Questions about quadratic formula

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Discussion Overview

The discussion revolves around the quadratic formula and the evaluation of double and triple negatives in the context of randomly generated second-degree equations. Participants explore the implications of defining "random" coefficients and the resulting probabilities of encountering specific sign combinations in the calculations.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant questions the percentage of equations that would yield a double negative for -b and a triple negative for -4ac, seeking a probabilistic approach.
  • Several participants express confusion over the concept of "percent of equations," with some suggesting that the infinite nature of real numbers complicates the question.
  • There is a discussion about defining "random" in the context of coefficients a, b, and c, with some suggesting a uniform distribution of signs.
  • One participant argues that without a specified probability distribution for the coefficients, the original question lacks a definitive answer.
  • Another participant proposes that the signs of a, b, and c could be treated as independent random variables with equal probability for positive and negative values.
  • Some participants suggest that the original inquiry could be better addressed through combinatorial problems rather than probabilistic percentages.
  • One participant attempts to analyze the problem by plotting parameters and discussing the nature of solutions based on the discriminant of the quadratic equation.
  • There is a recognition that the original poster's focus on double and triple negatives may have been overlooked in the broader discussion of parameter distributions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to define "random" in this context or how to approach the question of percentages related to double and triple negatives. Multiple competing views remain regarding the interpretation of randomness and the implications for the quadratic formula.

Contextual Notes

The discussion highlights limitations in defining randomness for real coefficients and the challenges in applying probabilistic reasoning to an infinite set of equations. There are unresolved mathematical steps regarding the distribution of coefficients and their impact on the evaluation of the quadratic formula.

  • #31
agentredlum said:
So 5 out of 8 times you are performing an un-necessary operation, because you have been taught a myth.

That's 62.5%

3 Things:

1) No one was taught a myth. The point of the QF is to have one formula that works for all quadratic equations, not just for one "form". No myth has been taught. Your forumula is (as I proved) a simple algebraic manipulation.

2) You are ignoring the fact that before I even think about using any QF, I'm going to try to factor it into two linear factors.

3) Those of us who have been using the QF for more than, say, 3 weeks don't actually write out explicitly the steps that you assume we do (at least, I am speaking for myself.) I see the quadratic and I know immeaditley what I need to plug into the QF to solve for x.
 
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  • #32
Assuming uniform probability distribution for the co-efficients.

What is the probability that you have something that factors nicely?

If it is more than 0 + delta(e) where delta(e) goes to zero, I would be surprised.

Attempting to factor is a waste of time.
 
  • #33
agentredlum said:
Assuming uniform probability distribution for the co-efficients.

What is the probability that you have something that factors nicely?

If it is more than 0 + e where delta(e) goes to zero, I would be surprised.

Attempting to factor is a waste of time.

Factoring is equivalent to solving the quadratic formula. Your point is moot.
 
  • #34
micromass said:
Factoring is equivalent to solving the quadratic formula.

Be that as it may, that's neither here nor there.

Factoring is not practical.
 
  • #35
agentredlum said:
Be that as it may, that's neither here nor there.

Factoring is not practical.

Proof for that, please?
 
  • #36
micromass said:
Proof for that, please?

Answer the question
 
  • #37
agentredlum said:
Answer the question

You didn't ask any question.
You made an assertion, which I ask you to prove.
 
  • #38
micromass said:
You didn't ask any question.
You made an assertion, which I ask you to prove.

Here it is AGAIN

What is the probability that you have something that factors nicely?
 
  • #39
agentredlum said:
Here it is AGAIN

What is the probability that you have something that factors nicely?

That probability is 1. I can factor each quadratic equation.

Your turn: prove your previous assertion.
 
  • #40
agentredlum said:
Assuming uniform probability distribution for the co-efficients.

What is the probability that you have something that factors nicely?

If it is more than 0 + delta(e) where delta(e) goes to zero, I would be surprised.

Attempting to factor is a waste of time.

This:
"If it is more than 0 + delta(e) where delta(e) goes to zero"
doesn't even make sense. What do you mean by "goes to zero".

