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If you were to ask me what the square root of 25 is, I would quickly say 5. But is that the correct answer?

Or is the true answer actually 5 OR -5?

In other words, does the square root of a real, positive number ALWAYS have two solutions?

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There are two solutions to the equation $x^2 = 25$, namely $+5$ and $-5$.

There is only one number corresponding to $\sqrt{25}$, namely $+5$. That's because we humans, in modern mathematics, have defined $\sqrt{25}$ to mean "The positive solution to $x^2 = 25$". There is no objective, mathematical reason to pick the positive over the negative, other than convinience. But we have to make a choice, so we've gone for the positive one, because overall that choice means more readable expressions and simpler use.

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You always have

$$\sqrt {25}=5$$

is is how this function is defined.

Though when you have to find a number $x$ such that:

$$x^2=25$$

there is two solutions $5$ and $-5$.

Don't forget the last one !

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The square root of x is usually not defined as the solution of $y^{2}=x$ but as the positive solution for this equation. Thus it is more correct, to use the positive number, but mostly by convention.

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It depends on your definition. See, in complex analysis, you have so called branch-cuts, where you define the domain of your function. You can aswell define the range of the function $\sqrt{\cdot}$ as $\mathbb{R}_{\leqslant 0}$. It does not matter. And yes, if $x$ solves $x^2 = y$ for positive $y$, so does $-x$.

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