Problem #PRU-65726

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.