Prove the equation has real roots.

(a) Discriminant
= (c - a)^2 - 4(b - c)(a - b)
= c^2 - 2ac + a^2 - 4(ab - ac - b^2 + bc)
= c^2 - 2ac + a^2 - 4ab + 4ac + 4b^2 - 4bc
= c^2 + 2ac + a^2 - 4ab - 4bc + 4b^2
= c^2 + ac + ac - 2bc + a^2 - 2ab - 2ab - 2bc - 2bc + 4b^2
= c^2 + ac - 2bc + ac + a^2 - 2ab - 2bc - 2ab + 4b^2
= c(c + a - 2b) + a(c + a - 2b) - 2b(c + a - 2b)
= (c + a - 2b)(c + a - 2b)
= (c + a - 2b)^2, which is non-negative.
Thus the roots are real.