Question

Root help?!

If `c = d^2`, does `d = sqrt c`?

Answer 1

Yes, but that's only half the story.

Explanation:

The thing to remember here is that every positive real number has two square roots

• a positive square root called the principal square root
• a negative square root

That is the case because the square root of a positive real number `c`, let's say `d` to use the variables you have in your example, is defined as the number that, if multiplied by itself, gives you `d`.

In other words, if you have

`d xx d = d^2 = c`

then you can say that

`d = sqrt(c)`

is the square root of `c`.

However, notice what happens if we multiply `-d` by itself

`(-d) xx (-d) = (d xx d) = d^2 = c`

This time, you can say that

`d = -sqrt(c)`

is the square root of `c`.

Therefore, for every positive real number `c`, you have two possible square roots denoted using a plus-minus sign

`d = +- sqrt(c)`

You can thus say that if

`c = d^2`

then

`d = +- sqrt(c)`

You can check that this is the case because if you square both side, you will end up with

`d^2 = (+sqrt(c))^2" "` and `" "d^2 = (-sqrt(c))^2`

which is

`d^2 = sqrt(c) * sqrt(c)" "` and `" " d^2 = (-sqrt(c)) * (-sqrt(c))`

`d^2 = sqrt(c) * sqrt(c)" "` and `" " d^2 = sqrt(c) * sqrt(c)`

`d^2 = c" "` and `" "d^2 = c`

So, for example, you can say that the square roots of `25` are

`sqrt(25) = +-5`

The principal square root of `25` is equal to `5`, which is why we always say that

`sqrt(25) = 5`

but do not forget that `-5` is also a square root for `25`, since

`(-5) * (-5) = 5 * 5 = 5^2 = 25`