The equation and centre/radius of the circle are random, in effect, they can be anything within a domain I specify.

I know I'm looking for 1 root, but surely solving this would give me that, I don't get why it's not possible to do it algebraic-ly, when all the letters represent are real numbers?

edit: When I come to use this, there will be real numbers in there, I need a case I can program in and run in a loop for a range of circle centres and a range of cubic co-efficients. So if I have a case with real numbers in it, for example the first set in my list - a=b=c=d=1, dc=0.9589, xp=0, yp=4. Substituting these in, and finding the roots of the 6th order eqn won't work