The equation

    29/36 - x/2 + x^2/4 + y/9 + y^2/36 - (4 z)/9 + z^2/9 == 1

is equivalent to

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

This describes an ellipsoid.
Using `Expand` on the second one give us the first one easily. But is there any way to go the other direction in Mathematica?