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Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.

Antonio Vargas
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user250868
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    Search on this site to find the answer. – Aaron Maroja Jun 26 '15 at 23:43
  • Why should $\sqrt{x^2}=x$? There are two things that square to $x^2$, there is only one thing we call $\sqrt{x}$, you've got to pick one, it may as well be $|x|$. – Adam Hughes Jun 26 '15 at 23:43
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    By convention, when we write $\sqrt{x}$ for some real, nonnegative $x$, we mean the positive square root. – qaphla Jun 26 '15 at 23:43
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    Here, we don't know if you mean $\sqrt{x}^2$ or $\sqrt{x^2}$. Use latex. This statement is wrong for the first but right for the other. – servabat Jun 26 '15 at 23:45
  • The square root symbol denotes a function. A function has a single value (by convention the positive one is taken). –  Jun 27 '15 at 00:02

4 Answers4

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It is conventional that the notation $\sqrt x$ means the non-negative square root of $x$.

There are indeed two square roots of $x$, and for non-negative numbers $x$, only one of the two is conventionally denoted $\sqrt x$.

Michael Hardy
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Don not confuse $x^2=a^2$ which is an equation that has two roots of opposite sign, $\pm\sqrt{a^2}$, and the expression $\sqrt{a^2}$, which is a positive number, equal to $|a|$.

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This is a very common question. Basically, people like to think, for example, that $\sqrt{9}$ is "the number such that when you square it, you get 9". So people think this must be $\pm 3$. But that's not what we are asking with square root. With $\sqrt{9}$, we are asking for "the positive number such that it squared equals $9$". That means $\sqrt{9} = 3$ (or $\sqrt{3^{2}} = |3|$).

The first (wrong) question applied to the square root should actually be used for solving the equation $x^{2} = 9$. To solve this, we need to "find the number such that it squared equals $9$", which means the solution is $x = \pm 3$.

layman
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because even if $x$ is negative $x^2$ is positive, that's why when you do $\sqrt{x^2}$ you first evaluate $x^2$ which is positive no matter if $x$ is negative or positive. Then you apply the square root which is also positive and so that's why $\sqrt{x^2} = |x|$ also notice. We evaluated the expression inside out.

$$\sqrt{\color{red}{inside}}$$

alkabary
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