The number of solutions of the equation z 2 + ¯ z = 0 is Options: A . &nbsp 1 B . &nbsp 2 C . &nbsp 3 D . &nbsp 4

Answer: Option D : D Let z = x+iy, so that ¯ z = x - iy, therefore z 2 + ¯ z = 0 ⇔ ( x 2 − y 2 + x ) + i ( 2 x y − y ) = 0 Equating real and imaginary parts , we get x 2 − y 2 + x = 0 .......(i) And 2xy - y = 0 ⇒ y = 0 or x = 1 2 if y = 0 , then (i) gives x 2 + x = 0 ⇒ x = 0 or x = -1 If x = 1 2 , Then x 2 − y 2 + x = 0 ⇒ y 2 = 1 4 + 1 2 = 3 4 ⇒ y = ± √ 3 2 Hence, there are four solutions in all.

The number of solutions of the equation z2 + ¯z = 0 is

Discussion Forum : Complex Numbers

Question -

The number of solutions of the equation $z^{2}$ + $¯ z$ = 0 is

Options:

A . 1

B . 2

C . 3

D . 4

Answer: Option D
:
D

Let z = x+iy, so that $¯ z$ = x - iy, therefore

$z^{2} + ¯ z = 0 \Leftrightarrow (x^{2} - y^{2} + x) + i (2 x y - y)$ = 0

Equating real and imaginary parts , we get

$x^{2} - y^{2} + x$ = 0 .......(i)

And 2xy - y = 0 ⇒ y = 0 or x = $\frac{1}{2}$

if y = 0 , then (i) gives $x^{2}$ + x = 0 ⇒ x = 0 or

x = -1

If x = $\frac{1}{2}$ ,

Then $x^{2} - y^{2} + x = 0 \Rightarrow y^{2} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4} \Rightarrow y = \pm \frac{\sqrt{3}}{2}$

Hence, there are four solutions in all.

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