Square root of a square

Question

I'm little puzzled with square roots basics which says that square root of a squared number is absolute value of that number. I was in a impression that it would have both positive and negative roots.For example

The Basic says: $\sqrt{2^2} = |2| = 2$

While I think that: $\sqrt{2^2} = \sqrt{4} = 2$ or $-2$.

Answer 1

The square root function is defined such that it returns the positive root of a square, otherwise it wouldn't be a function.

Answer 2

$\sqrt{a^2}=|a|=a$. This is because the square root function returns the principal (positive) root.

Note that both $a\times a$ and $-a\times-a$ equal $a^2$ though.

You may be wondering when do we use the above fact. An example is when solving a quadratic equation:

Solve for $a:$

$(a^2-2)=0\implies a^2=2\implies a=\sqrt{2}$ or $a= -\sqrt{2}$

So to summarize, when dealing with the square root, the principal root is returned, while solving for variables, you can consider all values for which the equation holds.

Answer 3

Compare the graph of $y^2 = x$:

with that of $y=\sqrt x$:

(Pictures courtesy of Wolfram Alpha.)

Answer 4

It is a function for the negative choice too. So the idea that the positive root is chosen to form a function is arbitrary. The better approach is to respect, in any root problems, the possibility of multiple answers. You can then apply your own preference to fit your application. But that is your choice, not the math's. Its like quantum mechanics, where more than one simultaneous state is possible, but the observation reduces to a specific instance.