I am starting a new thread to continue a discussion that started at the end of a long previous thread which contains much that is not germane to this issue.
The quadratic formula is often written:
\[x = \frac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}\] OR \[x = \frac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}\]
I asked if the OR should not be AND or AND/OR? The exchange followed as:
I responded:
Given: \[{x^2} + 3x + 1 = 0\]
the quadratic formula yields: \[\frac{{{\rm{ - }}\sqrt 5 }}{2} - \frac{3}{2}{\rm{ or }}\frac{{\sqrt 5 }}{2} - \frac{3}{2}\]
but at the same time the solution set of the equation is: \[\left( {\frac{{{\rm{ - }}\sqrt 5 }}{2} - \frac{3}{2},0} \right){\rm{ and }}\left( {\frac{{\sqrt 5 }}{2} - \frac{3}{2},0} \right)\]
No? So why not "and", or at least "and/or" ? Can you give me an example when the quadratic equation does not deliver two correct results even if they are the same? I know you are correct but I could not explain why if someone asked me to justify "or" as opposed to "and".
and received:
to which I reply, hopefully for further response and eluciadation:
I hate to appear dense, or worse, quibbling over nothing, but while I can understand the "and" regarding the solution set, I still don't understand why the quadratic formula should not be "and" since the quadratic formula seems to define and deliver a SET of two correct roots, i.e. -b/2a plus and minus two real or imaginary numbers, albeit in some cases the roots might be double roots, or conjugate pairs.
Is this a matter of custom or are there cases where one root works but the other doesn't in which case it still seems that "and/or" is preferable since most of the time anyway, each root is valid.
I have not been able to find a tutorial that doesn't gloss over this issue. Again, I know that text books and software programs always use "or", and that this seems like a no biggee deal, and probably isn't ... unless of course there are situations whose results preclude "and". I would think this is the sort of neophyte question that must repeatedly come up, and if someone should ask me someday I would like to know how to justify my position. Thanks all.
The quadratic formula is often written:
\[x = \frac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}\] OR \[x = \frac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}\]
I asked if the OR should not be AND or AND/OR? The exchange followed as:
I responded:
Given: \[{x^2} + 3x + 1 = 0\]
the quadratic formula yields: \[\frac{{{\rm{ - }}\sqrt 5 }}{2} - \frac{3}{2}{\rm{ or }}\frac{{\sqrt 5 }}{2} - \frac{3}{2}\]
but at the same time the solution set of the equation is: \[\left( {\frac{{{\rm{ - }}\sqrt 5 }}{2} - \frac{3}{2},0} \right){\rm{ and }}\left( {\frac{{\sqrt 5 }}{2} - \frac{3}{2},0} \right)\]
No? So why not "and", or at least "and/or" ? Can you give me an example when the quadratic equation does not deliver two correct results even if they are the same? I know you are correct but I could not explain why if someone asked me to justify "or" as opposed to "and".
and received:
to which I reply, hopefully for further response and eluciadation:
I hate to appear dense, or worse, quibbling over nothing, but while I can understand the "and" regarding the solution set, I still don't understand why the quadratic formula should not be "and" since the quadratic formula seems to define and deliver a SET of two correct roots, i.e. -b/2a plus and minus two real or imaginary numbers, albeit in some cases the roots might be double roots, or conjugate pairs.
Is this a matter of custom or are there cases where one root works but the other doesn't in which case it still seems that "and/or" is preferable since most of the time anyway, each root is valid.
I have not been able to find a tutorial that doesn't gloss over this issue. Again, I know that text books and software programs always use "or", and that this seems like a no biggee deal, and probably isn't ... unless of course there are situations whose results preclude "and". I would think this is the sort of neophyte question that must repeatedly come up, and if someone should ask me someday I would like to know how to justify my position. Thanks all.
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