10.1. The quadratic trinomial $y=a x^{2}+b x+c$ has no roots and $a+b+c>0$. Determine the sign of the coefficient $c$.
Answer: $c>0$.

Solution. The quadratic polynomial has no roots, which means its graph does not intersect the x-axis. Since $y(1)=a+b+c>0$, the graph is located in the upper half-plane (see figure), hence $c=y(0)>0$.

![](https://cdn.mathpix.com/cropped/2024_05_06_93d92577b59f6996e8edg-1.jpg?height=208&width=163&top_left_y=2089&top_left_x=1021)