Talk:Quadratic equation

Descriminate and factorisation

Added effect of descriminate on factorisation of quadratics. wolfie 17:19, 16 May 2006 (UTC)

Simplified eqn?

In school, I've been taught to solve quadratic equations by writing them on the form

$x^{2}+bx+c=0$

and using the formula

$x=-{\frac {b}{2}}\pm {\sqrt {\,{\frac {b^{2}}{4}}-c}}$

(which is the same as the quadratic formula but simplified for the special case of a = 1).

This always seemed to make more sense to me. Perhaps just because I'm all backwards but used to it? ;)

Is this method worth mentioning here? Fredrik (talk) 08:47, 8 Jun 2004 (UTC)

I'm afraid not. Your equation looks like a fairly obvious simplification to me. Plus, it fails on 0 = 3x² + 2x -1. 127 01:29, 5 November 2005 (UTC)

---

Actually, I've never seen it done that way before. I'm still in 8th grade, and I'm reviewing the quadratic formula for the upcoming EOC's, btw.

I believe the last bit of it (the square root of the fraction) can be simplified. I also remember hearing something about leaving the entire fraction in a square root bracket thing is improper. I really don't know though, and I don't feel like researching much.

I think what you might be refering to is that ${\sqrt {\frac {x}{y}}}$ does not neccessarily equal ${\frac {\sqrt {x}}{\sqrt {y}}}$. But putting a fraction in a square root is fine, as long as thats what you mean. Fresheneesz 05:14, 12 May 2006 (UTC)

Actually, those two expressions are necessarily equal. I think what the anonymous 8th grader was referring to was the fact that standard form for fractions with radicals is to rationalize, removing the radicals from the denominator. -lethe ^{talk +} 10:46, 14 May 2006 (UTC)

Well perhaps thats what he meant, but those expressions I gave aren't necessarily equal. Try x = 3 and y=-1. 3i doesn't = -3i. Fresheneesz 20:19, 20 May 2006 (UTC)

Moved from article - not sure what it means

(I know this is 1 year late, but here's an "inline" response to anyone who cares):

the quadratic equation also would mean that {x^2+(bx/a)=(-c/a)} reversed equals {((-c-bx)/a)=x^2}

This step is correct.

square rooting equals {sqrt/ ((-c-bx)/a)/}=x

This step is also correct.

if {sqrt/(-c-b)/a/}x = x, than {/sqrt-c-b=a/}

This step is not correct. Here is why, the x and the a are inside the squareroot; so they can not be taken out the way you have done.

if sqrt(-c-b)=a is inversed, than {sqrt/(-c-b)/}-a/}=0--67.49.12.102 06:25, 9 Nov 2004 (UTC)benjamin j. giglione

Hence, sqrt(-c-b) -a does not equal zero. A counter example: x² +2x +1 = 0. - 127 18:30, 29 November 2005 (UTC)

But what would you use it for in the real world?

$Insertformulahere$ I learnt about quadratic equations at school over 35 years ago but I still don't know what I would use them for in the real world. Ignorance is not bliss!

Well, it has several applications in the real world. For example, financing and physics. 127 01:29, 5 November 2005 (UTC)

---

I'd say it'd be useful if you go into a math-related career path, like NASA or something.

---

I work for NASA and I've had to arrange financing for several houses. Never used a quadratic formula, as far as I know. Still wonder what it is good for. Also, why is it called "quadratic"? 68.226.118.83 20:30, 24 November 2005 (UTC)

Sorry, I was thinking of economics not financing. Quadratic regression is used often in that field, but don't take my word for it (IANAES ~ I am not an economics student). The term quadratic comes from the word quadrate which means "a square object." According to Dictionary.com the Indo-European root related to this word means "four." The applications are endless. For example, in root finding, Muller's method uses quadratic equations. 127 17:48, 29 November 2005 (UTC)

