Shortcut Trick To Find The Square Root Of A Complex Number

I think most of us find it pretty difficult and tiresome to find the square root of a complex number, and in this blog post, I am gonna tell you a direct formula I made, to solve it very quickly, which helps a lot during competitive exams.

Given a Complex Number $z=a\pm ib$

The Square root will be given by the formula:

$\sqrt{z}=\sqrt{(\left | z \right |+ Re(z)))/2} \pm \sqrt{(\left | z \right |- Re(z)))/2})$

I know it looks very complex, but it becomes very easy to use after some practice.

Let's do some sample problems.

$z=3+4i$
We can easily calculate,

$Re(z)=3, \left | z \right |=\sqrt{3^{2}+4^{2}} => 5$

Now, let's put into our formula,

$\sqrt{z} = \sqrt{(5+3)/2} + i\sqrt{(5-3)/2},$

Isn't it easy to solve?

We get our answer as,

$\sqrt{z} = \pm(2+i)$

Let's prove the following formula.
(I tried using latex to write it, but it became too long and frustrating, so I decided to upload a pic clicked from my notebook. P.S, I am sorry for my bad handwriting)

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Shortcut Trick To Find The Square Root Of A Complex Number

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