Question

Can -5 be the square root of 25?

Answer 1

Yes

Explanation:

`5 xx 5 =25`
Hence 5 is the square root of 25.

Answer 2

`-5` is a square root of `25` in that `(-5)^2 = 25`

When we say the square root of a positive number, we usually mean the positive square root, also known as the principal square root.

Explanation:

Any non-zero number has exactly `2` square roots.

Actually in a technical sense `0` has `2` square roots, but they are both `0`.

If `x > 0` then `sqrt(x)` denotes the positive square root and `-sqrt(x)` denotes the negative one.

If `x < 0` then `sqrt(x) = i sqrt(-x)` denotes the square root with positive coefficient of `i` (the imaginary unit). `-sqrt(x) = -i sqrt(-x)` is the other square root.

If `z` is Complex, then it still has exactly two square roots, but different people use different conventions as to which one is the principal square root (i.e. the one represented as `sqrt(z)`).

In terms of polar representation, this boils down to choosing whether you prefer to use angles in the range `(-pi, pi]` or `[0, 2pi)` for `z`, resulting in angles in the range `(-pi/2, pi/2]` or `[0, pi)` for `sqrt(z)`.