---

I, too, would appreciate seeing some examples of real world applications. I also learnt about quadratic equations some 30(ahem)+ years ago (I'm a Physics Major, and have, in the past, been known to use my ability to recall the formula ,for calculating the roots of a quadratic equation, as way of determining how much alcohol I've had to drink - beats a breathalyzer anyday). Today, I discovered the formula for calculating the roots of a quadratic equation in one of my 9 year old daughter's reading books (Math Curse, published by Scholastic). I'd love to teach her all about quadratic equations but how do I convince her that it has use in the real world. Perhaps I need to pull out some of those Physics text books that are gathering dust in the garage.

Yea, the quadratic equation can be used to help solve simple physics things like distances involving accellerating objects:

$at^{2}/2+v_{0}t+d_{0}=\mathbf {distance}$

I'm an engineering major, and I've found plenty of uses for this equation - its a hell of alot more useful than learning how to factor. I doubt telling your 9yo about partial fraction expansion and imaginary number would help at all... but theres plenty of simple physics things that involve quadratics - basically anything involving accelleration, or any 2nd derivative. Fresheneesz 05:14, 12 May 2006 (UTC)

just for single variable quadratics?

I noticed that the Conic Section page links to this one with a remark that "In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section".

Yet this page only discusses quadratics in one variable!

So, should this page discuss quadratics in two variables (with a link to Conic Section), or is that a topic for a new page?

Parabola is the page you are looking for. 127 01:29, 5 November 2005 (UTC)

Are you talking about an equation in this form: $y=ax^{2}+bx+c\$ ? Fresheneesz 05:14, 12 May 2006 (UTC)

---

Consider the equation x*x – 5xy + 6y*y = 0. Isn’t this a quadratic equation? Yes it is, as it is a 2nd degree equation. What curve does it represent? Conic?? No. Its not that. It represents a pair of straight lines

(x – 2y) (x – 3y) = 0. Every quadratic equation has its own features.

Good point. 127 15:30, 30 December 2005 (UTC)

Not a good point, those straight lines are a conic section. Consider cutting the conic straight down the middle, you get an X shaped pair of lines. Fresheneesz 05:16, 12 May 2006 (UTC)

Maximum and minimum points

This section isn't about the quadratic equation, so it doesn't really belong in this article. I can't think of a good place, though. Fredrik | talk 17:20, 26 October 2005 (UTC)

This section is finding the min and max points of a quadratic equation, so I think it stays. (Unless I read it wrong. I'm in a rush!) 127 01:29, 5 November 2005 (UTC)

I thought it over. If you want to move it, you could put it in Maxima and minima or if you think its too much for an encyclopedia, we could dump it in Wikibooks. 127 15:55, 6 November 2005 (UTC)

Simple way to solve

Let me give a useful hint on how to solve easily a simple quadratic equation of the type:

$x^{2}+bx+c=0$

Well, if the roots of this equation, $x1$ and $x2$, are non-zero real numbers, then the following is true:

$x1+x2=-b$
$x1*x2=c$

Using this simple formula, the roots can be easily calculated in mind.

For example, let`s solve the following equation:

$x^{2}+7x-30=0$

According to the aforementioned formula,

$x1+x2=-7$
$x1*x2=-30$

That is, $x1=-10$, $x2=3$. Easy and quick!

You might know this as "factoring." It would only work, as stated above, if the coefficient of $x^{2}$ is 1.

Higher degrees

One of the issues I have with math text books is that at first I am thrilled to be learning this knowledge, and I am happy to be connected with truth, harmony, and the secrets of the universe, but then there's a quick switcheroo, and I am lost and confused, and I feel as if I am on the wrong side of an academic three card monte game. Now I see it, now I don't, and now I'm the chump.

For example, the into paragraphs on the quadratic equation are clear and make perfect sense, then this suddenly appears:

"Higher-degree equations may be quadratic in form, such as:

2x6th power + 3xthird power + 5 = 0

Note that the highest exponent is twice the value of the exponent of the middle term. This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring (also called factorising), the quadratic formula, or completing the square."

