is $\sqrt{x}$ always positive? I recently saw someone say:

$\sqrt{x}$ is, by definition, positive

Which didn't sit easily with me as we always say $\pm$ for any square root.
Then it got me thinking that if, by definition, the symbol $\sqrt{x}$ meant the positive or the negative root of x, then the $\pm$ symbol would be redundant.
So, have mathematicians defined the root symbol only to include the positive root, thereby necessitating the $\pm$ symbol?
EDIT: This question is different from the one here: Square root confusion? because my question focuses principally on the meaning/convention of the square root symbol rather than on the topic of their being two solutions to $x^2=4$
 A: I will explain the difference by example. 
The equation $x^2 = 4$ has solutions $x = \pm 2$. However, if I were to just say $\sqrt{4}$, that would be $2$ and not $-2$.
Also, $\sqrt{x}$ is nonnegative by definition.
A: It's correct that if look at, say, $4$, then both $2$ and $-2$ are square roots of $4$, and in general, for any positive $x$, that $x$ has two square roots, where each is the negative of the other. But in the case of $4$, because of that second thing I mentioned, if we know $2$ is a square root of $4$, then we know immediately that $-2$ is as well.
We use $\sqrt{x}$ in two ways. Sometimes, in the sense that you mentioned, we want to use $\sqrt{x}$ to refer to the family of all numbers, let's call them $t$, such that $t^{2} = x$. That is, we use it to refer to a collection of (two) numbers satisfying a particular condition.
However, we also note that if we have one root, then we can easily determine the other by taking the negative. So in a sense, we only actually need one of them to "communicate" the solution set. So we often choose a convenient "representative", and often pick the positive one. So here, we use $\sqrt{x}$ as a function that spits out the "seed" of the solution set for $t^{2} = x$.
If the use of "by definition" bugs you, then replace it with "by convention". 
