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Find the number of solutions of z2 + |z|2 = 0.

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Given: 

⇒ z2+|z|2=0 

Let us assume z=x+iy 

⇒ (x+iy)2 + \(\sqrt{(x^2+y^2)^2}\) = 0

⇒ x2+(iy)2+2(x)(iy)+x2+y2=0 

⇒ 2x2+y2+i2y2+i2xy=0 

We know that i2=-1 

⇒ 2x2+y2-y2+i2xy=0 

⇒ 2x2+i2xy=0 

Equating Real and Imaginary parts on both sides we get, 

⇒ 2x2=0 and 2xy=0 

⇒ x=0 and y\(\varepsilon\)

∴ z=0+iy where y\(\varepsilon\)R. i.e, Infinite solutions.

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