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?

SupersonicMan12 Jul 19, 2022

#2**0 **

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.

Voldemort Jul 19, 2022

#3**0 **

@You-Know-Who, i don't think the conjugate root theorem applies here, since the coefficients are complex too.

SupersonicMan12
Jul 20, 2022