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How to find the all real numbers $y$,$x$ such that :

$8x^2-2xy^2=6y=3x^2+3x^3y^2$

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Use the resultant of $8x^2-2xy^2-6y, 3x^2+3x^3y^2-6y$ with respect to the variable $y$. This gives a polynomial that is satisfied by the $x$-coordinate of any of the common solutions. In this case the resultant factors as $36x^3(x-1)(16x^6+16x^5+24x^4+24x^3+25x^2+10x+10)$. This has the real roots $x=0, x=1$ only. So we get the solutions $(0,0) (1,1)$.

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    To compute the resultant see http://en.wikipedia.org/wiki/Sylvester_matrix2012-08-11