Can the square root of a real number be negative? [duplicate]

Given a positive real number $a$, there are two solutions to the equation $x^2 = a$, one is positive, and the other is negative. We denote the positive root (which we often call the square root) by $\sqrt{a}$. The negative solution of $x^2 = a$ is $-\sqrt{a}$ (we know that if $x$ satisfies $x^2 = a$, then $(-x)^2 = x^2 = a$, therefore, because $\sqrt{a}$ is a solution, so is $-\sqrt{a}$). So, for $a > 0$, $\sqrt{a} > 0$, but there are two solutions to the equation $x^2 = a$, one positive ($\sqrt{a}$) and one negative ($-\sqrt{a}$). For $a = 0$, the two solutions coincide with $\sqrt{0} = 0$.

It is just a notational matter. By convention, for positive $x$ (real clearly), $\sqrt{x}$ denotes the positive square root of the real number $x$. Likewise we agree by way of notational convention that $-\sqrt{x}$ is the negative square root of $x$. Of course, every positive real number, $x$, has two square roots, $\sqrt{x}$ and $-\sqrt{x}$, positive and negative real numbers respectively.

I worry sometimes about what gets taught by way mathematics in secondary school these days.

I believe your confusion is coming from assuming that if $a^2 = b$ then $\sqrt{b} =a$, but this is actually not the case. The correct form of this would be $\sqrt{b} = \vert a \vert$. Because of this, $a^2 = b = (-a)^2$ however $\sqrt{b} = \vert a \vert \neq -\vert a \vert$. This can be proven by contradiction:

If we were to say that $a=\sqrt{b}=-a$, that would therefore imply that $a = -a$, and for example $1= \sqrt{1} = -1 \implies 1=-1$ which we know to be false. This contradiction does not show up when saying $a^2 = b = (-a)^2 \implies a^2 = (-a)^2$, because, if we were to use the same example, we would get $1^2 = 1 = (-1)^2 \implies 1^2 = (-1)^2 \implies 1 = 1$, which is true (since by definition, any number squared must be positive). This is why, if you were to evaluate $a^2 = b$, you would get two possible solution for $a$ (one positive, and one negative). However, if you were to evaluate the equation $a = \sqrt{b}$, $a$ can only have one solution at any given time, and for convention, a square root was defined to always be positive. This is an important distinction because it allows us to look at the equation $4=a^2$, and find that $a=2\oplus a=-2$, thus avoiding the contradiction $a=2 \wedge a=-2 \implies 2=-2$ by saying $2^2=4=2^2 \implies 2 = 2 \oplus (-2)^2=4=(-2)^2 \implies -2=-2$. This idea might seem to get lost when graphing equations such as a circle. The equation $x^2 + y^2 = 1$ appears to have 2 $y$ values for every $x$, and 2 $x$ values for every $y$. This can be better understood by instead looking at the parametric equation for a circle: $x=cos(t); y=sin(t)$. For any given value of $t$, there is only one corresponding value of $x$, and only one corresponding value of $y$. If you are given a value for $x$, and told to solve for $t$, the most you can do is find possibilities of $t$, since the $(x,y)$ points on the graph repeat themselves every $2\pi*t$. The same idea is true for square roots. When you square a number, it always creates a positive number, therefore it is impossible to reverse definitively. The most we can do is say that there are two possibilities of what the original number was. For convention, it has been established that for an equation $a^2 = b$, where $\sqrt{b}=c$, we say that $c=\vert a \vert$. It would work just as well to defined a square root by $c=-\vert a \vert$, but I guess the mathematicians that decided on it liked working with positive numbers more.

The point in all of this was to simply establish that taking the square root of a squared number, does not reverse its exponent, because it cannot be reversed definitively. As user86418 put it:

If a and b are real numbers, then the conditions $a^2=b$ and $a=\sqrt{b}$ are not logically equivalent; the second implies the first, but not conversely.

Therefor, for the purposes of convention, a square root has been defined to be the absolute value of the original number that was squared. This why, if you plug the functions $y^2 = x$ and $y = \sqrt{x}$ into a graphing calculator or Wolfram Alpha, you will find that you get two very different looking graphs. Notice how the the graph of $y=\sqrt{x}$ never goes below the $x$-axis. Had a square root been defined as always negative, the graph of $y=\sqrt{x}$ would simply be flipped about the $x$-axis.

Technically this statement is wrong. He could say, "The square root of a positive number is positive (by definition)". E.g. for 0 you get $\sqrt{0}=0$ which is neither positive nor negative. And for negative numbers you even get complex solutions which are neither positive nor negative nor 0.

The definite article and the singular in "the square root" is also important to imply the conventional definition of $\sqrt{}$. But more correctly he should say "the principal square root", because mathematically the expression "the square root" doesn't make sense, since there are two different roots in general.

THE square root is defined to be the positive root. In reality there are always 2 roots, one positive and one negative. Think about the sqrt of 4, what this is asking is "what two numbers that are the same multiply to 4?" The answer would be 2 or -2 because 2x2=4 and (-2)x(-2)=4.