Reduce quadratic form to canonical form

Question

Reduce the following quadratic form $x_1^2 + x_3^2 + 2x_{1}x_{2} + 2x_2x_{3}$ to canonical form. Also is this quadratic form positive definite?

I am familiar with approach for Partial differential equation conversion to canonical form including two variables $x$ and $y$ as derivatives of $z$ but unsure of how to proceed with such cases with $3$ variables.

Answer 1

\begin{multline}q(x_1,x_2,x_3)=x_1^2 + x_3^2 + 2x_{1}x_{2} + 2x_2x_{3}=\\(x_1^2 +2x_{1}x_{2}+x_2^2)+ (x_3^2 + 2x_2x_{3}+x_2^2)-2x_2^2=(x_1+x_2)^2+(x_2+x_3)^2-2x_2^2\end{multline}

Of course it is not positive definite, because $q(1,-2,1)<0$.