I'm learning about solving complex quadratic equations, in the examle I'm following, we start with:
This can be rewritten as:
The square can be rewritten as $w^2$ so:
Any complex number can be written on the form $a+bi $ so:
From this we learn that $2ab=4$ and that $a^2-b^2=3$ provided that $a$ and $b$ are real numbers. I'm prepared to accept all of this, but the last part of the solution really bothers me.
According to my book the following applies:
I can accept the first part, but why on earth would $|w|^2=|w^2|$?
This would mean that $a^2+b^2=a^2+2abi-b^2$, and why on earth would this be the case?
Also, why would $|w^2|=|3+4i|$?
If $|w^2|=|w|^2$ then by all rights, $|w^2|$ should equal $|3+4i|^2$?