... I've recently retired and it has been a very long time since I was exposed to a classroom learning about quadratic equations. But now finally jobless, I have more time to satisfy some personal curiousities. For my difficulty in not being able to ask a more sensible question, I apologise in advance. I do hope to get better at it. I seem to remember that quadratic solutions often resulted in more than one set of coordinates. The root sets might not seem related (quantity/ratio-wise) and it also seems we just merrily threw some of these sets away because, on the surface, they just didn't make sense. On the other hand, I believe math doesn't lie, and discarding perfectly good numbers seemed to me, wrong in principle then; and it still seems wrong today. All solutions should have some insight value. Since I also believe math must be firmly based in geometry, we might, therefore, simply throw away a set of assumed-useless coordinates that don't match the expected picture. If this is a true perspective, is there some sort of master rule that states, by some sort of deep reasoning, that some coordinate sets must be derived as incorrect? If no rule applies, has the casual abandonment ever fouled up the intepretation of an answer we should have used, or at least considered? It does seem as if, choosing the wrong basic set, of the various roots, might arbitrarily result in asymmetries that lead down entirely different paths in searching for reality. As an example, the math might continue to be right on, but the overall interpretation of complicated geometry ratios could be wrong. Perhaps the story of Galileo and geocentricity vs heliocentricity is an example. Thanks, Wes ...
The solving process can add new solutions to equations. They are not solutions to the original problem, so they are discarded afterwards. Simple, pointless example: x=4 Of course, we know the solution to that equation... but let's square both sides: x^2=16 Now, we have x=4 and x=-4 as solutions, but just one of them (x=4) is a solution to the original equation. In geometry, you often have additional requirements - distances between points should be positive, for example. If you care about those requirements in all steps, you don't get additional solutions. This can be a lot of work, however, so it is often easier to dicard negative solutions (as an example) afterwards.
Thanks, mfb. That's along the lines of what I thought prevailed. Perhaps the story of Galileo actually is relevant here. When one needed to describe geocentricity, solar planets did appear to move backwards in negative direction epicycles. Accepting this must have presented some Occam's Razor type problems in calculating planetary motion, all due perhaps because humans seem to instinctively prefer to think of themselves at rest I guess. It's a good thing Galileo did a little hand-wringing over it, for we have gained immeasurable insight and perspective. I suppose the next similar reversion problem, regarding simplified observation calculations, might be parallax with distance galaxies against local stars, where humans might have again found that simplifying rotational motion best requires converting the observing platform to independence above earth, the solar system and the galaxy to avoid computing negative distances. Perhaps another such problem will (or has) surface(d) one day. Thanks again for a very kind and succinct answer. Wes ...
I think observations of stellar parallaxes (together with abberation) would have established the heliocentric world view quickly, but parallaxes are small and hard to measure. The first successful measurement pf a parallax was done in 1838, more than 200 years after Galileo and Kepler.
Sometimes we discard solutions because a variable is constrained to be nonnegative by some physical process, or there is some other constraint. The mathematics doesn't lie, but the equation that is being used to model some physical phenomenon might not be a perfect fit at all times. In cases such as these, it's reasonable to discard solutions that don't fit the situation being modelled, by being too small (or negative) or too large or whatever. A classic example is the situation where someone is standing on a platform and throws a ball straight up. The usual question is to find when the ball strikes the ground, given the speed at which the ball is thrown. A simplified, and reasonably accurate modeling equation is that of a parabola, with h(t) = at^{2} + bt + c. Setting h(t) to 0 results in two values of t, one of which is negative. Since the question asks when the ball strikes the ground, we're not concerned with a time that is before the ball was thrown, so it's reasonable to discard.
Thanks, Mark. I think the proper perception of time finally does it for me. While it might be just arbitrary to insist that moving locations are merely a matter of perspective, it is pretty hard to think of the arrow-of-time going backwards under sane conditions. I did find a very-similar-to-your example of a "real world" time problem to be solved by a quadratic equation here. The webpage looks like a great site to brush up on some math. I once read an article by Isaac Asimov where he stated that both geocentricity and heliocentricity were valid if one were able to take into account Mach's Principle. I take that to mean that the coordinate locations can always be transformed to accurately describe the laws of our universe, excepting reversing time vectors and other possible absurdities. I know Mach's Principle has fallen into somewhat lower regard since Einstein first used it in developing Relativity, but I suppose it might still be the basis for him saying, "according to general relativity all co-ordinate systems are equally valid". Thanks again, Wes ...