Maximum number of intersections of two quadratics

Hi there,

I am wondering what is the maximum number of points of intersection of 2 quadratics of the form:

$\displaystyle ax^2 + bxy + cy^2 + dx + ey + f = 0$

I think the answer is 4, because expressing y in terms of x has a $\displaystyle \pm$ sign, and then subbing each of these y-values in to the other quadratic leads to 2 solutions each through use of the quadratic formula. Thus 2 x 2 = 4 is the maximum number of intersections.

This also seems intuitive, if we imagine for example a hyperbola being intersected 4 times by an ellipse. I can't picture in my head any 2 quadratics who intersect more than 4 times.

Anyway if someone could please verify or disprove my assertion I'd be very grateful!