I think the idea is that, while a and b are uniquely determined, given any ol' quadratic, there's no way to tell how much of the coefficient of x^2 should be c, and how much should be d. Example:
quadratic is x^2+x-6. Obviously, a = -6, b = 1/2. But with c and d, I could have c=1, d=0, or c=0, d=1, or any other of an infinite number of possibilities.
To put it on a more rigorous footing, you could always write the RHS in terms of a standard basis for P2.