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Question

If z is a complex number and ¯¯¯z is it's conjugate, then the number of solution(s) of z3+¯z=0 is

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Solution

Let z=x+iy, x,yR, then
z3+¯¯¯z=0(x+iy)3+(xiy)=0x3+(iy)3+3ixy(x+iy)+xiy=0(x3+x3xy2)i(y33x2y+y)=0x3+x3xy2=0, y33x2y+y=0
So, solving
x3+x3xy2=0(1)x=0 or x23y2+1=0y33x2y+y=0(2)

When x=0, using equation (2), we get
y(y2+1)=0y=0(yR)

When x23y2+1=0 using equation (2), we get
y33x2y+y=0y(y23x2+1)=0y(y2+13(3y21))=0y(8y2+4)=0y=0, y=±12
But when y=0
x2+1=0, we don't get real solutions, so
y=±12x=±12

Therefore,
(x,y)=(0,0), (12,12), (12,12), (12,12), (12,12)

Hence, the number of solution is 5.

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