Question 125
We call two quadratic trinomials $P(x) = x^2 + a x + b$ and $Q(x) = x^2 + c x + d$ *friendly* if each of them has distinct real roots and $x_1 + x_3$, $x_2 + x_4$ are the roots of the quadratic trinomial $x^2 + (a + c)x + b + d$, where $x_1 \lt x_2$ are the roots of $P(x)$, $x_3 \lt x_4$ are the roots of $Q(x)$.
Let $M$ be the set of pairwise *friendly* trinomials consisting of at least three trinomials.
Prove that $0$ is a root of every trinomial from the set $M$.
(E. Barabanov)
Let $M$ be the set of pairwise *friendly* trinomials consisting of at least three trinomials.
Prove that $0$ is a root of every trinomial from the set $M$.
(E. Barabanov)