How many integer pairs \((x,y)\) satisfy \(x^{2}\)\(+\)\(4y^{2}\)\(-\)\(2xy\)\(-\)\(2x\)\(-\)\(4y\)\(-\)\(8\) \(=\) \(0\)? Also,how can we solve this using the method of 'completing the squares'?

Completing the square in \(x\), and then completing the square in \(y\), this equation becomes
\[ \begin{array}{rcl}
(x-y-1)^2 + 3y^2 - 6y - 9 & = & 0 \\
(x-y-1)^2 + 3(y-1)^2 & = & 12
\end{array} \]
Thus \(|y-1| = 1\) or \(2\), and we obtain the six solutions \((4,3)\), \((0,-1)\), \((6,2)\), \((4,0)\), \((0,2)\) and \((-2,0)\).

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## Comments

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TopNewestCompleting the square in \(x\), and then completing the square in \(y\), this equation becomes \[ \begin{array}{rcl} (x-y-1)^2 + 3y^2 - 6y - 9 & = & 0 \\ (x-y-1)^2 + 3(y-1)^2 & = & 12 \end{array} \] Thus \(|y-1| = 1\) or \(2\), and we obtain the six solutions \((4,3)\), \((0,-1)\), \((6,2)\), \((4,0)\), \((0,2)\) and \((-2,0)\).

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Thanks a lot.

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