How to represent the given quadratic equation into the tensor notation and find the solutions?

Question

I have derived an equation which exists as $$p^2+{2 p \cos\alpha} \, {\chi} +\chi^2=\left( \frac{v^2} r\right)^2$$ which I hope could be transformed analogously into the form of function $F(x,y)=ax^2+2bxy+cy^2$ where, $(x,y)$ implies $(p,\chi)$. But how to represent it into tensor notation and solve for $p$ and $\chi$?