Bringing in a non-second degree equation is a good idea, but only as an example against the previous and following equations. The note confuses me. Is it my imagination, or has that +5 also made this equation devilishly difficult? Also, could I see just exactly how one would solve "This equation..." especially with the quadratic formula?

I think what the author had in mind was to introduce another variable $u=x^{3}$. Then the equation becomes $2u^{2}+3u+5=0$ which is quadratic in $u$. We could use the quadratic formula to solve for $u$ and then get $x$ since $x=u^{1/3}$. 68.123.46.200 03:04, 18 January 2006 (UTC)

Above explanation is perfect and should be in the orginal text. So the text might be amended to read:

You can solve this non-second degree equation directly, or if you choose not to solve it directly, because the highest exponent is twice the value of the exponent of the middle term, you may instead substitue a variable for $x^{2}$, say "u", and use quadratic methods, such as factoring (factorizing),the quadratic formula, or completing the square to solve for $2u^{2}+3u+5=0$.

Simplifed way?

i'm in gr 9 and i'm doing gr. 10 math. could u guys who work on all these math related articles have a simplified version in a small corner for those who want to get ahead but need to get a more simple explanation, not a year's worth of knowledge in a short paragraph?

C++ program

The following C++ program computes the quadratic formula:

//quadratic formula
//(C) 2006 42istheanswer
//Released under the GNU GPL or GFDL.
#include <iostream>
#include <cmath>
using namespace std;
int qformula(long double a,long double b,long double c,long double *output){
long double answer1;
long double answer2;
answer1=(-b+sqrt(b*b-4*a*c))/(2*a);
answer2=(-b-sqrt(b*b-4*a*c))/(2*a);
output[0]=answer1;
output[1]=answer2;
}
int main(){
long double a,b,c;
long double answer[2];
cout<<"A:";
cin>>a;
cout<<"B:";
cin>>b;
cout<<"C:";
cin>>c;
cout<<"Solving...\n";
qformula(a,b,c,answer);
cout<<"Answer 1 is:"<<answer[0]<<"\n";
cout<<"Answer 2 is:"<<answer[1]<<"\n";
return 0;
}

This is just a simple example, as it does not make sure that a is not zero.

a=0???

Why does a have to = 0? I'm pretty sure it doesn't have to. One can use the quadratic formula with ANY expression with the form $ax^{2}+bx+c/$, regardless if a is 0 or if a, b, and c are 0. It just becomes more work than neccessary in those cases. I think we should remove the condition that a=0. Fresheneesz 04:50, 12 May 2006 (UTC)

The condition is "a is NOT equal to zero". In the formula given, the roots are not defined if a = 0 as you are dividing by zero. Of course if a = 0, then bx + c = 0 and x = -c/b. There is only one root. --Bduke 05:15, 12 May 2006 (UTC)

I'm sorry, i meant "why does a have to NOT = 0" - my fault. I see what you're saying, my mistake again. But thats a condition put on the quadratic formula, not on quadratic equations - however the condition is also put on a quadratic equation (at the top of the article), thats what i'm trying to take issue with. Fresheneesz 05:25, 12 May 2006 (UTC)

Well, if a = 0, you do not have a quadratic equation. i.e. a polynomial of order 2. You have a linear equation - a polynomial of order 1. --Bduke 05:30, 12 May 2006 (UTC)

I see your logic, but I'm a big fan of generaliation. Instead of using forms for solving first order DEs, I'd rather use the method for solving 2nd Order DEs, as they require me to remember less - and they still work even for first order DEs.

