Originally Posted by JeffM & lookagain edit

What does $x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a}$ mean exactly? It means

$x = \dfrac{-\ b + \sqrt{b^2 - 4ac}}{2a} \text { OR } x = \dfrac{-\ b - \sqrt{b^2 - 4ac}}{2a}.$

Now $x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{(-\ 1)(-\ b \pm \sqrt{b^2 - 4 ac})}{(-\ 1)(2a)} = \dfrac{b \pm (-\ \sqrt{b^2 -4ac})}{-\ 2a}\implies$

$x = \dfrac{b + (-\ \sqrt{b^2 - 4 ac})}{-\ 2a} \text { OR } x = \dfrac{b - (-\ \sqrt{b^2 - 4ac})}{-\ 2a} \implies$

$x = \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \implies$

$x = \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \implies$

$x = \dfrac{b \pm \sqrt{b^2 - 4ac}}{-\ 2a}.$

It comes from the definition of $\pm$ as plus OR minus and the reversibility of "or." The two formulas mean EXACTLY THE SAME THING as tkhunny correctly pointed out.
The plus-minus sign is a mathematical symbol with multiple meanings across mathematics and non-mathematics areas. And, it is used in different ways within mathematics itself,
sometimes, depending on which topic/problem in mathematics one is covering. But within this context of the Quadratic Formula here, the plus-minus symbol means one thing:
"plus or minus."
.
https://en.wikipedia.org/wiki/Plus-minus_sign

2. ... depending on the Domain, it could be inclusive or exclusive, as far as "or" goes.
...still, as state above, one of those "plus or minus" has to mean "minus or plus" if you want the corresponding solutions to match up.

aka "squishy".

3. Originally Posted by tkhunny
... depending on the Domain, it could be inclusive or exclusive, as far as "or" goes.
...still, as state above, one of those "plus or minus" has to mean "minus or plus" if you want the corresponding solutions to match up.

aka "squishy".
I don't know what correspondence issue you're referring to. "Plus or minus" is the same as "minus or plus," given that there is the
connector word "OR."

Example:

Show $\pm\dfrac{f}{g} \ \$ means the same as $\ \ \pm\dfrac{f}{-g} \ \$.

$\ \ \pm\dfrac{f}{g} \ \implies$

$\ \ \dfrac{f}{g} \ \$ OR $\ \ \dfrac{-f}{g} \ \$

Switch order:

$\ \ \dfrac{-f}{g} \ \$ OR $\ \ \dfrac{f}{g} \ \implies$

$\ \ \bigg(\dfrac{-1}{-1}\bigg)\bigg(\dfrac{-f}{g}\bigg) \ \$ OR $\ \ \bigg(\dfrac{-1}{-1}\bigg)\bigg(\dfrac{f}{g}\bigg) \ \implies$

$\ \ \dfrac{f}{-g} \ \$ OR $\ \ \dfrac{-f}{-g} \ \implies$

$\ \ \pm\dfrac{f}{-g} \ \$

4. Originally Posted by lookagain
I don't know what correspondence issue you're referring to. "Plus or minus" is the same as "minus or plus," given that there is the
connector word "OR."
"or" excessively defines the situation when a single root is considered separately.

$\dfrac{-b+\sqrt{b^{2}-4ac}}{2a} = \dfrac{b-\sqrt{b^{2}-4ac}}{-2a}$

If you want this equality, you must switch the sign when moving left to right. Having picked the sign on the LHS, there is no "or" about it. The sign on the RHS is unique and determined. Thus, writing simply $\pm$ on both sides is at least somewhat misleading. That's why we use the minus-plus symbol on one side in the formula for the cosine of a sum of two angles. It's a "degrees of freedom" sort of thing. If we have a restricted Domain, it is likely that failing to recognize that "+" leads to a valid solution in one case and an invalid solution in the other case will lead to an error. This is what I mean by correspondence. We all would like both formulations to lead to the same, consistent, correct results.

Would you write "6 + 7 = 13 OR 14"? It's certainly true, but why would you ever write that (unless you were teaching an addition class and wanted the student to circle the correct result)?

5. Originally Posted by tkhunny

Would you write "6 + 7 = 13 OR 14"? It's certainly true, but why would you ever write that (unless you were teaching an addition
class and wanted the student to circle the correct result)?
Why would I ask that, when it's not an appropriate analogy?

I might write "x = 13 OR 14" for the solutions to a quadratic equation, where the plus sign of $\pm$ corresponds
to the solution of 14, and the negative sign of $\pm$ corresponds to the solution of 13.

$(a^3 \pm b^3) \ = \ (a \pm b)(a^2 \mp ab + b^2) \ \ \ \$ This dual formula necessitates switching corresponding signs.

The Quadratic Formula is a dual formula that just gives the signs a turn using the "$\pm$" symbol, because the variable,
unlike your "6 + 7" can equal either one or another different value, in certain cases.

Said another way:
If you entered "6+7=" in your AOS calculator, would you be happy with the response "13 or 14"?

7. Originally Posted by tkhunny

Said another way:
If you entered "6+7=" in your AOS calculator, would you be happy with the response "13 or 14"?
Nope. You are talking about your question which doesn't apply. You are off track and ignoring my points.

8. Again, you didn't answer my question.

To sum up:

1) Opinions vary.
2) $\pm$ is a little squishy, or opinions wouldn't vary.

When one writes $\pm$ in the STANDARD version of the quadratic formula, we all are pretty sure we know what it means. On the other hand, if one were to invent a non-standard version (which is where we started), one might need to think about it a little. This is EXACTLY why the cosine of a sum/difference and factoring of cubes STANDARD formulas mandated the minus-plus symbol, because using plus-minus a second time wouldn't quite mean the same thing. There is a difference between "or" (it's one of them) and "if you picked that one, you now have to pick this one" (there is a correspondence we should be aware of).

Good discussion.

9. ## This is just more quackery** from tkhunny.

** Please see five-digit numbers being possible by him when the
problem mentoned dealt only with four-digit numbers in another
I'm surprised posts got this far in this thread. Nonsense words,
such as his "squishy," (the mathematical sense) have no business
in the mathemat-
ical discussion threads. In these specific instances, he
managed to go off on some fantasy discussions that are not based
in reality
to the problems posed by the original posters.

aka He doesn't know what he's talking about (again) in those
instances.