RagvaloD wrote: The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this square trinomial has? It’s obvious that the graphs of $f(x)=ax^2+bx+c$ and $y=2ax+b$ must be tangent, meaning that the equation $ax^2+(b-2a)x+c-b=0$ has only one solution. So $(b-2a)^2-4a(c-b)=0 \Longrightarrow 4ac=b^2+4a^2$. The equation $ ax^2+bx+c=0 $ has no solutions as $ \delta=-4a^2 <0$.

Graphs are not nessesary tangent. It can be, that graphs intersect in one point.

RagvaloD wrote: Graphs are not nessesary tangent. It can be, that graphs intersect in one point. if two graphs intersect at 2 points, the plane devided into 5 parts. if two graphs intersect at 1 points(means be tangent), the plane devided into 4 parts. if two graphs intersect at 0 points, the plane devided into 3 parts.