Solve the equation \((x + 1)^3 = x^3\).
The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) intersect at two points, bounding a certain figure.
Find the equation of the vertical line dividing the area of this figure in half.
The quadratic trinomials \(f (x)\) and \(g (x)\) are such that \(f' (x) g' (x) \geq | f (x) | + | g (x) |\) for all real \(x\). Prove that the product \(f (x) g (x)\) is equal to the square of some trinomial.
Is it true that, if \(b>a+c>0\), then the quadratic equation \(ax^2 +bx+c=0\) has two roots?
One of the roots of the equation \(x^2 + ax + b = 0\) is \(1 + \sqrt 3\). Find \(a\) and \(b\) if you know that they are rational.
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.
The equations \[ax^2 + bx + c = 0 \tag{1}\] and \[- ax^2 + bx + c \tag{2}\] are given. Prove that if \(x_1\) and \(x_2\) are, respectively, any roots of the equations (1) and (2), then there is a root \(x_3\) of the equation \(\frac 12 ax^2 + bx + c\) such that either \(x_1 \leq x_3 \leq x_2\) or \(x_1 \geq x_3 \geq x_2\).
The expression \(ax^2+bx+c\) is an exact fourth power for all integers \(x\). Prove that \(a=b=0\).