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Number of solutions of the equation z^3+...

Number of solutions of the equation `z^3+[3(barz)^2]/|z|=0` where z is a complex number is

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`Z^3+(3(overlineZ)^2)/|Z|=0`
`r^3e^(i3theta)+(3r^2e^(i(-2theta)))/r=0`
`r^3e^(i3theta)+3re^(i(-2theta)=0`
`r/e^(i2theta)(r^2e^(i5theta)+3)=0`
`r!=0`
`|r^2e^(i5theta)|=|-3|`
`r^2=3`
`r=sqrt3`
...
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