Question

Why are solutions to square roots positive and negative?

Answer 1

Given a positive real number a, there are two solutions to the equation `x^2=a`, one is positive, and the other is negative. We denote the positive root (which we often call the square root) by `\sqrt{a}`. The negative solution of `x^2=a` is `−\sqrt{a}` (we know that if `x` satisfies `x^2=a`, then `(−x)^2=x^2=a`, therefore, because `\sqrt{a}` is a solution, so is `−\sqrt{a}`). So, for `a>0, \sqrt{a}>0`, but there are two solutions to the equation `x^2=a`, one positive `(\sqrt{a})` and one negative `(−\sqrt{a})`. For `a=0`, the two solutions coincide with `\sqrt{a}=0`.

As we all know a square root is occurrence when an integer n is multiplied to itself to give us an integer n* n. We also know when 2 integers with the same signs are multiplies it gives a positive integer .

with a these facts in mind we can say that n can be negative or positive and still give us the same perfect square.
PS . note that something like `sqrt{-1}` wouldn't exist as we know that 2 integers with opposite symbols will not give a negative number.And for it to be a square number both the nos . have to be same.

Hopefully this helps