Problem #PRU-65726

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial

Problem

Author: A. Khrabrov

Do there exist integers \(a\) and \(b\) such that

a) the equation \(x^2 + ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + ax + b = 0\) does have roots?

b) the equation \(x^2 + 2ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + 2ax + b = 0\) does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.