How might I find linear combinations $$\begin{align*} A&=a_1x+a_2y+a_3z\\ B&=b_1x+b_2y+b_3z\\ C&=c_1x+c_2y+c_3z \end{align*}$$
Such that I can transform the two polynomials
$$2x^2+3y^2-2yz+3z^2\text{ and }x^2+6xy+3y^2+2yz-6zx+3z^2$$
into
$A^2+B^2+C^2$ and $\alpha A^2+\beta B^2+\gamma C^2$ respectively for some $\alpha, \beta,\gamma\in \mathbb R$?
I think I should be completing the square, but I can't see how to.