How To Solve Quadratic Equation With Integer X & Y Solution

[april19]

To all,
Thank you in advance for any help you can provide.
I am not sure if this question belongs here. If not, please let me know where I should post this question.

y = (1/2) x² - (1187/34) x + 602/17
The solution for x and y must be integers. By plugging in numbers, I found out one solution to be (69,7).

Is there a better way than plugging in numbers to find integer (x,y) solutions?

 

[HallsofIvy]

You could use the quadratic formula. Since the denominator will be "2a" and, here, a= 1/2, that will be
$\displaystyle \dfrac{1187}{34}\pm\sqrt{\frac{1408969}{1156}- \frac{1204}{17}+ 2y}$

Of course, in order that this be an integer, the "discriminant", inside the square root, must be a perfect square. Further, it is useful to note that 1156= 34 squared and 34= 2(17).

 

[april19]

Thank you for your post.

After playing around with your formula,



I ruled out the minus in the formula because the answer becomes negative as y gets bigger.
There are a few things I notice here:
- Like you said the "discriminant" has to be a perfert square
- the square root of that number has to be an odd number
- the numerator has to be divisible by 34

I still end up with one formula with 2 variables (x,y).

Any suggestion on how to solve x and y without guessing?

 

[JeffM]

Thank you for your post.

After playing around with your formula,

View attachment 3484

I ruled out the minus in the formula because the answer becomes negative as y gets bigger.
There are a few things I notice here:
- Like you said the "discriminant" has to be a perfert square
- the square root of that number has to be an odd number
- the numerator has to be divisible by 34

I still end up with one formula with 2 variables (x,y).

Any suggestion on how to solve x and y without guessing?

I do not get 1156y

$\displaystyle y = (1/2)x^2 - (1187/34)x + (602/17) \implies 34y = (34x^2/2) - (34 * 1187x)/34 + (34 * 602 / 17) = 17x^2 - 1187x + 1204 \implies$

$\displaystyle 17x^2 - 1187x + (1204 - 34y) = 0 \implies x = \dfrac{- (-1187) \pm \sqrt{(-1187)^2 - 4(17)(1204 - 34y)}}{2 * 17} \implies$

$\displaystyle x = \dfrac{1187 \pm \sqrt{1408969 - 81872 + 2312y}}{34} = \dfrac{1187 \pm \sqrt{1327097 + 2312y}}{34}$

 

[lookagain]

I do not get 1156y


$\displaystyle x = \dfrac{1187 \pm \sqrt{1408969 - 81872 + 2312y}}{34} = \dfrac{1187 \pm \sqrt{1327097 + 2312y}}{34}$

I confirmed this equation above (not counting any possible restrictions on any solutions regarding the plus or minus sign).

 

[april19]

Oops. You are right. I missed a step to multiply y by 2. Sorry.

Here is the part I don't understand. I hope someone can explain it to me or point me to some website for an explanation.

I know if I plug 1 or 7 in y in 1327097+2312y, the answer is a perfect square but how do I know 1 and 7 are the answers? How do I find these 2 numbers in the first place without plugging numbers in to test it?

Is there a way to mathematically show/prove what y is so that 1327097+2312y is a perfect square?