How do I get the square root of a complex number? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?

Here is a direct algebraic answer.
Suppose that $z=c+di$, and we want to find $\sqrt{z}=a+bi$ lying in the first two quadrants.  So what are $a$ and $b$?
Precisely we have
$$a=\sqrt{\frac{c+\sqrt{c^{2}+d^{2}}}{2}}$$ and
 $$b=\frac{d}{|d|}\sqrt{\frac{-c+\sqrt{c^{2}+d^{2}}}{2}}.$$  (The factor of $\frac{d}{|d|}$ is used so that $b$ has the same sign as $d$) To find this, we can use brute force and the quadratic formula.  Squaring, we would need to solve $$a^2-b^2 +2abi=c+di.$$  This gives two equations and two unknowns (separate into real and imaginary parts), which can then be solved by substitutions and the quadratic formula.
I hope that helps!
