Find all the values of z (complex number) that satisfy the equality:? #z^2+|z|=0#

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Answer 1 · 2018-05-14T12:21:11

# z=0, +-i #

We seek a solution of:

# z^2 + |z| = 0# where #z in CC#

Let us explicitly denote the real and imaginary components of #z# by #z=x+iy# (where #x,yin RR#) so that:

# z^2 = (x + iy)^2 #
# \ \ \ = x^2 + 2xyi + (iy)^2 #
# \ \ \ = x^2 - y^2 + 2xyi #

And:

# |z| = sqrt(x^2+y^2) #

So we can write the equation as:

# z^2 + |z| = 0 iff x^2 - y^2 + 2xyi + sqrt(x^2+y^2) #

Equating real and imaginary components we have:

# Re: \ x^2 - y^2 + sqrt(x^2+y^2) = 0 => x=0, y=+-1#
# Im: \ 2xy = 0 => x,y=0#

Thus we gain three solutions:

# z=0, z=+-i #