2020 AMC 12A Problems/Problem 15
Problem
In the complex plane, let be the set of solutions to
and let
be the set of solutions to
What is the greatest distance between a point of
and a point of
Solution
Realize that will create an equilateral triangle on the complex plane with the first point at
and two other points with equal magnitude at
.
Also, realize that can be factored through grouping:
will create points at
and
Plotting the points and looking at the graph will make you realize that and
are the farthest apart and through Pythagorean Theorem, the answer is revealed to be
~lopkiloinm