<s>[INST] In simplifying the formula that I've derived for finding the square root of a complex number to the standard formula. So by easy means, I derived 
$\sqrt{a+ib} = \sqrt{\frac{a(a+1)+b^2}{2}}+i(\frac{b}{\sqrt{2}\sqrt{a(a+1)+b^2}})$
But then I checked for the actual formula it is this;
$\sqrt{a+ib} = \sqrt{\frac{\sqrt{a^2 + b^2}+a}{2}}± i(\sqrt{\frac{\sqrt{a^2 + b^2}-a}{2}})$
So how do you simplify that to this?
Btw here is the derivation;

Let $\sqrt{a+ib} = x+iy$
$a+ib=x^2 - y^2 + i2xy$
We know, $(x^2-y^2)^2 + (2xy)^2 = x^2 + y^2$
or, $x^2 + y^2 = a^2 + b^2$
Hence we get $x = \sqrt{\frac{a+a^2+b^2}{2}}$ and
  $y=\frac{b}{\sqrt{2}\sqrt{a^2+a+b^2}}$
Therefore;
$\sqrt{a+ib} =
> \sqrt{\frac{a(a+1)+b^2}{2}}+i(\frac{b}{\sqrt{2}\sqrt{a(a+1)+b^2}})$

 [/INST] You made a mistake in your derivation.  You should have $(x^2+y^2)^\color{red}2=a^2+b^2$.  
This follows from $a=x^2-y^2$ and $b=2xy$ (or from known properties of complex modulus).
Thus, $a=x^2-\dfrac {b^2}{4x^2}$;
solving this quadratic equation in $x^2$ yields $x^2=\dfrac{a+\sqrt{a^2+b^2}}2$ as the correct answer.
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