Something having to do with roots and i and stuff

Algebra Level 5

\[\large[(x-1)^2+4x][(x-1)^2+x][(x-1)^2+2x]=8x^3\]

Suppose \(a_i\) where \(i={{1,2,3,4,5,6}}\) are the 6 complex roots of the equation above.

And \(\arg(x_j)\leq \arg(x_k)\) for \(j\leq k\).

In which quadrant does \(x_1x_6\) lie in?

Details and Assumptions:

  • \(0\leq \arg(x_i)<2\pi\)

  • You may want to use a calculator at the end after you had found the exact values of the roots.


All credit for this problem goes to Sean Ty.
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