mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2004-07-30, 00:35   #1
dsouza123
 
dsouza123's Avatar
 
Sep 2002

2·331 Posts
Default Equality of quadratic equations

Does anybody know what math techniques are available to solve
two quadratic equations for equality ?

The only one I am aware of is substituting in ordered pairs until
the result of the two equations are equal.

An example:

What ordered pair y,x solves y^2 + 1393y + 463 = x^2 + 6x + 407

What (if any) direct methods exist ?

There is also the issue of how many solutions 0, 1, 2+, infinite ?

If the answer involves software, what program is required ?
If it involves a branch of mathematics, what one(s) ?

I tried graphing (in a spreadsheet), but didn't find any solutions,
maybe there is a specific technique to use that I'm not aware of.
Different software ?

By plugging in the following ordered pair a solution is found,
y = 3550
x = 4186
a solution with both quadratics equal = 17548119

There is at least one more solution for this particular example.
dsouza123 is offline   Reply With Quote
Old 2004-07-30, 02:44   #2
jfollas
 
Jul 2004

168 Posts
Default

Check out Dario Alpern's "Generic Two Integer Variable Equation Solver":

http://www.alpertron.com.ar/QUAD.HTM

He offers a lot of background information about this subject.

BTW: You might not have typed your equation correctly because your solution doesn't appear to work. See the following solution generated by his applet.

============= APPLET OUTPUT (STEP-BY-STEP MODE) ====================

x^2 - y^2 + 6 x - 1393 y - 56 = 0
by Dario Alejandro Alpern

First of all we must determine the gcd of all coefficients but the constant term, that is: gcd(1, 0, -1, 6, -1393) = 1.

Dividing the equation by the greatest common divisor we obtain:
x^2 - y^2 + 6 x - 1393 y - 56 = 0

We try now to solve this equation module 9, 16 and 25.

There are solutions, so we must continue.

We want to convert this equation to one of the form:
x´^2 + B y^2 + C y + D = 0

Multiplying the equation by 4:
4 x^2 - 4 y^2 + 24 x - 5572 y - 224 = 0

4 x^2 + ( 24)x + ( - 4 y^2 - 5572 y - 224) = 0

To complete the square we should add and subtract:
( 6)^2

Then the equation converts to:
( 2 x + 6)^2 + ( - 4 y^2 - 5572 y - 224) - ( 36) = 0

( 2 x + 6)^2 + ( - 4 y^2 - 5572 y - 260) = 0

Now we perform the substitution:
x´ = 2 x + 6

This gives:
x´^2 - 4 y^2 - 5572 y - 260 = 0

Multiplying the equation by -1:
- x´^2 + 4 y^2 + 5572 y + 260 = 0

-x´^2 +( 4 y^2 + 5572 y) + 260 = 0

-x´^2 +((-2)^2 y^2 + 2*(-2)*(-1393) y) + 260 = 0

Adding and subtracting (-1393)^2:
-x´^2 +((-2)^2 y^2 + 2*(-2)*(-1393) y + (-1393)^2) + 260 - (-1393)^2 = 0

-x´^2 +( - 2 y - 1393)^2 - 1940189 = 0

Making the substitution y´ = - 2 y - 1393:
- x´^2 + y´^2 - 1940189 = 0

( y´ + x´) ( y´ - x´) = 1940189

Now we have to find all factors of 1940189.


Since 1940189 is equal to 1 times 1940189, we can set:
y´ + x´ = 1
y´ - x´ = 1940189
x´ = -970094
y´ = 970095
x = -485050
y = -485744

and also:
x = 485044
y = -485744

and also:
x = -485050
y = 484351

and also:
x = 485044
y = 484351


Since 1940189 is equal to 163 times 11903, we can set:
y´ + x´ = 163
y´ - x´ = 11903
x´ = -5870
y´ = 6033
and also:
x = -2938
y = -3713

and also:
x = 2932
y = -3713

and also:
x = -2938
y = 2320

and also:
x = 2932
y = 2320
jfollas is offline   Reply With Quote
Old 2004-07-30, 09:03   #3
dsouza123
 
dsouza123's Avatar
 
Sep 2002

2×331 Posts
Default

Thank You

I did have a typing error, + 463 should have been + 469
dsouza123 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Basic Number Theory 18: quadratic equations modulo n Nick Number Theory Discussion Group 4 2017-03-27 06:01
MPQS b-parameter and Pell-like equations Till Abstract Algebra & Algebraic Number Theory 13 2017-01-20 21:12
Can anyone explain/prove this equality? davieddy Math 9 2009-11-07 07:42
solving 2nd order differential equations Joshua2 Homework Help 9 2009-10-30 07:37
Navier-Stocks equations mfgoode Math 1 2006-10-09 16:02

All times are UTC. The time now is 08:30.


Tue Jan 18 08:30:46 UTC 2022 up 179 days, 2:59, 0 users, load averages: 1.07, 1.00, 1.03

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