Q(X) = $x_1^2 + x_2^2 + 2x_3^2 - 2x_1x_3 -2x_2x_3$
$∀x=(x_1,x_2,x_3)^T ∈ R^3$
Actually, I try to transform this equation to $(x1 - x3)^2 + (x2 - x3)^2$
then I think I should prove it that it is positive definite from my understanding.
For that, I am not sure how to solve this
$$Q(x,y,z)=x^2+y^2+2z^2-2xz-2yz$$
Write down the above in matrix form:
$$Q(x,y,z)=(x,y,z)\begin{pmatrix}1&0&-1\\0&1&-1\\-1&-1&2\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}$$
and now check the above matrix: is it positive/negative (semi) definite, or none? This will mean your quadratic form is the same (further hint: the matrix is singular...)
Yet amd already hinted strongly about the solution to your question, in case you don't want to work with matrices.