My classmates and I have been locked in a debate on whether this simplification is allowed (we are linearizing a continuous time system about an equilibrium solution at $x=0$).
I say it is not allowed because doing so changes the domain of the expression, but others say it should be ok because representing $x^2$ as $|x||x|$ is mathematically allowed.
The answer to this question determines whether the nonlinear system can be linearized about this solution, so we are very keen to find out which camp is correct!