Solve the equation \((x + 1)^3 = x^3\).
Is it true that, if \(b>a+c>0\), then the quadratic equation \(ax^2 +bx+c=0\) has two roots?
In a numerical set of \(n\) numbers, one of the numbers is 0 and another is 1.
a) What is the smallest possible variance of such a set of numbers?
b) What should be the set of numbers for this?
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.