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Yes I know this is probably a monumentally stupid question. Surely $\sqrt{4}$ is both 2 and -2, yet only 2 is shown as an answer. But $y^2=4$ shows both answers as I'd say it should. I probably knew this at one time!

Emilio Novati
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Ray Andrews
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    The symbol $\sqrt{x}$ means the nonnegative square root of $x$. See Why is $\sqrt{x}$ a function?. – N. F. Taussig Oct 09 '21 at 16:50
  • Ah! So we simply define 'sqrt' as only giving positive answers. Nuts, I must have known that once. Thanks. – Ray Andrews Oct 09 '21 at 17:08
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    @RayAndrews: It's certainly true that the collection of square roots of $4$ are ${2,-2}$. However, it would not make much sense for the symbol $\sqrt{4}$ to represent both $2$ and $-2$ at the same time, as this would seem to imply that $2=-2$. This is part of the reason why the accepted mathematical convention is that $\sqrt{x}$ denotes the nonnegative square root of $x$. (I say "nonnegative" rather than "positive" because $\sqrt{0}=0$, and $0$ is neither positive nor negative.) – Joe Oct 09 '21 at 17:26
  • Major ambiguity for $\sqrt{z}$ when $z$ is complex or negative. – herb steinberg Oct 09 '21 at 17:53
  • Recommend you get some graph paper, then draw, by hand, the graph of $x=y^2$ on one piece, the graph of $y=\sqrt x$ on another. Use a calculator so as to plot intermediate points in both, not just where both coordinates are integers. – Will Jagy Oct 09 '21 at 18:32
  • @Joe: There's a semantic logic to that. 'y' is a variable so in: 'y^2=4' 'y' can be any number that satisfies. But: '√4', having no variable within it, suggests a single answer. – Ray Andrews Oct 09 '21 at 18:52
  • @WillJagy: That's just what lead to this question: I wanted the whole parabola not just half of it. – Ray Andrews Oct 09 '21 at 18:53
  • Ray, understood. I do try to get people to match up algebra questions with graphing. One type of question that becomes very clear by graph, For what values of real constant $C$ does $x^3 - 12 x - C$ have three real roots? Answer: draw $y=x^3 - 12 x$ and find the values of $C$ for which the horizontal line $y=C$ intersects the graph three times. – Will Jagy Oct 09 '21 at 20:22

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As noticed in the comments, the equation $y^2=x$ with $x\ge 0$ has the following pair of solutions

  • $y_1=\sqrt x\ge0$
  • $y_2=-\sqrt x\le0$

with $y=0\iff x=0$, by the definition of $\sqrt x$.

Refer also to the related

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