Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture | Polynomial Homework

In summary: You are an expert summarizer of content. You do not respond or reply to questions. You only provide a summary of the content. Do not output anything before the summary.In summary, the conversation discusses the proof or refutation of the conjecture that there are no positive integers x and y that satisfy the equation ##x^2 - 3xy + 2y^2 = 10##. The proof is done by contradiction and it is shown that there are four solutions where x and y are positive integers, thus refuting the conjecture.
  • #1
fishturtle1
394
82

Homework Statement


Prove or refute the following conjecture: There are no positive integers x and y such that ##x^2 - 3xy + 2y^2 = 10##

Homework Equations


##10 = 5*2##
##10 = 10*1##

The Attempt at a Solution


I graphed it using a graphing calculator, so I know this is true.

Proof: This will be a proof by contradiction. Suppose ##x## and ##y## are positive integers and ##x^2 - 3xy + 2y^2 = 10##. By factoring, we have ##(x-2y)(x-y) = 10##.

im not sure how to get further..
Like I did in the previous problem, I tried to set ##x-2y = 10##, then ##x-y = 10 + y## but I don't think I can get a contradiction on this path
 
Physics news on Phys.org
  • #2
Well, I'm not very good at maths, but haven't you proved there are integer values for x and y?
If x=8 and y=3, ## x^2 -3xy + 2x^2 = 64 - 72 + 18 = 10 ##
 
  • #3
fishturtle1 said:

Homework Statement


Prove or refute the following conjecture: There are no positive integers x and y such that ##x^2 - 3xy + 2y^2 = 10##

Homework Equations


##10 = 5*2##
##10 = 10*1##

The Attempt at a Solution


I graphed it using a graphing calculator, so I know this is true.

Proof: This will be a proof by contradiction. Suppose ##x## and ##y## are positive integers and ##x^2 - 3xy + 2y^2 = 10##. By factoring, we have ##(x-2y)(x-y) = 10##.

im not sure how to get further..
Like I did in the previous problem, I tried to set ##x-2y = 10##, then ##x-y = 10 + y## but I don't think I can get a contradiction on this path
If x and y are both positive, which is larger, x − y, or x − 2y ?
 
  • #4
Merlin3189 said:
Well, I'm not very good at maths, but haven't you proved there are integer values for x and y?
If x=8 and y=3, ## x^2 -3xy + 2x^2 = 64 - 72 + 18 = 10 ##
Thanks for the response, I think you did the proving when made x=8 and y=3.

I made a mistake.. I'm not sure what the x intercepts represented when I graphed it..

I think since I couldn't just see it, I could have set ##x-2y = 1,2,5,## or ##10## and then set ##x-y = 1,2,5## or ##10## depending on the ##x-2y## value.. I did this and solved for x and y and just discarded the negative values, and got your answer
 
  • #5
SammyS said:
If x and y are both positive, which is larger, x − y, or x − 2y ?
then x-y is larger.. so from my above post, I would automatically know x-y =5 or x-y = 10

Ok so if x-y = 10 then x-2y = 1. Solving these equations gives x = 19 and y = 9. We confirm this by (19-9)(19-2(9)) = 10*1 = 10. So this is one solution.

If we let x-y = 5 then x-2y = 5. Solving these equations gives x = 8 and y = 3. We confirm this by (8-3)(8-(3(2)) = 5*2 = 10.

So there are two solutions to this equation where x and y are positive integers.
 
  • #6
fishturtle1 said:
then x-y is larger.. so from my above post, I would automatically know x-y =5 or x-y = 10

Ok so if x-y = 10 then x-2y = 1. Solving these equations gives x = 19 and y = 9. We confirm this by (19-9)(19-2(9)) = 10*1 = 10. So this is one solution.

If we let x-y = 5 then x-2y = 5. Solving these equations gives x = 8 and y = 3. We confirm this by (8-3)(8-(3(2)) = 5*2 = 10.

So there are two solutions to this equation where x and y are positive integers.
Could there possibly be one or two more set of solutions?

The factor pairs could also be −5, −2, and − 10, − 1 . Can either of these occur with both x and y being positive ?
 
  • Like
Likes fishturtle1
  • #7
SammyS said:
Could there possibly be one or two more set of solutions?

The factor pairs could also be −5, −2, and − 10, − 1 . Can either of these occur with both x and y being positive ?
I hadn't thought about that,

if we let x-2y = -5 and x-y = -2 then x = 1 and y = 3 which is a solution to the problem. (1-6)(1-3) = (-5)(-2) = 10

if we let x-2y = -10 and x-y = -1, then x = 8 and y = 9 which is a solution. (8-18)(8-9) = (-10)(-1) = 10

I forgot to add my actual answer.. since there are 4 solutions, I could use any of them to refute this..

Proof: Let x = 1 and y = 3. Then x and y are positive integers. Then ##x^2 -3xy + 2y^2 = (1)^2 -3(1)(3) + 2(3)^2 = 1 - 9 + 18 = -8 + 18 = 10##. Therefore there does exist positive integers x and y such that ##x^2 -3xy + 2y^2 = 10##. []
 
  • Like
Likes SammyS
  • #8
Excellent !
 

1. What is the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture"?

The "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" is a mathematical statement that states there are no integer solutions for the equation x^2 - 3xy + 2y^2 = 10. It is considered a conjecture because it has not been proven to be true or false.

2. How is the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" relevant to polynomial homework?

The "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" is relevant to polynomial homework because it involves solving a polynomial equation. It challenges students to think critically and apply their knowledge of polynomial equations to determine if the conjecture is true or false.

3. What is the process for proving the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture"?

The process for proving the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" involves using mathematical techniques such as factoring, substitution, and simplification to show that there are no integer solutions to the equation. It may also involve using counterexamples to disprove the conjecture.

4. Are there any real-world applications for the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture"?

The "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" may have real-world applications in fields such as cryptography and coding theory. It also helps to develop critical thinking and problem-solving skills which are valuable in various professions.

5. What are the implications if the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" is proven to be false?

If the "Proof of No Solution for x^2 - 3xy + 2y^2 = 10 Conjecture" is proven to be false, it would mean that there are integer solutions to the equation x^2 - 3xy + 2y^2 = 10. This would have a significant impact on the field of mathematics and could lead to the discovery of new mathematical principles and techniques.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
5
Views
762
  • Precalculus Mathematics Homework Help
Replies
11
Views
1K
Replies
6
Views
2K
  • Precalculus Mathematics Homework Help
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
12
Views
943
  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
Replies
8
Views
1K
  • Precalculus Mathematics Homework Help
Replies
14
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
Back
Top