Because and
I agree that there are two square roots of any positive real number.
But, I disagree completely that . That is simply an abuse of notation!
This is a standard discussion in any elementary mathematics course.
The two square roots of 16 are: .
Therefore, .
Completely agree with that !
If you see the graph of the function y=sqrt(x), you will see that y can't be negative.
Actually, working with the graph y=x² is a mistake because sqrt(x) is not the inverse function of x² !!!!
If one has x²=16, then for sure x=+ or - sqrt(16) because x²=16 --> x²-16=0 --> x²-(sqrt(16))²=0 --> (x-sqrt(16))(x+sqrt(16))=0 and the rest follows.
Actually, I'd say that these two messages are contradictory. You state clearly that sqrt(x²)=|x|, which is true.
So since 16=(-4)²=4², sqrt(16)=|4|=|-4|=4 !