Aside from that, let's just say that the coefficients of a general quadratic equation have a uniform distribution across the reals (in text-book problems, I'd bet $1 Million that this is not the case) and that I just won't try to factor the quadratic into linear factors without using the QF. You are still missing the point that it is much easier for me (and I'd imagine for people who have been using the QF for more than 3 weeks) to look at the equation and determine without writting anything down what I need to put into my QF than it is to say "gee, ok, let me see, this is the first form which means I need to use this QF."

Again, your QF is simply a minor algebraic tweak (AS I PROVED) of the normal QF. You have done nothing.
 
  • #41
micromass said:
That probability is 1. I can factor each quadratic equation.

Your turn: prove your previous assertion.

Uh-hmmmmm

I think you have some problems understanding the English language.

It is a waste of time answering your challenges, you will disagree anyway.
 
  • #42
agentredlum said:
Uh-hmmmmm

I think you have some problems understanding the English language.

It is a waste of time answering your challenges, you will disagree anyway.

Wow, start to insult me? Way to prove your point! :smile:
 
  • #43
May I ask a question? What math classes have you taken, agentredlum? Have you taken the Calc. sequence, yet?
 
  • #44
Robert1986 said:
This:
"If it is more than 0 + delta(e) where delta(e) goes to zero"
doesn't even make sense. What do you mean by "goes to zero".

Aside from that, let's just say that the coefficients of a general quadratic equation have a uniform distribution across the reals (in text-book problems, I'd bet $1 Million that this is not the case) and that I just won't try to factor the quadratic into linear factors without using the QF. You are still missing the point that it is much easier for me (and I'd imagine for people who have been using the QF for more than 3 weeks) to look at the equation and determine without writting anything down what I need to put into my QF than it is to say "gee, ok, let me see, this is the first form which means I need to use this QF."

Again, your QF is simply a minor algebraic tweak (AS I PROVED) of the normal QF. You have done nothing.

I posted the 'trick'

You keep claiming it's your proof

That's not honest.
 
  • #45
agentredlum said:
I posted the 'trick'

You keep claiming it's your proof

That's not honest.

No, he's claiming that your trick is absolutely trivial and useless. He didn't claim that it's his proof.
 
  • #46
Robert1986 said:
May I ask a question? What math classes have you taken, agentredlum? Have you taken the Calc. sequence, yet?

I have completed calc 1, 2, 3 all A using Stewarts book

completed Linalg with A

Every math course i have taken, i did not get less than A

I worked 10 years as co-ordinator of math learning center at university.

Not only have i helped students but all tutors would come to me when they got stuck.

I have used the QF thousands of times, pen to paper so your comments about 3 weeks and 4th grade math are hurtfull
 
  • #47
agentredlum said:
I posted the 'trick'

You keep claiming it's your proof

That's not honest.

I never ever said I came up with the "trick"; I gave a very simple proof that your trick worked. There is nothing at all dishonest about that. The proof is, in fact, mine.

Who gets credit for the proof of FLT? Fermat or Wiles?

I rest my case.
 
  • #48
micromass said:
No, he's claiming that your trick is absolutely trivial and useless.

That's not the impression one gets when reading his posts.

Whether it's trivial or not is a matter of opinion.

I'm sure someone will find a use for it, especially in computer programming.
 
  • #49
agentredlum said:
I have completed calc 1, 2, 3 all A using Stewarts book

completed Linalg with A

Every math course i have taken, i did not get less than A

I worked 10 years as co-ordinator of math learning center at university.

Not only have i helped students but all tutors would come to me when they got stuck.

I have used the QF thousands of times, pen to paper so your comments about 3 weeks and 4th grade math are hurtfull

What I cannot understand is this: how can somebody who completed calc I,II,III and linear algebra think that your trick is nontrivial and useful?? I really cannot grasp that.
Everybody I know would call your trick a trivial result and wouldn't defend it as hard as you do.

It's like saying that the quadratic equation

ax^2+bx+c=0

has a root given by

\frac{\sqrt{b^2-4ac}-b}{2a}

and somehow claim that above formula does not follow trivially from the quadratic formula and that the formula is useful somehow...
 
  • #50
agentredlum said:
I have completed calc 1, 2, 3 all A using Stewarts book

completed Linalg with A

Every math course i have taken, i did not get less than A

I worked 10 years as co-ordinator of math learning center at university.

