The absolute value for x2

Answer 1

As long as the range of values for x stays within the set of real numbers, then:

|x2| = x2

As the square of a real number will always be positive anyway.

If on the other hand, x belongs to the set of imaginary numbers, then:

|x2| = -x2

The reason for that is that in the case of imaginary numbers, x2 will give you a negative number. Its absolute value then would be the negative of that.

And with complex numbers it's, well... complex:

(ai + b)2 = -a2 + 2abi + b2

So:

|(ai + b)2| = a2 + 2abi + b2