I need to find ALL INTEGER solutions for
0 = 67291851x^2 + 337901589xy − 79475826y^2 − 74334x + 1287y + 3
i think i broke it down correctly, but stuck
= (2823x + 14811y − 3)(23837x − 5366y − 1)
need (x,y) solutions
If you broke it down correctly then you are left with :
$\displaystyle 2823x + 14811y - 3 = 0$
$\displaystyle 23837x - 5366y - 1 = 0$
(Then solve the system)
EDIT : how could I not think about system of equations ? I lack imagination so badly ...
Hello, AwesomeDesiKid!
Find all integer solutions for:
$\displaystyle 67291851x^2 + 337901589xy - 79475826y^2 - 74334x + 1287y + 3 \:=\:0$
i think i broke it down correctly, but stuck:
. . $\displaystyle (2823x + 14811y - 3)(23837x - 5366y - 1) \:=\:0$
Need (x,y) solutions.
Set each factor equal to zero and solve the system of equations.
(And hope there are integer solutions.)
Nice job factoring. I have no idea how you did that.
You aren't going to get a set (x, y) for these. From the first factor you get
$\displaystyle 0 = 2823x + 14811y - 3$
$\displaystyle y = -\frac{2823}{14811}x + \frac{3}{14811}$
(I'll let you take care of the cancellations.)
This linear relationship gives you a list of zeros of this equation. The other factor gives you a similar solution.
You may, of course, pick out a specific zero. For example, we can pick out x = 0 and we have the zero
$\displaystyle \left ( 0, \frac{3}{14811} \right )$
-Dan
If you're looking at solving your linear Diophantine equations, here's a site that might help
LINEAR DIOPHANTINE EQUATIONS