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Question
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
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Solution
Let \[z = x + iy\].
Then,
\[\left| z \right| = \sqrt{x^2 + y^2}\]
\[\therefore z^2 + \left| z \right|^2 = 0\]
\[ \Rightarrow \left( x + iy \right)^2 + \left( \sqrt{x^2 + y^2} \right)^2 = 0\]
\[ \Rightarrow x^2 + i^2 y^2 + 2ixy + x^2 + y^2 = 0\]
\[ \Rightarrow x^2 - y^2 + 2ixy + x^2 + y^2 = 0\]
\[ \Rightarrow 2 x^2 + 2ixy = 0\]
\[ \Rightarrow 2x(x + iy) = 0\]
\[ \Rightarrow x = 0 \text { or } x + iy = 0\]
\[ \Rightarrow x = 0 \text { or } z = 0\]
For
\[x = 0, z = 0 + iy\]
Thus, there are infinitely many solutions of the form
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