Conics, standard form

[frusic]

convert 9x^2 + 4y^2 + 54x - 8y +49 = 0

what I have in my notes as the first step is :

9 (x^2 + 6x + 9 - 9) + 4 (y^2 - 2y + 1 - 1) + 49 = 0

except I don't know where I got the + 9 - 9 and the +1 - 1 from??

 

[Gene]

You want ax² so you add 9 to get
x²+6x+9-9 = (x+3)²-9 and
(y-1)²-1
Since you added 9 & 1 you have to subtract them.
Next step is to bring in the multipliers for
9(x+3)² + 4(y-1)²-(9*9)-(4*1)+49 = 0
9(x+3)² + 4(y-1)² = 36
(x+3)²/2²+(y-1)²/3² = 1
One of those last two should be the standard form of an an elipse for which you are looking.