Why is the answer to a square root problem always positive and negative?
SAMPLE:
sqrt{16} = -4 and +4
Why two answers?
They both lead to the same answer when squared...
$\displaystyle (-4)^2 = 16$
$\displaystyle 4^2 = 16$
$\displaystyle \implies \sqrt{16} = \pm 4$
Visually it may appear clearer, consider the quadratic graph of $\displaystyle y = x^2$ (Shown Below), for every $\displaystyle \pm x$ value, it is mapped onto the same $\displaystyle y$ value.
I agree that there are two square roots of any positive real number.
But, I disagree completely that $\displaystyle \sqrt {16} = \pm 4$. That is simply an abuse of notation!
This is a standard discussion in any elementary mathematics course.
The two square roots of 16 are: $\displaystyle \sqrt {16} = 4\,\& \, - \sqrt {16} = - 4$.
Therefore, $\displaystyle \sqrt {x^2 } = \left| x \right|$.
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 !