Our result is something very alien. A multivariable equation that graphing tells us is our ellipse, but the original features of that ellipse have been buried. Is it possible to reverse this process? Can we build on what we know about the format of an ellipse equation to “undo” all of that multiplying?
The discussion here is fascinating as we dissect each piece. Fresh off the expansion, students start to see various elements pop back out, and they have an intuition about what could be repurposed to help restore our equation.
First students find a common factor for the x and y terms. Right away they notice these numbers look familiar. How did a random 10 and 3 find its way to the front? They must have originated from the denominator. And aren’t my quadratics “missing” a term? Is there something we remember about how these quadratics were constructed that could offer a hint? And what about the 187? We remember it was the combination of two constants, but how in the world can we split it correctly? Do we have any direction on that?