I was also wondering, since the quadratic formula doesn't work when a = 0, does this mean that the quadratic formula is an *approximation* whose error becomes greater and greater as a approaches 0? I just did some calculations, with the equation $ax^{2}+3x+2=0\$. I found that when a is 1 the quadratic formula works perfectly - but when a is .00000000000001, then at the supposed zero $ax^{2}+3x+2=.00159856$ rather than equaling 0. I also found that if a is not 1, the quadratic formula also doesn't work very perfectly - but may give complex number with non-zero magnitude. This leads me to believe that the quadratic formula is an approximation. Fresheneesz 08:32, 14 May 2006 (UTC)

Perhaps it has something to do with the "loss of significance" ?? Fresheneesz 08:36, 14 May 2006 (UTC)

If you try to use the quadratic formula with a = 0, you get division by zero. A linear equation should not be thought of as a quadratic equation with a =0 ; they are different beasts. -lethe ^{talk +} 10:42, 14 May 2006 (UTC)

I think that just such thinking is healthy. Not only does it allow a person to notice the transition between -a, a line, and +a, but it forces the thinker to think about the realities of handling 0 in real mathematics. People avoid the odities of zero whenever possible, but it is a fact of life that is sometimes harder to reconcile than imaginary numbers. People aren't knowlegable enough about 0, I know I'm not. Fresheneesz 19:44, 14 May 2006 (UTC)

I don't understand. The quadratic equation simply doesn't work with a = 0. I don't know any way of thinking that will save you. -lethe ^{talk +} 19:55, 14 May 2006 (UTC)

I can accept that. But I would guess that the quadratic equation also doesn't work very well when a is very close to 0. Fresheneesz 18:36, 15 May 2006 (UTC)

Derivation of quadratic formula

I really do think that Fresheneesz should have discussed splitting off the derivation to another page here before doing it. What do others think? --Bduke 06:12, 12 May 2006 (UTC)

Sorry, I was being bold. Derivations are useful - only if you're looking for them. Do you think it was a bad idea? Fresheneesz 08:21, 14 May 2006 (UTC)

I think it is unnecessary spliting stuff that is best kept together unless the article is getting too long, but I'm not going to get worked up about it. Let's leave it and see whether others agree or disagree. --Bduke 10:40, 14 May 2006 (UTC)

"generalized" form vs standard form

The definition at the top says the "generlized form is "ax^2+bx+c = 0 where a does not = 0" - but I think this is *standard* form. General form should be "y=ax^2+b+c" (or "f(x)="). Comments? Fresheneesz 08:39, 14 May 2006 (UTC)

I agree. There is nothing generalized about the form. Perhaps the author was an eightgh grader who thought that using variables instead of numbers for the coefficients makes it "generalized". I don't agree with that reasoning though. I've changed it to say "standard form". -lethe ^{talk +} 11:03, 14 May 2006 (UTC)

Minus plus sign

"Note also that the signs distinguishing the two roots are reversed."

What does this mean? Whats the significance of a minus-plus sign vs a plus-minus sign? On Plus-minus_sign, it says the minus-plus sign is used to mean that the minus-plus sign is *minus* when the plus-minus sign is *plus*, and vice versa. From what I could understand, minus-plus only takes on significant meaning when used *with* a plus-minus sign (in the same expression or equation). Fresheneesz 08:46, 14 May 2006 (UTC)

The minus/plus has meaning when compared to the plus/minus of the other expression. -lethe ^{talk +} 11:03, 14 May 2006 (UTC)

Yes, however the quadratic formula is not compared to any other expressions on this page. The equation could use a plus-minus sign until it is compared with another equation involving an opposite plus-minus sign. But I do think its a good thing to introduce here, and so I'll just add a note about it saying something like: "The minus-plus sign only takes on a different meaning than the plus-minus sign when it is used with a plus-minus sign - see plus-minus sign for more detail."

