Question

Find all complex numbers z satisfying:

Answer 1

\[\small (\frac{2z+1+\sqrt{3}}{2z+1-\sqrt{3}})^3+(\frac{2z+1+\sqrt{3}}{2z+1-\sqrt{3}})^2+(\frac{2z+1+\sqrt{3}}{2z+1-\sqrt{3}}+1=0?\]

Answer 2

You have
\[ w^3 + w^2 + w + 1 = 0 \]
where w is that scary looking expression.

Now do you know what solutions are to the equation above in w?

Answer 3

Ah yes, stupid of me not to see that. can get (w^2 +1)(w+1) so roots are -1, +i or -i of that. but then how do i sub z back in?

Answer 4

Right. Now solve the equation
w = stuff/stuff
for z. You'll see it simplifies.

Answer 5

thanks very much! getting on that now