How can this equation be solved?

Question

I have no idea how to solve this equation.

$$x^2y^2+324y^2+64x^2-36xy^2-16x^2y+144xy = 0 $$ Sorry $x,y \in \mathbb{+Z}$

Answer 1

I am just converting @labbhattacharjee comment into an answer.

$x^2y^2+324y^2+64x^2−36xy^2−16x^2y+144xy$

$=x^2y^2+324y^2+64x^2−36xy^2−16x^2y+144xy+144xy-144xy$

$=x^2y^2+324y^2+64x^2−36xy^2−16x^2y+288xy-144xy$

$=(xy)2+(18y)2+(8x)2−2xy⋅18y−2⋅xy⋅8x+2⋅18y⋅8x-144xy$

$=(8x+18y-xy)^2-144xy$

Equating this to $0$, we get

$(8x+18y-xy)^2=144xy$

Can you take it from here?

Answer 2

Consider the equation as a quadratic in $y$, we obtain

$$ y^2 (x^2-36x+324)+y (144x-16x^2) + 64x^2 = 0 $$

First, check that if the coefficient of $y^2$ is 0, then we must have $ x = 18$, which gives $y=8$ (1 solution).

Otherwise, using the quadratic equation, we get that (for $x\neq 18$),

$$y = \frac { 8(x^2-9x \pm 3\sqrt{x^2( 2x - 27)})} {(x-18)^2} $$

Hence, we must have $2x-27 = (2a+1)^2$ for some non-negative $a$. This gives $x = 2a^2 + 2a + 14$. Substituting it into $y$ above, we get

$$ \begin{array} {l l }y & = \frac{ 8 (2a^2 + 2a + 14) ( 2a^2 + 2a + 14 - 9 \pm 3 (2a+1) ) } { (2a^2+2a-4)^2} \\ & = \frac{ 4 (a^2 + a + 7) ( 2a^2 + 2a + 14 - 9 \pm (6a+3) )}{[(a-1)(a+2)]^2} \end{array}$$

Let's first deal with the $+$ case. The numerator actually will cancel out with $(a+2)^2$ in the denominator. Then, using partial fractions, we get

$$ y = 8 + \frac{24}{a-1} + \frac{72}{(a-1)^2}$$

Since $72 = 2^3 \times 3^2$, this means that $(a-1) = -1, 1, 2, 3, 6$.

The $-$ case is similar, and left to you.