Well, just remember that the minus/plus in that expression does have a meaning. Don't change the text to make it seem like it doesn't. -lethe ^{talk +} 19:41, 14 May 2006 (UTC)

Yea, I understand its significance now. Do you think that text is misleading? How would you reword it? I do think that some note to that effect is neccessary. Fresheneesz 19:46, 14 May 2006 (UTC)

I changed the sentence to be more clear (I hope). Fresheneesz 19:48, 14 May 2006 (UTC)

I've changed it even more. I think maybe it's best to just say exactly what the symbol means, which should make it clear the reason for using the opposite symbol here. What do you think of my version? -lethe ^{talk +} 19:54, 14 May 2006 (UTC)

Well, technically, it doesn't matter in what order you calculate the solutions, as long as the minus plus sign is always the opposite of the plus minus sign. I'm not saying your version is worse, but I wouldn't have understood it. I think of plus minus signs as short hand notation for two different solutions (an unordered pair). Fresheneesz 07:00, 15 May 2006 (UTC)

If you consider the solution set to be unordered, then there can be no distinction between the two roots, and the plus/minus minus/plus distinction has no meaning. But of course, this solution set is ordered, as indicated by the subscripts. I've an idea, which may make the idea even clearer. Tell me what you think. -lethe ^{talk +} 10:17, 15 May 2006 (UTC)

"If you consider the solution set to be unordered, then ... the plus/minus minus/plus distinction has no meaning" - not true, the solution set can be unordered and the disctinction between the two is just as important. For example, $1\pm 2\mp 3$ equals 0 or 2, but its not an ordered pair. It doesn't matter if you get 2 then 0, or 0 then 2. In the quadratic equation, the roots aren't the "first" and "second", even though some might call them that - first vs second has no significance in any mathematical way for solutions sets.

"of course, this solution set is ordered" - I don't know why the ordering would matter, as long as the plus-minus sign is always calculated as the opposite of minus plus. Fresheneesz 18:34, 15 May 2006 (UTC)

As sets, $2\pm 1$ and $2\mp 1$ are the same. Both sets are $\{1,3\}$. As ordered sets, the first is $(3,1)$ and the second is $(1,3)$. As you see, they are different. -lethe ^{talk +} 19:20, 15 May 2006 (UTC)

If they *were* ordered sets - yes I can see how they're different. But I don't see how they are in any way ordered, except by your arbitrary convention. What exactly does this ordering *mean* about those sets (which, if i'm not mistaken, are by definition unordered lists, as it says in the article on sets, "Set identity does not depend on the order in which the elements are listed"). Fresheneesz 21:44, 15 May 2006 (UTC)

Put a different way - how can you say one integer root for 4 is "first" and another "second" ? Is -2 first? or is 2 first? My opinion is that they aren't ordered solutions. Fresheneesz 21:46, 15 May 2006 (UTC)

This isn't a matter of opinion. It's a matter of notation. Either you want to denote the solution set as ordered, or you don't. The notation used in the article indicates ordered solution sets. This has the benefit that you have a correspondence between the two forms of the solution. -lethe ^{talk +} 22:12, 15 May 2006 (UTC)

Sets are not ordered. That is part of a definition of a set. You can *think* of a plus-minus sign as having an "order", but this only has to do with the way you learned it - it does not have to do with the reality of the math. The only thing that matters when calculating something like $a\pm b\mp c$ is that the plus-minus sign is always rendered as the opposite of the minus-plus. It has nothing to do with order. a-b+c is not first nor second, and a+b-c is also neither. They are both solutions of equal importance. Fresheneesz 19:13, 17 May 2006 (UTC)

It's true that sets are not ordered. Ordered sets (also called tuples) are ordered, of course. The notation in the article denotes ordered sets (that's what subscripts do for you. The solution set x₁ = 3, x₂ = 5 is distinct fromx₁ = 5, x₂ = 3 because of the subscripts, even though the two unordered sets coincide. So let me just say this again: the notation used in the article denotes ordered sets. With the order, the two plus/minus signs are distinct). Now, I'm not really interested in being taught set theory by you. I think you seem to have a fundamental misunderstanding, but I don't know how to correct it without repeating myself. Do you still have a question about why the minus/plus sign is distinct? Because this conversation seems to be going in circles. -lethe ^{talk +} 19:29, 17 May 2006 (UTC)

