Sorry - I don't see how you got this equation: (2q)^2-4r, which you simplified to 4(q^2-r). Could you explain further?
Solution of a quadratic equation ax^2 + bx + c is
X = (-b +/- sqrt(b^2 â€“ 4ac)
In this case, a=1 b=2q and c=r
Substituting into the bold (2q)^2 - 4(1*r) = 4q^2 - 4r = 4(q^2 - r)
If you can determine this value of (2q)^2 - 4(1*r) then you know how many intercepts there are - I think.
Tennisball - care to elaborate?