When do we use the positive vs. negative square root of numbers?

Question

When we take the square root of a number or in a equation, where do we get only the positive root, and where do we get both the positive and negative (both answers)?

For example, when taking the square root of $4$ or solving an equation like $x^2 = 9$.

Answer 1

By convention, in order to make the square root function $x \mapsto \sqrt{x}$ a function, we choose the positive root to be the output of it. However, if you consider the solutions to quadratic equations of the form $x^2 = a$, then both the positive root and the negative root work.

So essentially, the solutions tend to be both the positive and the negative root, but in order to make the square root a function, we have to "pick" one side - and we picked the positive side.

Answer 2

Usually, the use of the symbol $\surd$ denotes the positive root, so $\sqrt 9=+3$ (for example).

If you have an equation such as $x^2=9$ however, you are presumably interested in finding all possible solutions. In that case, both the positive and the negative roots work, i.e., both $\pm\sqrt9=\pm3$ are solutions.