# How do I solve this equation :$\bar{z}-iz²=-\sqrt{3}-3i$ without using identity way?

let $z$ be a complex number and $\bar{z}$ it's conjugate ,i would like

to solve this equation :$$\bar{z}-iz²=-\sqrt{3}-3i$$ without using identity way ?

Note :by identity way the solution is clear and is: $z= -\sqrt{3}$

Thank you for any help

• What is the "identity way"? Mar 20, 2016 at 13:49
• I meant by identity way The real part of equality of RHS is the same the real part of the LHS ,the same with imaginary part Mar 20, 2016 at 13:51
• You could do the same by using the absolute value and the argument. Mar 20, 2016 at 13:53

Hint you can do it by arguments . now argument if RHS is $2\pi-\tan^{-1}(\sqrt{3})=2\pi-\pi/3$ so argument of lhs needs to be the same so $\tan^{-1}(z^2/\bar{z})=2\pi-\pi/3$ but $z^2=z \times \bar{z}$ can you continue from here?