Question

Is `+-5i` the square root of `-25` ?

Answer 1

Yes and no.

Explanation:

The notation:

`+-5i`

is shorthand for the two values:

`5i" "` and `" "-5i`

So we can say that the polynomial:

`x^2+25 = 0`

has roots:

`x = +-5i`

meaning that it has roots:

`x = 5i" "` and `" "x = -5i`

In other words, `-25` has two square roots, namely `5i` and `-5i`.

By common definition and convention, the symbols `sqrt(-25)` denote `5i` (which is called the "principal square root"). So `-sqrt(-25) = -5i`. Both `sqrt(-25) = 5i` and `-sqrt(-25) = -5i` are square roots of `-25`.

`color(white)()`
Remarks

The `+-` symbol is very useful for shortening expressions that describe multiple values, but can be unclear when overused.

For example, we might write:

`(a+-b)^2 = a^2+-2ab+b^2`

which is true, provided that the chosen signs match.

Then again, we might say that the roots of:

`x^4-10x^2+1 = 0`

are:

`x = +-sqrt(2)+-sqrt(3)`

but in that case we would intend all `4` possible combinations.