Simplifying $|a+b|^2 + |a-b|^2$

Question

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.

Answer 1

$|z|^2=zz'$ 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.