Question 269783
Here is our equation
{{{9x^2 + 4y^2 - 36x + 24y + 36 = 0}}}
step 1 - rearrange so that x's are together and y's are together as
{{{9x^2 - 36x + 4y^2 + 24y + 36 = 0}}}
step 2 - complete the square on both x's and y's to get
{{{9(x^2 - 4x + ______) + 4(y^2 + 6y + _____) = -36 + _____ + ______}}}
The blanks are waiting for numbers . . . 
step 3 - take (1/2) middle term and square it. PUt these into the left side of blanks as
{{{9(x^2 - 4x + (-2)^2) + 4(y^2 + 6y + (+3)^2) = -36 + _____ + ______}}}
we added 4 into the x's, but there is a 9 in front, so we really added 36. add 36 to the right side to keep it balanced.
we added 9 into the y's, but there is a 4 in front, so we really added 36. add another 36 to the right side to keep it balanced. we get
{{{9(x^2 - 4x + (-2)^2) + 4(y^2 + 6y + (+3)^2) = -36 + 36 + 36}}}
step 4 - rewrite the left side as 2 binomials squared and simplify the right side as
{{{9(x-2)^2 + 4(y+3)^2 = 36}}}
step 5 - divide by 36 to get
{{{(x-2)^2/4 + (y+3)^2/9 = 1}}}