The number of solutions of $z^{3} + z + ¯ z = 0$ is

3

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4

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5

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6

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Solution

The correct option is C $5$
$z^{3} + z + ¯ z = 0$
$\Rightarrow {(x + i y)}^{3} + 2 x = 0$
$\Rightarrow (x^{3} + 2 x - 3 x y^{2}) - i (y^{3} - 3 x^{2} y) = 0$
$x^{3} + 2 x - 3 x y^{2} = 0$
$y^{3} - 3 x^{2} y = 0 \Rightarrow y = 0 o r y^{2} - 3 x^{2} = 0$
case-I: if y = 0 ; $\Rightarrow$ x = 0 z = 0
case-II: if $y^{2} = 3 x^{2}$
$\Rightarrow x^{3} + 2 x - 3 x (3 x^{2}) = 0$
$\Rightarrow, 8 x^{3} = 2 x$
$\Rightarrow x = 0$ or $x^{2} = \frac{1}{4}$ $\Rightarrow x = \pm \frac{1}{2}$ &
$y = \pm \frac{\sqrt{3}}{2}$
so , $Z = 0, \frac{1}{2} + i \frac{\sqrt{3}}{2}, - \frac{1}{2} + \frac{i \sqrt{3}}{2}, \frac{1}{2} - i \frac{\sqrt{3}}{2}, - \frac{1}{2} - i \frac{\sqrt{3}}{2}$

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