Proving symmetric matrix has positive eigenvalues

Question

I'm trying to solve the following problem

Let $A$ be $n \times n$ symmetric matrix such that $A^2-5A+6Id_n = 0$

where $Id_n $ stands for the identity matrix.

Show that $A$ has only positive eigenvalues.

I know what eigenvalues are, what a symmetric matrix is, and everything else in the problem. I'm just pretty confused on how to start this proof/what it would look like. Any help would be great!

Answer 1

Guide:

Let $v$ be an eigenvector,

$$A^2v-5Av + 6v = 0$$

Hence

$$(\lambda^2-5\lambda+6)v=0$$

Can you solve for $\lambda$?