Square roots -- positive and negative

It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.

I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified?

If you want your square-root function $\sqrt x$ to be a function, then it needs to have the properties of a function, in particular that for each element of the domain the function gives a single value from the codomain. If you take a function to be a set of ordered pairs, then each of the initial values of the pairs must appear exactly once.

So to be a function, square-root needs to be single valued; the multi-valued version is really a relation, at which point you might get into issues of principal values.

For convenience, the square root of non-negative real numbers is usually taken to be the non-negative real value, but there is nothing other than practicality to stop you from taking some other pattern. Such arbitrary choices can raise significant issues when considering, for example, cube-root functions defined on the real and complex numbers.

For positive real $x$, $\sqrt x$ denotes the positive square root of $x$, by definition. Wikipedia agrees with me on this.

The radical sign '√' means we are taking the positive square root of given equation

if we simply say taking square roots on both sides,then we apply a '±' before radical('√') sign,as I said '√' sign means positive square root,so in order to get negative one also we apply that '±' sign.

as you can see '(±√x)^2' gives result as x, i.e (+√x)(+√x)=x and (-√x)(-√x)=x

The simplest way to understand this is by the following expression

if 𝑥^2=9

taking square root on both sides

±√x^2=±√9 So, ±|x|=±|3|,so +|x|=3,-|x|=-3,,i.e in order to define √ positive,mathematicians added | |,this,called modulus function,which makes everything positive So,x=3 or x=-3 so x=±3 or we can say x=±√9 as i said again √9 is always positive

  notice I have used word **Square root** not the symbol,means we are taking both positive square root and negative square root

but when we say √x^2,notice here is no ± symbol,so here,it is asked for the positive square root only

conclusion:We conclude that √ is defined to be positive

you can also see this in Quadatic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$$

there is written ± in order to include negative root too!

hope it helped you......