Triangle Inequality and Pseudometric

[fabbi007]

Homework Statement

<br /> 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}<br />

where a&gt;0, b&gt;0, c&gt;0 and 4ac-b^2&lt;0

Show whether d(x,y) exhibits Triangle inequality?

Homework Equations

(M4) 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.

d^2(x,y); [d(x,z)+d(z,y)]^2. separately

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?

 

[fabbi007]

Any help is greatly appreciated!