Square Root of Complex Number

To find $\sqrt{a+bi}$, assume the answer has the form $x + yi$ and square both sides:

$(x+yi)^2 = x^2 - y^2 + 2xyi = a + bi$

Matching real and imaginary parts gives $x^2 - y^2 = a$ and $2xy = b$. Combined with the modulus relation $x^2 + y^2 = \sqrt{a^2+b^2}$, you can solve for $x^2$ and $y^2$ directly, then pick the signs of $x$ and $y$ so that $2xy$ matches the sign of $b$.

Example: $\sqrt{3+4i}$. Here $a=3, b=4$, modulus $=5$. Then $x^2 = \tfrac{5+3}{2} = 4$ and $y^2 = \tfrac{5-3}{2} = 1$, so $x=2, y=1$ (same sign, since $b>0$). Check: $(2+i)^2 = 4+4i-1 = 3+4i$. ✓