## Task B-2.5.

A regular hexagon is inscribed in a parabola such that two of its vertices are on the $x$-axis, and the remaining four vertices are on the parabola, as shown in the figure. How far apart are the intersections of the parabola with the $x$-axis, if the side length of the regular hexagon is 2?

![](https://cdn.mathpix.com/cropped/2024_05_30_bcbf631b6726be7fa637g-12.jpg?height=339&width=466&top_left_y=1047&top_left_x=1252)
## First Solution.

The distance between points $x_{1}$ and $x_{2}$, or the difference $x_{2}-x_{1}$, will not change by moving the parabola horizontally. Therefore, without loss of generality, we can move the parabola along the $x$-axis so that the midpoint of the side of the hexagon that lies on the $x$-axis is at the origin of the coordinate system.

![](https://cdn.mathpix.com/cropped/2024_05_30_bcbf631b6726be7fa637g-12.jpg?height=810&width=1214&top_left_y=1748&top_left_x=341)

The new parabola has an equation of the form $y=a x^{2}+c$.

1 point

Let's determine the coordinates of two points on the parabola.

The abscissa of point $A$ is equal to the negative half of the side length of the regular hexagon, i.e., $x_{A}=-1$. The ordinate of point $A$ is equal to the negative length of the diagonal $\overline{A E}$, i.e.,

$$
y_{A}=-|\overline{A E}|=-2 \cdot \frac{2 \sqrt{3}}{2}=-2 \sqrt{3}, \quad \text { i.e., } \quad A(-1,-2 \sqrt{3}) \text {. }
$$

Similarly, we obtain the coordinates of point $F$.

$$
x_{F}=-2, y_{F}=-\frac{2 \sqrt{3}}{2}=-\sqrt{3}, \quad \text { i.e., } \quad F(-2,-\sqrt{3})
$$

Substituting the coordinates of points $A$ and $F$ into the equation of the parabola, we get the system of equations

$$
\begin{aligned}
-2 \sqrt{3} & =a+c \\
-\sqrt{3} & =4 a+c
\end{aligned}
$$

Its solution is $a=\frac{\sqrt{3}}{3}, c=-\frac{7 \sqrt{3}}{3}$, so the equation of the parabola is

$$
y=\frac{\sqrt{3}}{3} x^{2}-\frac{7 \sqrt{3}}{3}
$$

Then, from $\frac{\sqrt{3}}{3} x^{2}-\frac{7 \sqrt{3}}{3}=0$, we get $x_{1}=-\sqrt{7}, x_{2}=\sqrt{7}$

so the distance $x_{2}-x_{1}=2 \sqrt{7}$ is the desired distance.