With a known format in mind, students can recognize not only what the “missing” term in the expanded binomial is, but how to “unexpand” that binomial back into (x+a)^2 format. By knowing the features of the equation we’re trying to create, it helps fill in some of the blanks about how to work backwards to that point. They develop the algorithm for complete the square on their own, many observing that the square of half the middle term is the number they need. At present we are missing denominators, but we know multiplication was used to clear them, so division can restore them. Working backwards from our expanded quadratic to its origin binomial yields the other version of our ellipse equation.
Struggle is minimum, complete the square isn’t a magic trick, and students have a better grasp of reversibility.