Complex Numbers

Given the equation $z^3+\overline{z}=0$, where $z=x+iy$ is a complex number, we can find the distinct solutions as follows:Rewrite the Equation:Start with:$z^3+\overline{z}=0$This implies:$z^3=-\overline{z}$Equate Magnitudes:Taking the magnitude of both sides, we have:$|z^3|=|-\overline{z}|$This simplifies to:$|z|^3=|\overline{z}|$Since $|\overline{z}|=|z|$, the equation becomes:$|z|^3=|z|$Solve for Magnitude:This equation leads to: $|z|(|z|^2-1)=0$Therefore, either $|z|=0$ or $|z|^2=1$.Consider Each Case:If $|z|=0$, then $z=0$. This gives one distinct solution: $z=0$.If $|z|^2=1$, then $|z|=1$ and consequently $z\overline{z}=1$. This implies:$z=\frac{1}{z}$Substitute Back:Substituting $\overline{z}=\frac{1}{z}$ into the original equation:$z^3+\frac{1}{z}=0$Multiply through by $z$ to clear the fraction:$z^4+1=0$Solve $z^4+1=0$ :This equation has four roots:$z=e^{i(\pi/4)}, e^{i(3\pi/4)}, e^{i(5\pi/4)}, e^{i(7\pi/4)}$Count Distinct Solutions:Including $z=0$, we have: One solution for $z=0$Four solutions from $z^4+1=0$Therefore, the number of distinct solutions to the equation is 5 .