If both side of the equation have variable and we take the square root of both sides, do we add the plus and minus sign?

Question

For example:

$a^2 = b^2 + c^2$ with $a, b$ and $c$ are real numbers.

1. $a = \sqrt{b^2 + c^2}$
2. $a = ±\sqrt{b^2 + c^2}$

Should the answer be 1 or 2?

I know this sounds obvious, but I ask because in the event that we remove the square root sign from $\sqrt(b^2 + c^2)$, we would need to use an absolute value sign anyways, so putting $±$ here seems to neglect the purpose of absolute value when taking variable out of square root.

Answer 1

The use of variables is not the key issue here, but it is correct (in this instance) to use the $\pm$.

The reason is that $\sqrt{x^2} = |x|$, with this being $+x$ or $-x$, depending on the value of $x$.

Equivalently, squaring $x$ and $-x$ both give $x^2$, so if we're trying to solve an equation for all possible values, we need to account for differences in signs.

Hence, whether we have variables like in

$$y = x^2 \implies \sqrt y = |x| \text{ or rather } x = \pm \sqrt y$$

or with numbers, as in

$$x^2 = 4 \implies |x| = \sqrt 4 \text{ or equivalently } x = \pm 2$$

we will be dealing with the $\pm$.

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My guess is you're somewhat getting wires crossed with the fact that $\sqrt x$ is always nonnegative, i.e. $\sqrt 4 = 2$ but not $-2$ nor $\pm 2$.

That arises from the definition of the square root function, and is an aspect moreso of computation, rather than solving equations. (That is, when solving equations and taking square roots, you may end up needing to include a $\pm$ to account for all possible solutions, whereas when just calculating the root itself, you always take the nonnegative root.)

Answer 2

Personally, I have benefited from not using the plus-or-minus sign at all. I think it adds a layer of potential confusion. When using it, it is not difficult to make a mistake.

So for example, the following statement is what I prefer to use:

If $\ a^2 = b^2 + c^2,\ a,b,c\in\mathbb{R},\ $ then $\ a = \sqrt{b^2 + c^2}\ $ or $\ a = -\sqrt{b^2 + c^2}.\ $

The statement can be shortened to the following:

If $\ a^2 = b^2 + c^2,\ a,b,c\in\mathbb{R},\ $ then $\ a = \pm\sqrt{b^2 + c^2}.$

One benefit of writing the statement the first way rather than the second way is that you won't forget to deal with the two cases - 'plus' and 'minus' separately - which is sometimes necessary to do in certain questions, and also, you keep track of the logic: the "or" - some people mistakenly think that $\ \pm x\ $ says "plus and minus $\ x\ $", when it actually says, "plus or minus $\ x."\ $

The second way of writing it hides the real meaning of the whole statement beneath the symbol "$\ \pm\ $", and I don't see how this really helps.

In conclusion, symbols that sacrifice clarity for brevity aren't worth it. However, if you're super clear and confident on what the $\ \pm\ $ symbol means, and are super clear and confident that you won't make a "silly mistake" when using it, then it's fine to use it.