Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why.

Question

I looked up the solution guide and found out: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49+4$ $=53$

Why the unknown "$i$" just disappeared$?$ I supposed it might be: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49-4i$ does it?

May someone tell me which one is right and tell me the reasons? Thank you so much.

Answer 1

We have $(7-2i)(7+2i)=49+14i-14i-4i^2=49-4i^2.$ Note that $i=\sqrt{-1}\implies i^2=(\sqrt{-1})^2=-1.$ Therefore, $$(7-2i)(7+2i)=49-4i^2=49-4(-1)=49+4=53$$

Answer 2

What you did wrong is that $(2i)^2$ is not $4i$, but rather $4i^2$, which is equivalent to $-4$.