I'm just not convinced that the convention of "ordered sets" is standard or most used. In fact, could I see some sort of references for such thinking? Also, could you show me exactly what meaning of "ordered set" you're talking about, because there are different meanings. Fresheneesz 00:38, 19 May 2006 (UTC)

I don't know off the top of my head any reference which explains the notational significance of subscripts on variables. I think it's a very basic notational concept. I assure you that the use of subscripts is quite standard, and so is the fact that subscripts entail ordered sets. I don't know what "different meanings" you could be referring to. There is only one possible meaning that one can assign to a collection of indexed variables.-lethe ^{talk +} 00:57, 19 May 2006 (UTC)

Check out ordered set for a couple different meanings. I tried looking up ordered set, and none of them mention the type you're talking about. Fresheneesz 03:24, 19 May 2006 (UTC)

I don't see anything in that article relevant to this discussion. -lethe ^{talk +} 04:43, 19 May 2006 (UTC)

Thats my point. Whatever "ordered set" you're talking about doesn't seem to exist whenever I try to look for it. Fresheneesz 22:04, 19 May 2006 (UTC)

So wait, are you trying to show me meanings of the term "ordered set", or are you asking for me to show you something? Your first comment "check out ordered set" makes me think the former, while your latter comment "no meaning seems to exist" makes me think the latter. -lethe ^{talk +} 08:39, 20 May 2006 (UTC)

Its both. I showed you some definitions, and I'm asking what definition *you're* talking about. Fresheneesz 20:21, 20 May 2006 (UTC)

Well, I'd point you to the article subscript, but it doesn't say anything at all. Instead I'll suggest that you check out some introductory math books from the library which explain what different notations mean. I don't have any particular reference. -lethe ^{talk +} 21:12, 20 May 2006 (UTC)

Ok... i'm not going to go to the library. Something so "basic" would surely be mentioned *somewhere* online. I don't have much of a problem with the notation though. I figured that the 1 and 2 meant that there were two solutions to the equation - but given that such notation seems to be extremely non-standard, I think it wouldn't be a bad idea to remove that notation. Along with the fact that the term "ordered sets" is never used to refer to the order in which its members exist, and not only that - sets are never ordered - I'm going to change the plus-minus thing to not imply ordered solution sets.

I hope you think thats logical too, because it really does seem like ordered solution sets aren't mentioned anywhere, and aren't implied by the minus-plus sign. If such notation and thinking was very common, I would have no problem agreeing with you - but it seems as though the notation isn't common, and probably won't be very well understood by people refering to this page. Also, I think it would mislead students in thinking that sets are ordered. Fresheneesz 11:06, 21 May 2006 (UTC)

You're wrong that this notation isn't very common, and you're wrong that it is not standard. It's OK to correct articles that are not clear, but it is not OK to rewrite articles to address your own fundamental lack of understanding. Your failure to understand what subscripts mean is quite atypical. -lethe ^{talk +} 13:26, 21 May 2006 (UTC)

I know what subscripts mean, don't insult me. I would be happy to admit that I'm wrong if you show that the notation is

1. common,
2. denotes unordered sets,
3. that unordered sets mean what you say they mean.

