Find all values of the real number \(a\) so that the four complex roots of \(z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\) form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas.
Thanks Rom but how did you find it ?
I coded up a Mathematica sheet that plotted the 4 roots on the complex plane as I adjusted the parameter a.
Having seen it was right about 3 I then checked the roots.
Unless there's some trick this is an absurdly difficult problem w/o software.