Here is the solution which contents quantifiers and logical connectives that express the fact that a quadratic polynomial with real number coefficients has at most two real roots and I am pretty sure you will find it helpful :
∀a∀b∀c[∃x ∃y ((ax2+ bx+c=0 )∨ (ay2+ by+c=0)) ∧ (∀w(aw2+ bw+c=0) →→ (w=x ∨ w=y)]
Hope you will agree with me.
Regards!
