In the period following the publication of Algebra: Themes, Tools, Concepts, we started asking our Math 1 students to write and illustrate a short report or poster, tying in four representations of a trinomial in the form x^2+bx+c. (Math 1 is Urban School's Math Department's version of Algebra 1, sort of.) I just posted a version of this assignment on my Web site.
One year, while grading this assignment, it occurred to me that it should be possible to use the "constant sums, constant products" setup as a path towards the quadratic formula. And sure enough, it is. I wrote an article about this for the "Delving Deeper" department of The Mathematics Teacher. ("A New Path to the Quadratic Formula", February 2008.) The proof involves neither completing the square, nor parabolas!
A worksheet guiding you through the proof has been on my site for a long time, but I just added a presentation of it on slides. (No audio, but the steps are spelled out sufficiently, I hope.) Let me know if it doesn't work on your computer.
You can download the presentation (3.2MB, in .mov —QuickTime— format) if you want to have it in your own computer. In fact, this version has helpful animations which do not work in the online version.
Finally, as long as I was adding these things, I thought I might as well organize the Parabolas and Quadratics page better, dividing the many links into three approximate groups: Algebra 1, Algebra 2, teachers' mathematics.
I still have a little more to add to the quadratic part of the site, but it will have to wait.