<p>The number of solutions of the equation \( z^2 + |z|^2 = 0 \), where \( z \) is a complex number, is:</p>
Step-by-Step Solution
Key Concept: Write z = x + iy; z^2 + |z|^2 = (x^2-y^2+x^2+y^2) + 2xyi = 2x^2 + 2xyi = 0 \Rightarrow x=0 and y=0, giving infinite solutions (the imaginary axis? No: x=0 gives z=iy, then z^2=-y^2 and |z|^2=y^2, so z^2+|z|^2=0 ✓ for all real y). So infinitely many solutions.
<p>Let $z = x+iy$. $z^2 + |z|^2 = (x^2-y^2+2xyi)+(x^2+y^2) = 2x^2+2xyi = 0$. Real part: $2x^2=0 \Rightarrow x=0$. Imaginary part: $2xy=0$, satisfied for $x=0$. So $z = iy$ for any real $y$ — infinitely many solutions.</p>
Correct Answer: D