## Task B-2.3.

The equation $y=x^{2}+b x+c$ defines a set of parabolas whose roots are consecutive integers. To which set of points do the vertices of these parabolas belong? Determine its equation.
## Solution.

Let $x_{1}=n$ and $x_{2}=n+1, n \in \mathbb{Z}$ be the roots of the parabola from the given set. Then,

$$
y=(x-n) \cdot(x-(n+1))=x^{2}-(2 n+1) x+n^{2}+n
$$

We calculate the coordinates of the vertex:

$$
x_{T}=\frac{2 n+1}{2}, y_{T}=\frac{4 \cdot\left(n^{2}+n\right)-\left(4 n^{2}+4 n+1\right)}{4}=\frac{-1}{4}
$$

Thus, the vertices of the parabolas from the given set have the same ordinate, so the vertices of all such parabolas lie on the line $y=\frac{-1}{4}$.