discriminant of $ a|z_{1}|^2+b |z_{2}|^2+c|z_{3}|^2 +2d Re(z_{1} \bar{z_{2}}) +2e Re(z_{2} \bar{z_{3}}) +2f Re(z_{3} \bar{z_{1}}) \geq 0$

Question

I want to know about discriminant of $ a |z_{1}|^2+b |z_{2}|^2+c|z_{3}|^2 +2d Re(z_{1} \bar{z_{2}}) +2e Re(z_{2} \bar{z_{3}}) +2f Re(z_{3} \bar{z_{1}}) \geq 0 $ where $a,b,c,d,e,f \in \mathbf{C}$. I tried searching for it, in wikiepedia, I came to know this, ax² + 2bxy + cy² + 2dxz + 2eyz + fz² = 0 ----(1)

The discriminant of this quadratic form is determined by evaluating the following determinant: \begin{vmatrix} a & b & d \\ b & c & e \\ d & e & f \\ \end{vmatrix} Is it right to say that discriminant is negative if (1) takes only positive values like we do for quadratic in real coefficients. Please help me in this regard. Thanking you in advance.