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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 $x^2 = 9$.

Caleb Stanford
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Dewansh Chaturvedi
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2 Answers2

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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.

Luke Collins
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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.

It'sNotALie.
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  • Wouldn't it be more precise to say that if one is given no other information, by convention, a square root problem with a positive integer is assumed to be in the domain of Natural numbers $\mathbb{N}$. If one is solving a quadratic, this assumption no longer holds. So whether the square root function is a one to one function is a consequence of which domain it is applied to. – Mark Sep 12 '22 at 11:27
  • @Mark If one is solving a quadratic, one introduces the symbol $\pm$ (or something more explicit) to account for the case of the negative square root. – David K Jun 16 '23 at 12:00