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I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get $ |a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right) $ I'm trying to understand the steps involved in arriving at this result: $\begin{eqnarray*} |a+b|^2 + |a-b|^2 &=& |(a+b)^2| + |(a-b)^2| \\ &=& | a^2 + 2ab + b^2 | + | a^2 - 2ab + b^2 | \end{eqnarray*} $ But I'm at a loss as to how to continue from here; I find it hard to work symbolically with absolute values.

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    Oh, I did not notice that we work with complex numbers. (I should have read the post more thoroughly.) Since adding complex numbers is the same as adding vectors, this is basically the [Parallelogram law](http://en.wikipedia.org/wiki/Parallelogram_law).2012-08-08

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$|z|^2=zz'$ where $z'$ stands for the complex conjugate of $z$.

$|a+b|^2+|a-b|^2=(a+b)(a'+b')+(a-b)(a'-b')=aa'+ab'+ba'+bb'+aa'-ab'-ba'+bb'=2aa'+2bb'$ and you're pretty much there.