We solve problems

Problem #PRU-111345

Problems Calculus Integral Applications of integration Area by integration Algebra Polynomials Quadratic polynomials Quadratic equations. Vieta's rule.

16–16 3.0

The numbers $p$ and $q$ are such that the parabolas $y = - 2 x^{2}$ and $y = x^{2} + p x + q$ intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.

Problem #PRU-111763

Problems Calculus Derivative Calculating a derivative Algebra Absolute value Inequalities containing absolute values Polynomials Quadratic polynomials

15–16 4.0

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.

Problem #PRU-65318

Problems Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial Mean values Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables) Probability and statistics Statistics

14–16 3.0

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?

Problem #PRU-77989

Problems Methods Examples and counterexamples. Constructive proofs Algebra Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–16 4.0

The equations $\begin{matrix} (1) & a x^{2} + b x + c = 0 \end{matrix}$ and $\begin{matrix} (2) & - a x^{2} + b x + c \end{matrix}$ 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 $\frac{1}{2} a x^{2} + b x + c$ such that either $x_{1} \leq x_{3} \leq x_{2}$ or $x_{1} \geq x_{3} \geq x_{2}$ .