# Integer solutions to ax^2 + bx - cy^2 - dy = 0

• Puma
In summary, the conversation discusses a method for finding solutions to an equation with integer variables. The suggested method involves looking at the numbers produced by the equation and finding a relationship between them. The conversation also mentions the use of modular arithmetic and suggests a way to find solutions for y as well. The conversation concludes with a discussion about the set of solutions.
Puma

## Homework Statement

I am a hobbyist looking for solutions to ax^2 + bx - cy^2 - dy = 0 where all variables are integers and are non-zero. Is there a method of doing this effectively?

## The Attempt at a Solution

I can look at the numbers produced by ax^2 + bx vs cy^2 + dy and see that they have a relationship: what I mean is if I manually find a pair of close numbers, difference = d, I find the next set of values is d+2 apart, then d + 4 and so on. So it looks as though there should be a method in algebraic terms for doing this.

Of course, the trivial solution is (0, 0). Otherwise, rewrite the equation as $x(ax+b)-y(cy+d)=0$.

Puma
Sorry I know the variables a, b, c, and d but I don't know x, y. It looks as though I might be able to do something with modular arithmetic given that both x(ax+b) and y(cy+d) now seem to both be integer multiples in other words either x or ax+b must necessarily contain some factors in common with y and cy+d. Is there a good way to find x,y? Thanks!

http://www4a.wolframalpha.com/Calculate/MSP/MSP100420ag0a9de184i4e300006aa5e486371cg88e?MSPStoreType=image/gif&s=23&w=258.&h=46.
You can find solutions of y in the same way. Its a bit silly though as it requires you to know all but x. You can see intuitively the set of solutions from the form Svein put it in
x(ax+b)−y(cy+d)=0
Yeah you could do what your saying and then write out the set of solutions

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Puma

## 1. What are integer solutions to the equation ax^2 + bx - cy^2 - dy = 0?

Integer solutions to this equation are values of x and y that satisfy the equation and are also whole numbers. In other words, when substituted into the equation, x and y result in an integer value on both sides.

## 2. How do I find integer solutions to this equation?

One way to find integer solutions is by using a technique called "completing the square." This involves manipulating the equation into the form (x + m)^2 = n, where m and n are integers. By solving for x in this form, you can find integer solutions for x and then solve for y using the original equation.

## 3. Can all quadratic equations with integer coefficients have integer solutions?

No, not all quadratic equations with integer coefficients have integer solutions. For example, the equation x^2 + 2x + 1 = 0 has a solution of x = -1, which is an integer. However, the equation x^2 + 2x + 2 = 0 has no integer solutions.

## 4. Are there any patterns or rules that can help me find integer solutions?

Yes, there are some patterns and rules that can help you find integer solutions. For example, if the coefficient of the x^2 term is a perfect square and the constant term is a perfect square, then the equation is likely to have integer solutions. Additionally, if the coefficients of the x and y terms are relatively prime (have no common factors), then the equation is more likely to have integer solutions.

## 5. Can I use a graphing calculator to find integer solutions?

Yes, you can use a graphing calculator to find integer solutions. By graphing the equation and using the "table" function, you can find integer values of x and y that satisfy the equation. However, this method may not always work for more complex equations or equations with large coefficients.

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