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Question Number 187 by 123456 last updated on 25/Jan/15
solve for integer x,y 4x^2 −9y^2 =6
$$\mathrm{solve}\:\mathrm{for}\:\mathrm{integer}\:{x},{y} \\ $$$$\mathrm{4}{x}^{\mathrm{2}} −\mathrm{9}{y}^{\mathrm{2}} =\mathrm{6} \\ $$
Answered by ghosea last updated on 16/Dec/14
(2x−3y)(2x+3y)=2∙3 or 1.6 or −1,−6 or−2,−3 since (2x−3y) and (2x+3y) are both integers both must be a factor of 6, like 1,6 or 2,3 none of the factors give an integer solution for x and y so the given equation has no integer solution.
$$\left(\mathrm{2}{x}−\mathrm{3}{y}\right)\left(\mathrm{2}{x}+\mathrm{3}{y}\right)=\mathrm{2}\centerdot\mathrm{3}\:{or}\:\mathrm{1}.\mathrm{6}\:{or}\:−\mathrm{1},−\mathrm{6}\:{or}−\mathrm{2},−\mathrm{3} \\ $$$$\mathrm{since}\:\left(\mathrm{2}{x}−\mathrm{3}{y}\right)\:{and}\:\left(\mathrm{2}{x}+\mathrm{3}{y}\right)\:\mathrm{are}\:\mathrm{both}\:\mathrm{integers} \\ $$$$\mathrm{both}\:\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{6},\:\mathrm{like}\:\mathrm{1},\mathrm{6}\:{or}\:\mathrm{2},\mathrm{3} \\ $$$$\mathrm{none}\:\mathrm{of}\:\mathrm{the}\:\mathrm{factors}\:\mathrm{give}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{solution}\: \\ $$$$\mathrm{for}\:{x}\:{and}\:{y} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{given}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{no}\:\mathrm{integer}\:\mathrm{solution}. \\ $$$$ \\ $$

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