11.1. The graphs of the functions $y=a x^{2}, y=b x$ and $y=c$ intersect at a point located above the x-axis. Determine how many roots the equation $a x^{2}+b x+c=0$ can have.
Answer: No roots.

Solution. From the condition of the problem, it follows that the graphs intersect at the point $(m ; c)$, where $c$ > 0. Then the equalities $b m=c$ and $a m^{2}=c$ hold, so $m \neq 0$. Therefore, the discriminant of the given equation $D=b^{2}-4 a c=\frac{c^{2}}{m^{2}}-\frac{4 c^{2}}{m^{2}}=-\frac{3 c^{2}}{m^{2}}<0$, which means that this equation has no roots.

Evaluation criteria.

"+" A complete and justified solution is provided

“士” A generally correct solution is provided, containing minor gaps or inaccuracies

“Ғ” The correct line of reasoning is provided, but a computational error is made in the final phase of the solution

“-" Only the answer is provided

"-" An incorrect solution is provided or it is missing