write the equation 9x^2+4y^2+18x-16y-11=0 in standard form

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First we get the x variables and the y variables separated.

`9x^2 + 18x + 4y^2 - 16y = 11`

Now we need to complete the square, first we factor the leading coefficients

`9(x^2 + 2x) + 4(y^2 - 4y) = 11`

Now complete both squares with half the middle term squared

`9(x^2 + 2x + 1 - 1) + 4(y^2 -4y + 4 - 4) = 11`

We had to add and subtract so we do not change the equation, now we distribute to get

`9(x^2 + 2x + 1) - 9 + 4(y^2 - 4y + 4) - 16 = 11`

Complete the square and move the constants to the other side and we get

`9(x+1)^2 + 4(y-2)^2 = 11 + 16 + 9`

Now we simplify and put in standard form

`9(x+1)^2 + 4(y - 2)^2 = 36`

Divide both sides by 36

`(x+1)^2/4 + (y-2)^2/9 = 1`

This is an elipse center at (-1,2) major axis (y) is 9 and minor axis (x) is 4

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