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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 $x_{1} \geq x_{3} \geq x_{2}$ .

Problem #PRU-86508

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other) Algebra Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–14 2.0

Solving the problem: “What is the solution of the expression $x^{2000} + x^{1999} + x^{1998} + 1000 x^{1000} + 1000 x^{999} + 1000 x^{998} + 2000 x^{3} + 2000 x^{2} + 2000 x + 3000$ ( $x$ is a real number) if $x^{2} + x + 1 = 0$ ?”, Vasya got the answer of 3000. Is Vasya right?