Not only have i helped students but all tutors would come to me when they got stuck.

I have used the QF thousands of times, pen to paper so your comments about 3 weeks and 4th grade math are hurtfull

Well, it has been my experience (again, this is just my experience) that after using the QF for about 3 weeks, it is no longer necessary for most students to explicitly write out what they are doing.
 
  • #51
Robert1986 said:
I never ever said I came up with the "trick"; I gave a very simple proof that your trick worked. There is nothing at all dishonest about that. The proof is, in fact, mine.

Who gets credit for the proof of FLT? Fermat or Wiles?

I rest my case.

To be honest you can't mention Wiles without mentioning Fermat

You used the 'trick' i posted without mentioning where you got it.
 
  • #52
Robert1986 said:
Well, it has been my experience (again, this is just my experience) that after using the QF for about 3 weeks, it is no longer necessary for most students to explicitly write out what they are doing.

Well, if you work at a learning center where every day you have to show students STEP BY STEP how to use a formula... you might start thinking about some shortcuts.

My own personal opinion of the matter is...If the people writing the texts knew it could be done that way... they would show it.

It is folly to do with more when you could do with less- Occam's Razor
 
  • #53
agentredlum said:
To be honest you can't mention Wiles without mentioning Fermat
That wasn't my question. My question was who gets credit for the proof. It was Fermat's idea but Wiles gave a proof. The "trick" is your idea, but I gave a proof. Therefore, I get credit (but not much credit because it is, after all, a trivial thing to prove) for the proof I gave. It is pretty simple.


agentredlum said:
You used the 'trick' i posted without mentioning where you got it.

I never use your trick. And it is pretty obvious where I got it, man. I never claimed that I came up with the trick; I only claimed that I gave a proof of it. Do you dispute this? Do you need me to go to the thread and post my proof?

I think it is you who has a difficult time understanding the English language.
 
  • #54
agentredlum said:
If the people writing the texts knew it could be done that way... they would show it.

So you think you are smarter than the people who write the math texts?? Ok...

Anyway, the reason that the original quadratic formula is used because of the definition of a root.
 
  • #55
agentredlum said:
Well, if you work at a learning center where every day you have to show students STEP BY STEP how to use a formula... you might start thinking about some shortcuts.

My own personal opinion of the matter is...If the people writing the texts knew it could be done that way... they would show it.

Yeah, James Stewart missed your little trick; that's why he didn't mention it.


agentredlum said:
It is folly to do with more when you could do with less- Occam's Razor

Exactly. This is why I prefer to remember one QF and not 4.
 
  • #56
micromass said:
What I cannot understand is this: how can somebody who completed calc I,II,III and linear algebra think that your trick is nontrivial and useful?? I really cannot grasp that.
Everybody I know would call your trick a trivial result and wouldn't defend it as hard as you do.

It's like saying that the quadratic equation

ax^2+bx+c=0

has a root given by

\frac{\sqrt{b^2-4ac}-b}{2a}

and somehow claim that above formula does not follow trivially from the quadratic formula and that the formula is useful somehow...

You are using the word trivial to mean useless.

They don't mean the same thing.

The result is trivial in the sense that you get it quickly without higher math involved.

Useless is a matter of opinion.
 
  • #57
agentredlum said:
You are using the word trivial to mean useless.

No, I didn't.
 
  • #58
Robert1986 said:
This is why I prefer to remember one QF and not 4.

You keep saying this, even though I keep correcting you.

I don't use 4 formulas.

I only use 1 formula, the formula that minimizes operations.

You refuse to understand this-Occam's Razor
 
  • #59
micromass said:
No, I didn't.

Many times, in many posts you and others said there was no use for it. Even only as a curiosity it is useful. It has computational advantages as well.

So what are you saying now?

It is trivial but useful?
 
  • #60
agentredlum said:
Many times, in many posts you and others said there was no use for it. Even only as a curiosity it is useful. It has computational advantages as well.

So what are you saying now?

It is trivial but useful?

No, I'm saying it's trivial AND useless. I'm not mixing up the two words.
 

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