Question

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

Short answer

The fact that a quadratic polynomial with real number coefficients has at most two real roots can be determined.

Step-by-step solution

Stating quadratic polynomial

The quadratic polynomial with real coefficient can be \(a{x^2} + bx + c = 0\,where\,a \ne 0.\)There are quadratic polynomial with real coefficient has two real roots.

Finding required expression

The statement can be denoted as,

\(\begin{aligned}{}\forall a\forall b\forall c\exists x\\\exists y\{ (a{x^2} + bx + c &= 0) \cap (a{y^2} + by + c = 0) \cap (\forall z((x \ne z) \cap (y \ne z)) \to (a{z^2} + bz + c \ne 0))\} \end{aligned}\)

The domain for a is R-{0} and domain is R for b, c, x, y and z.