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Please help in this N.T Prob.

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'?

Note by Bhargav Das
4 years, 1 month ago

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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 \((-2,0)\).

Mark Hennings - 4 years, 1 month ago

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Thanks a lot.

Bhargav Das - 4 years, 1 month ago

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