solving equation (b-c)x² + (c-a)x + (a-b) = 0 folling roots are obtained
Just use the quadratic formula:
x = (-(c-a)±√((c-a)^2 - 4(b-c)(a-b)))/(2(b-c))
You can play around with that some, but I don't see much in the way of simplification.
http://www.wolframalpha.com/input/?i=%28b-c%29x%C2%B2+%2B+%28c-a%29x+%2B+%28a-b%29