Get the x terms together and the y-terms together and
the constant term on the right
Factor the coefficient of the squared terms out of the first two terms
and the last two terms on the left:
We want to add a positive number inside each parentheses to make
it factorable as a perfect square. To find that number
In the first parentheses,
Half the coefficient of x, which is -8, getting -4, then square
-4 getting +16 and add that inside the first parentheses on the right. Since
there is a coefficient of 36, adding +16 inside the first parentheses is
the equivalent of adding 36x16 or 576 to the left side. So we must add
576 to the right side as well:
In the second parentheses,
Half the coefficient of y, which is +8, getting +4, then square
+4 getting +16 and add that inside the second parentheses on the right. Since
there is a coefficient of 25, adding +16 inside the second parentheses is
the equivalent of adding 25x16 or 400 to the left side. So we must add
400 to the right side as well:
Now we factor each parentheses and combine the numbers on the right side.
We notice that they factored into perfect squares, so perhaps you can skip
the preceding step, and have gone straight to
Now we must get 1 on the right side, so we divide through by 900
The coefficients of the squares of binomials must be 1, so divide
the top and bottom of the first term by 36 and the second by 25.
compare to 
That's a vertical ellipse (looks like the number "0") because the larger
denominator a2 is under the term in y.
It has center (h,k)=(4,-4).
It has semi-major axis a=6, semi-minor axis b=5.
Vertices (4,-10) and (4,2) and co-vertices (-1,-4) and (9,-4).
Edwin
