Question 83203
{{{3x^2+4y^2-6x+4y-4=0}}}


{{{3x^2+4y^2-6x+4y=4}}} add 4 to both sides


{{{(3x^2-6x)+(4y^2+4y)=4}}} Group like terms


{{{3(x^2-2x)+4(y^2+y)=4}}} Factor out the GCF


{{{3(x^2-2x+1)+4(y^2+y+1/4)=4+3+1}}} Complete the squares of each parenthesis (remember to add to both sides)


Since we completed the squares, we get


{{{3(x-1)^2+4(y+1/2)^2=8}}}


{{{(3(x-1)^2+4(y+1/2)^2)/8=8/8}}} Now divide both sides by 4 to get the equation into standard form {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}


{{{3(x-1)^2/8+(y+1/2)^2/2=1}}}


So the equation is actually an ellipse (check your problem again) where the radii are {{{sqrt(8/3)}}} and {{{sqrt(2)}}} and the center is (1,{{{-1/2}}})