Square root - primitive question The answer to my question might be obvious to you, but I have difficulty with it. 
Which equations are correct:
$\sqrt{9} = 3$
$\sqrt{9} = \pm3$
$\sqrt{x^2} = |x|$
$\sqrt{x^2} = \pm x$
I'm confused. When it's right to take an absolute value? When do we have only one value and why? When two and why? 
Thank you very much in advance for your help!
 A: The confusion about the sign is understandable. The square root symbol applied to a positive number always yields a positive number (disregarding the case of zero for the sake of simplicity here). The problem arises when you don't know ahead of time whether $x$ is positive or not. It is true that one of the numbers $x$ and $-x$ must be positive, though. So you can write with certainty that
$$\sqrt {x^2}=|x|$$
since $|x|$ is precisely the one of these two numbers that is positive--it's just another way to say the same thing more concisely.
It is also true that "either $\sqrt{x^2}=x$ or $\sqrt{x^2}=-x$" is true, which is often abbreviated as "$\sqrt{x^2}=\pm x$". But be very careful what this says. It is a disjunction, a compound statement that at least one of the two component statements must be true. It does not say that both must be true. So it is also correct to write
$$\sqrt{x^2}=\pm x$$
if you understand that it means "or" but not necessarily "and".
So to answer your question: they are all correct.
A: By definition the square root of a number is the positive number whose square is the original number. So we have $\sqrt9=3$ and $\sqrt{x^2}=|x|$ and no doubt about either.
There is no number whose square root is $-3$ (even if we move to complex numbers and consider principal square roots).
What can create confusion is that we sometimes have an equation such as
$$ x^2 = 9 $$
and say something like "now let's take the square root on both sides" to get
$$ x = \pm 3 $$
which can look like we're saying taking the square root of $9$ gives $\pm 3$. But what really happens is that the square roots give us
$$ |x| = 3 $$
and then there's an implicit invisible step that replaces the absolute value sign with a $\pm$ to get $x=\pm 3$ instead.
A: The mathematical symbol 
√
refers to positive number of the two possible square roots.
If the question is written as "What is the square root of 9?", then the answer is both 3 and -3. 
However, if the question is "Evaluate √9," the answer would only be 3. Consequentially, -√9  = -3
A: In the real numbers, $\sqrt x$ is defined to be positive.
In the complex numbers, $\sqrt z$ is a multivalued function that indeed yields 2 values. In that case we have a principal value of $\sqrt 9$ that is $3$.
