The four complex roots of
\(2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0\)
when plotted in the complex plane, form a rhombus. Find the area of the rhombus.
Can I have a hint?
The area of the rhombus is 2*sqrt(3).
I believe that each pair of complex roots are conjugates of each other, no?
But think about it, if they are conjugates, they are mirrors across the x axis. If the distance between two roots conjuagates of each other are different from the other pair of roots which are conjugates, then this would form a trapezoid. Could you see what I mean? Can you continue from here?
What if some fo the roots are real, however? If the roots are real, what does z have to be? Is there even a root that is real? I have to go real soon, so Ill try to post the solution when I am back.