is an equation. Meaning it has to be solved.
is a function. It represents the relationship between the x-coordinate and y-coordinate.
Diffrence of square means,
and they can always be factored.
Why use perfect squares?
You mean why bring everything to perfect square in a conic? Because after you used perfect squares the shape of the conic is easily recognizable. And also it can be graphed easily by simply doing a translation of the axes.