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# Really really difficult problem

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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?

Jul 19, 2022

#1
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The area of the rhombus is 2*sqrt(3).

Jul 19, 2022
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Can you show us the steps?

SupersonicMan12  Jul 20, 2022
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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.

Jul 19, 2022
edited by Voldemort  Jul 19, 2022
edited by Voldemort  Jul 19, 2022
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@You-Know-Who, i don't think the conjugate root theorem applies here, since the coefficients are complex too.

SupersonicMan12  Jul 20, 2022