Questions like 10.3 #41...

There are a few of these where we are meant to simplify equations like 41x^2-24xy+34y^2-25=0 and then graph them. It's possible to do this if you pretend the equation is all one big quadratic in x or y (for example, if this one was in the form Ax^2+Bx+C=0, A=41, B=-24y, and C=(34y^2-25)). However, it's easiest to just solve them on the calculator for y and then graph the resulting solutions in function mode (there should always be two values for y, so graph both). #s 41-49odd are all either ellipses, parabolas, or hyperbolas, and they are all rotated. However, I had trouble figuring out what the shapes were until I graphed them. CJ explained some sort of trick below for determining the shape before graphing, but I don't really understand how finding the discriminant tells you the shape... does anyone have a different explanation?