Solve $7 x^{2} y^{2}+4 x^{2}=77 y^{2}+1260$ for $x, y \in \mathbb{Z}$
Andrew Henderson 
Attempt:
$$4 x^{2}=7(11 y^{2}-x^{2} y^{2}+180)$$
$$\implies x=7u\quad(u\in \mathbb{Z})$$
$$11 y^{2}-49 u^{2} y^{2}+180=28 u^{2}$$
$$y^{2}=\frac{28 u^{2}-180}{11-49 u^{2}} \geqslant 0$$
$$u=0, \pm 1 ; \pm 2 \quad u^{2}=1 \implies y^{2}=4$$
$$ (x, y) \in\{(7,2) ;(-7,-2) ;(-7,2) ;(7,-2)\}.$$
Is my solution correct?
$\endgroup$ 111 Answer
$\begingroup$Your solution would be improved by explaining what your solution is. As it stands, it is simply a sequence of formal expressions. It is not always clear how these relate to each other. The greatest leap is from $y^2\geq0$ to $|u|\leq2$. As a general rule of thumb, every expression you write down is in fact a claim, which usually requires some sort of argument.
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