Why does $|x|^2$ not equal $x^2$

Question

A friend told me that the two are unequal. I had a though that maybe one has two solutions on the same point, but from what I can tell that's not the case.

Edit: LaTeX is hard :).

Answer 1

Consider the case when $x$ is a complex number.

Let's express $x$ in the form $x=a+b\mathrm{i}$, where $a\ne 0$ and $b\ne 0$. Then, we have $|x|^2=a^2+b^2$, but $x^2=a^2-b^2+2ab\mathrm{i}\ne a^2+b^2$. This demonstrates that $|x|^2$ and $x^2$ are not equal.

Answer 2

The fundamental problem of your question is that you haven't defined what $x$ is. If $x$ is a real number, $|x|^2$ and $x^2$ are equal.