Non-negative Integral solutions

I have a nice question for you all: Find all Non-negative integral solutions to the following equation:

$4x^4+12x^3y+4x^3-7x^2y^2+4x^2y-25x^2-48xy^3-15xy^2-25x-36y^4-18y^3+100y^2+50y=0$

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hint

Calvin Lin Staff - 8 years, 3 months ago

I think there will be infinitely many solution of the form x=2y.........

kiran patel - 8 years, 3 months ago

But you need to show all of them. (2a,a), where a is a natural number, is one possibility. Think of others.

Shourya Pandey - 8 years, 3 months ago

It means that (x-2y) is a factor of the given equation.

Shourya Pandey - 8 years, 3 months ago

1 is a sol.......and also -1 i guess

superman son - 8 years, 3 months ago

Well 1 is a possibility, but not -1, because -1 is a negative integer!!

Shourya Pandey - 8 years, 3 months ago

yeah

superman son - 8 years, 3 months ago

Everyone should try this question.....you may take y=kx and see that for which value of k is the lhs always zero. Then y-kx becomes a factor of the lhs.

Shourya Pandey - 8 years, 3 months ago

are 1 and 0 the only solution?

Ekdeep SIngh Lubana - 8 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$