Does the square root spit out a negative result?

Question

I have being told all my life that $\sqrt{9}$ equals to $\pm3$. That all changed when I saw a video talks about it. It said that the square root does not spit out a negative number. I wanted to see if it's true, and if it is true, then why? Logically, $(-3)(-3)$ equals to $+9$ too. Thanks in advance.

Answer 1

It is a choice. The most convenient approach seems to be to consider $f(x)=\sqrt x$ as a function, which implies a choice. The canonical choice is that of the positive square root.

Answer 2

A function must be defined so that each input is mapped to only one output. In this case the function $f(x) = \sqrt{x}$ must be chosen so that the result is only one number. While the square root of a number has two possible results, it seems more natural to go with the positive solution for the square root function. Thus, we ignore the negative solution.

Answer 3

No. This is a common misconception about square roots that I carried for most of my life. The problem occurs when people think about the meaning square roots; normally if we see something like $\sqrt{9}$ we think "what is the number that squared gives us 9?" and following this logic we say $3$ and $-3$. However this is not the rigorous definition of square root, instead what we are doing is solving the equation $x^2=9$ which is NOT equivalent to $\sqrt{9}$. The square root of a positive number is always a positive number.

• If anybody were to graph the function $y=\sqrt{x}$ on a $x$ an $y$ plot the result will be a line that is not defined for $x<0$.

• Since we can express the square root of a number as the number elevated to $\frac12$ it follows that the result cannot be positive because a number elevated to any real power is always positive (note that $-x^{\frac12}$ is different from $(-x)^{\frac12})$

Answer 4

The radical sign √ refers to the principal (positive) square root only. source: Square Root Calculator