Find a general formula for $x_k$ Suppose that the sequence $(x_k)$ is defined by $x_0 = 0, x_1 = 6, x_2 = 1$ and $$x_{k+3} = −x_{k+2}+17x_{k+1}−15x_k\quad \text{for }\, k\geq0.$$ Find a general formula for $x_k$. 
I have this answer to this question but I have no idea how to do it. Can anyone show me without omitting any steps? Thanks!

$U_k=\begin{bmatrix} x_{k+2} \\ x_{k+1} \\ x_k  \end{bmatrix}$
then
$$U_{k+1}=\begin{bmatrix} -1 & 17 & -15 \\ 1 & 0 & 0 \\ 0 &1 &0\end{bmatrix}U_k=AU_k$$
so
$$U_k=A^kU_0$$
characteristic polynomial of $A$
$$\lambda^3+\lambda^2-17\lambda+15$$
has three roots $1,3,-5$,  $A$ is diagonalizable.
so 
$$x_n=a+b3^n+c(-5)^n$$
$x_0 = 0, x_1 = 6, x_2 = 1$, so
$$a+b+c=0,a+3b-5c=6,a+9b+25c=1$$
