A big pillar of Algebra II is the quadratic formula. It's not the first introduction, but quadratics get an exalted position in the course, so there you go. Quadratics as a topic is a bit of a mess. There are so many solution methods that it's no surprise a student would freeze when presented with a problem. Do I need to factor? Could I try to complete the square? Or is a square root simplification all I need?
If that wasn't enough, this giant formula gets thrown at them in a "I know we already went over 5 methods, but this could've done the job the entire time." I paired down the number of solution options in my Algebra II Pivot Project and even then I feel like I might have done too many. However, after I got a feel for the graphing unit, I actually found a way to motivate the use of the quadratic formula.
My students' progression has been 6 weeks of algebra followed by 5 weeks of proving that algebra via graphs, and the last 2 weeks doing special topics in quadratics. Specifically, calculating a vertex, proving the validity of vertex form, and calculating a focal distance. Along the way, we were finding solutions to equations that we did not have the algebra equipment for (not yet anyway):
I used these as an example for what was possible for graphing. But then out of curiosity I wondered, how do you get those intersection points to surface? And then DUH, there was my introduction for the quadratic formula.
How did I miss this for so many years? Well, textbooks do a terrible job of putting context around it:
Every problem has the same terrible set up. Find the roots/zeros/solutions. Why so much emphasis on the zeros? Who cares? [I have another rant about the limited worth of the discriminant, but some other time] To me, this pigeon holes the quadratic formula. It is only useful for finding x-intercepts, say the textbooks. Let it be the nuclear option it was born to be. Don't force a kid to wait and never see the usefulness until doing Heat Transfer homework. The power of showing the method with the picture cannot be understated.
Why is the quadratic formula so valuable? Well, it really does work with everything. And a student should understand this. Case in point: complex numbers. There was a whole discussion panel on the topic at TMC13. I never got the idea until I sat there looking at my graphing example and thought: what happens if you force the issue? What if we try to prove two parabolas intersect when they clearly don't?
It's this kind of experimentation that discovered complex numbers in the first place. Which is such a cool feature of math! Why restrict it? Why is it constrained to the "finding zeros" silo? These silo techniques are what kill student motivation. If it seems like a unique one-off, they will dutifully memorize the formula, answer your silly question about roots and move on. But if they see WHY you need such a thing. That it has general purpose applications and is just an extension of their prior work with equations, that has value. Math courses should have a flow and a universal theme. Too many textbooks have turned Algebra II into a mess of unrelated drive-by topics. The quadratic formula deserves better.