Q.Find a square root of the complex number $3+4i$.

Write $\sqrt{3+4i}=x+iy$, square both sides, match real/imaginary parts, and use the modulus as a third equation to pin down $x,y$.

Let $\sqrt{3+4i}=x+iy$ where $x,y$ are real. Squaring,

$x^2-y^2+2ixy = 3+4i.$

Comparing real and imaginary parts:

$x^2-y^2=3 \qquad\text{and}\qquad 2xy=4 \Rightarrow xy=2.$

A third relation comes from equating moduli, $|x+iy|^2=|3+4i|$:

$x^2+y^2=\sqrt{3^2+4^2}=\sqrt{25}=5.$

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