And once again, the subscripts are not a problem for me - only the "ordered set" idea. Fresheneesz 19:06, 21 May 2006 (UTC)

I gave an example earlier in this thread of how subscripts entail ordered sets. This happens every time you use subscripts in your notation; it's what the subscripts mean. So I guess you are not disputing that subscripts are common, but rather you're disputing that subscripts imply an ordering of the values. To that, I can only respond: you should think about the example a little. -lethe ^{talk +} 19:23, 21 May 2006 (UTC)

Look, thats not "what subscripts mean". Subscripts are used to further specifiy a variable, and double subscripts are used to refer to something between two points. Subscripts are simply part of a variable to help further explain the meaning of that variable. I've seriously never seen subscripts to denote two ordered values of a set. I understood the example, (I assume you're talking about "...solution set x₁ = 3, x₂ = 5 is distinct fromx₁ = 5, x₂ = 3 because of..."), but it doesn't prove that thats the way solution sets are thought of. Also, I went back and looked at the page on tuples, which are apparently not ever called "ordered sets", but are called "ordered lists".

Please, I don't want this to turn into a fight. I'm just asking for some sort of evidence that what you say is true. Thats how wikipedia works. One of the main policies at wikipedia is verifiability, and this subscript notation lacks that without some sort of evidence that its used in that way elsewhere. Fresheneesz 20:10, 21 May 2006 (UTC)

I think the subscripts are being used to point out a certain relation between the two equations. A cookie for anyone who tells me what the relation is. :D -- 127.*.*.1 20:29, 21 May 2006 (UTC)

It certainly doesn't prove that "that is the way solution sets are thought of". Indeed, solution sets are usually not thought of that way. But as I have said, it is a matter of notation. The original author of this page chose to label the solutions with subscripts. It is, as I have said, merely a matter of notation. -lethe ^{talk +} 00:23, 22 May 2006 (UTC)

Ok. However, my argument is that that notation is non standard. Don't get me wrong, I fully see how the idea of ordered solutions can be used, but I don't find it to be either a useful idea or a common idea. My whole problem with it is that it seems to simply not exist. Why is this idea that you admit is non-standard so important to keep on this page? I like good non-standard ideas as much as the next crackpot, but I don't see the use of using ordered sets when the minus-plus sign is much more easily explained as "the opposite of the plus-minus sign", plain and simple. Fresheneesz 06:10, 22 May 2006 (UTC)

Well, I am going to continue replying to you, even though everything I say here will be something I've already said in this thread. We've fallen into the pattern of the repetitive circular argument which I was hoping to avoid. Perhaps if I discuss my points at greater length, then you'll be more willing to consider them? So the notation used on this page has an obvious advantage. It allows you to have a correspondence between the roots written in the form

$x_{\pm }={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

and the form

$x_{\pm }={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac\ }}}}$

if these two solution sets were unordered, then there would be no way to denote which of the two roots from the first solution set was which root from the second solution set. But with subscripts, we have the two equations

${\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}=x_{+}={\frac {2c}{-b-{\sqrt {b^{2}-4ac\ }}}}$

and

${\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}=x_{-}={\frac {2c}{-b+{\sqrt {b^{2}-4ac\ }}}}$

It is my opinion that seeing the correspondence between these two equations is useful. Knowing which root is which when switching between the two methods for calculating roots is useful. This cannot be done without some ordering of the two roots. So I disagree with your assessment that the idea isn't useful. This is the point which I made at the top of this thread with the comment "This has the benefit that you have a correspondence between the two forms of the solution."

I guess lethe gets the cookie... -- 127.*.*.1 15:43, 22 May 2006 (UTC)

As for your claim that this is nonstandard; well it's my opinion that the notational device of ordering elements of a set by denoting them with subscripted variables is very common. It might be a little odd to force an ordering on a set which has no natural order, but I think it's useful in this context. Here at wikipedia, we are allowed to make our own choices of notation, this does not violate our original research policy. As I have been saying all along, this is a matter of notation. If a notation is useful to illustrate a point, then let's use it. Changes of notation are trivial enough that we do not have to worry about them being standard or not. The obvious exception is of course new notation. We cannot invent new notation here at wikipedia, but of course subscripts are not new notation; they are one of the most common notations in mathematics. -lethe ^{talk +} 15:38, 22 May 2006 (UTC)