Question 83213
I assume you mean {{{3 * x ^ 2 + 4 * y ^ 2 + 1 8 * x - 3 2 * y - 5  =0}}}


{{{3 * x ^ 2 + 4 * y ^ 2 + 1 8 * x - 3 2 * y - 5  =0}}} Start with the given conic




{{{3 * x ^ 2 + 4 * y ^ 2 + 1 8 * x - 3 2 * y = 5 }}}Add 5 to both sides




{{{( 3 * x ^ 2 + 1 8 * x ) + ( 4 * y ^ 2 - 3 2 * y ) = 5 }}} Group like terms





{{{3 ( x ^ 2 + 6 * x ) + 4 ( y ^ 2 - 8 * y ) = - 5 }}} Factor a 3 out of the first parenthesis
and factor a 4 out of the second parenthesis.





Now we must complete the individual squares inside the parenthesis:




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 For instance to complete {{{3 ( x ^ 2 + 6 * x )}}}, take half of 6 and square it (ie {{{(6/2)^2=(3)^2=9}}} to get 9. Now add 9 inside the parenthesis like this: 

{{{3 ( x ^ 2 + 6 * x + 9 )}}}

Since you really added {{{3*9}}} to the entire left side, we must add {{{3*9}}} to the right side also. Now lets complete  {{{4 ( y ^ 2 - 8 * y ) }}}


In order to complete {{{4 ( y ^ 2 - 8 * y ) }}}, take half of -8 and square it (ie {{{(-8/2)^2=(-4)^2=16}}} to get 16. Now add 16 inside the parenthesis like this: 

{{{4 ( y ^ 2 - 8 * y + 1 6 ) }}}

Since you really added {{{4*16}}} to the entire left side, we must add {{{4*16}}} to the right side also.


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{{{3 ( x ^ 2 + 6 * x + 9 ) + 4 ( y ^ 2 - 8 * y + 1 6 ) = - 5 + 9 * 3 + 1 6 * 4 }}} Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).


{{{3 ( x ^ 2 + 6 * x + 9 ) + 4 ( y ^ 2 - 8 * y + 1 6 ) = 8 6 }}} Combine like terms on the right side





{{{3 ( x + 3 ) ^ 2 + 4 ( y - 4 ) ^ 2 = 8 6 }}} Factor the individual groups on the left side


{{{( 3 ( x + 3 ) ^ 2 + 4 ( y - 4 ) ^ 2 ) / 8 6 = 8 6 / 8 6 }}} Divide both sides by 86





{{{3 ( x + 3 ) ^ 2 / 8 6 + 4 ( y - 4 ) ^ 2 / 8 6 = 1 }}} Break up the fraction into standard form {{{(x-h)^2/a^2+(y-$k)^2/b^2=1}}}


So the equation is an ellipse where the center is (-3,4) 

Here's the graph

{{{drawing (500,500,-10, 10, -10, 10,
   ellipse(-3,4,2*5.35412613473634,2*4.63680924774785),
   circle(-3,4,0.08),
   circle(-3,4,0.05),
   graph( 500, 500, -10, 10, -10, 10,0)
   )
}}} graph of {{{3 ( x + 3 ) ^ 2 / 8 6 + 4 ( y - 4 ) ^ 2 / 8 6 = 1 }}} with the center (-3,4)