root of periodic continued fraction

Question

Let $\xi \in \mathbb{R}$ be a periodic continued fraction expansion with $\xi =[\overline{c_0,\dots,c_n}], c_0 \neq0 $ and $\frac{ k_i}{l_i}$ the approximate fraction with $1 \leqslant i \leqslant n$. Show $\frac{-1}{\xi'}$ with $\xi'=[\overline{c_n,\dots,c_0}]$ is polynomial root of $f(x)=l_nx^2+(l_{n-1}-k_n)x-k_{n-1}$. Use: $[{c_n,\dots,c_0}]= \frac{k_n}{k_{n-1}}$ and $[{c_n,\dots,c_1}]= \frac{l_n}{l_{n-1}}$.