Geometry of discriminant

The square triceps X2 + PX + Q form a bic parametric family: each of them corresponds to the point of the plane with the coordinates (P, G) The discriminant condition P2 - 4Q = 0 can be considered as an equation of a curve that separates the points of this plane corresponding to polynomials with a different number of roots. Similar (but more complicated) separating sets are also available for equations of higher degrees, as well as for equations systems. Knowing their geometry is very useful for studying parameters equations and for solving many other problems. The text of the brochure is a record of a lecture read by the author on February 14, 2015 at a small mehmate of Moscow State University for schoolchildren in grades 9-11