integral solution

juan · 2011-12-16 01:50:45

find Integral solution of the equation

TheDude · 2011-12-16 02:40:58

Do you mean integer solution?

bobbym · 2011-12-16 03:10:35

Hi juan;

If you want integral solutions there is a rote procedure for finding them. That particular one has no solutions.

juantheron · 2011-12-16 04:35:55

bobbym how can i get integer solution

thanks

bobbym · 2011-12-16 04:52:24

Hi;

As I said there aren't any. This is as much as I know about the mechanical way to solve these.

A=21
B=0
C =-10
F = - 9
B^2 - 4AC = 840

because 4F^2 is less than the discriminant (840) you need the continued fraction of the root of:

Solve

The solutions are among the convergents of the continued fraction of t.

Meaning that (2,4,2,2) repeats.The nice part is that a theorem by
Lagrange assures us that every square root like this will always have a repeating
form.

Compute the convergents which is done by two linear recurrences.

You pick the fourth convergent: 9/13. You try 9 and 13 as solutions, they are not! If they were they would have been the smallest ones. So there are no solutions.

TheDude · 2011-12-16 04:52:28

Try to solve for y:

For y to be an integer we need sqrt( (21x^2 - 9) / 10) to be an integer. We can first see that 21x^2 - 9 must be divisible by 10, so the last digit of 21x^2 must be 9. This occurs whenever the last digit of x^2 is 9, which occurs when the last digit of x is 3 or 7. So we can substitute x = 3 + 10h, x = 7 + 10h where h is any integer. This gives us

Similarly for 7 we get 210h^2 + 294h + 102. Since this must be a perfect square we should be able to factor them into the form (ah + b)^2, but you can see that this is impossible, so there are no solutions.

juantheron · 2011-12-16 05:55:03

thanks bobbym and dude got it

bobbym · 2011-12-16 05:59:19

Hi juantheron;

You are welcome. I think the TheDude's solution is superior in this case. Mine is for the general problem and does not take advantage of some of the properties of this particular case.

TheDude · 2011-12-16 06:17:38

bobbym wrote:

Hi juantheron;
You are welcome. I think the TheDude's solution is superior in this case. Mine is for the general problem and does not take advantage of some of the properties of this particular case.

Ha, and I would say the opposite. General solutions all the way

bobbym · 2011-12-16 07:57:14

Hi;

I remember a funny story about an instructor saying the good thing about the quadratic formula is that it takes about 3 minutes to solve any quadratic. The bad thing about it is, that it takes about 3 minutes to solve any quadratic. Factoring can be much quicker but is not always possible.

Of course he lost his credibility when he tried to prove that all his students when asked to solve (x+3)^2 = 0, expanded it and plugged into the quadratic formula.

Mathsyperson once suggested that one of my solutions used too much heavy machinery. He was right. Confucius says,"Do not use a cannon to kill a mosquito." On that basis mine is a howitzer.

Math Is Fun Forum

#1 2011-12-16 01:50:45

integral solution

#2 2011-12-16 02:40:58

Re: integral solution

#3 2011-12-16 03:10:35

Re: integral solution

#4 2011-12-16 04:35:55

Re: integral solution

#5 2011-12-16 04:52:24

Re: integral solution

#6 2011-12-16 04:52:28

Re: integral solution

#7 2011-12-16 05:55:03

Re: integral solution

#8 2011-12-16 05:59:19

Re: integral solution

#9 2011-12-16 06:17:38

Re: integral solution

#10 2011-12-16 07:57:14

Re: integral solution

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