Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots.

There is really no need for derivatives. Here is the answer:

∀a, b, c ∈ ℝ,

[[∃x, y, z ∈ ℝ, ax^2 bx c=ay^2 by c=az^2 bz c=0] ⇒ [ (a-b) (b-c) (c-a) = 0]].

Note: " (a-b) (b-c) (c-a) = 0" is simply the same to saying "[a=b] ∨ [b=c] ∨ [c=a]", but a few keystrokes shorter.