MathDB

In the diagram,

ABCDEF

is a regular hexagon with side length 2. Points

E

and

F

are on the

x

axis and points

A

B

C

, and

D

lie on a parabola. What is the distance between the two

x

intercepts of the parabola?
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.3215445204635294, xmax = 7.383669550094284, ymin = -4.983460515387094, ymax = 6.688676116382409; pen zzttqq = rgb(0.6,0.2,0); pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);
draw((2,0)--(4,0)--(5,1.7320508075688774)--(4,3.4641016151377553)--(2,3.4641016151377557)--(1,1.732050807568879)--cycle, linewidth(1));
Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true);
draw((2,0)--(4,0), linewidth(1)); draw((4,0)--(5,1.7320508075688774), linewidth(1)); draw((5,1.7320508075688774)--(4,3.4641016151377553), linewidth(1)); draw((4,3.4641016151377553)--(2,3.4641016151377557), linewidth(1)); draw((2,3.4641016151377557)--(1,1.732050807568879), linewidth(1)); draw((1,1.732050807568879)--(2,0), linewidth(1)); real f1 (real x) {return -0.58*x^(2)+3.46*x-1.15;} draw(graph(f1,-3.3115445204635297,7.373669550094284), linewidth(1)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /*yes i used geogebra fight me*/ [/asy]

A5

Problems(1)