrix}\right)=Vect\left(\begin{pmatrix}\cfrac{1}{\psi}\\1\end{pmatrix}\right)$ because $\cfrac{1}{\psi}\begin{pmatrix}-\psi\\ 1 &\end{pmatrix}+1\begin{pmatrix} 1\\ 1-\psi\end{pmatrix}=\begin{pmatrix} 0\\0\end{pmatrix}$ so we can take $e_2=\psi\begin{pmatrix}\cfrac{1}{\psi}\\ 1\end{pmatrix}=\begin{pmatrix}1\\ \psi\end{pmatrix}$
And we get $P=\begin{pmatrix}1 & 1\\ \varphi & \psi\end{pmatrix}$
Then we need $P^{-1}=\cfrac{\begin{pmatrix}\psi & -1\\ -\varphi & 1\end{pmatrix}}{\psi-\varphi}$
Then $M^n=(PDP^{-1})^n=PD^nP^{-1}$ and $D^n=\begin{pmatrix}\varphi & 0\\ 0 & \psi\end{pmatrix}^n=\begin{pmatrix}\varphi^n & 0\\ 0 & \psi^n\end{pmatrix}$
So $\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}=V_n=PD^nP^{-1}V_0=\cfrac{1}{\psi-\varphi}\begin{pmatrix}1 & 1\\ \varphi & \psi\end{pmatrix}\begin{pmatrix}\varphi^n & 0\\ 0 & \psi^n\end{pmatrix}\begin{pmatrix}\psi & -1\\ -\varphi & 1\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}=\cfrac{1}{\psi-\varphi}\begin{pmatrix}1 & 1\\ \varphi & \psi\end{pmatrix}\begin{pmatrix}\varphi^n & 0\\ 0 & \psi^n\end{pmatrix}\begin{pmatrix}-1\\1\end{pmatrix}=\cfrac{1}{\psi-\varphi}\begin{pmatrix}1 & 1\\ \varphi & \psi\end{pmatrix}\begin{pmatrix}-\varphi^n\\\psi^n\end{pmatrix}=\cfrac{1}{\psi-\varphi}\begin{pmatrix}1 & 1\\ \varphi & \psi\end{pmatrix}\begin{pmatrix}-\varphi^n\\\psi^n\end{pmatrix}=\cfrac{1}{\psi-\varphi}\begin{pmatrix}-\varphi^n+\psi^n\\-\varphi^{n+1}+\psi^{n+1}\end{pmatrix}=\begin{pmatrix}\cfrac{\psi^n-\varphi^n}{\psi-\varphi}\\ \cfrac{\psi^{n+1}-\varphi^{n+1}}{\psi-\varphi}\end{pmatrix}$
From which we can deduce $F_n = \cfrac{\psi^n-\varphi^n}{\psi-\varphi}$
It's more often written as $\cfrac{\varphi^n-\psi^n}{\varphi-\psi}$ because $\varphi-\psi=\sqrt{5}$ whereas $\psi-\varphi=-\sqrt{5}$.
And we finally get $\boxed{F_n = \cfrac{\varphi^n-\psi^n}{\sqrt{5}}}$
