The problem statement, all variables and given/known data

$\displaystyle d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2

$

where a>0, b>0, c>0 and $\displaystyle 4ac-b^2<0$

Show whether exhibits Triangle inequality?

Relevant equations:

(M4) $\displaystyle d(x,y) \leq d(x,z)+d(z,y) $ (for all x,y and z in X)

The attempt at a solution

I started my solution by solving by squaring the both sides of the equation separately.

$\displaystyle d^2(x,y); [d(x,z)+d(z,y)]^2$

I am tending to think it does not satisfy the triangle inequality any other simple way to prove it? Also is this a pseudometric? if it does not satisfy the triangle inequality?