Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch
Friday, May 15, 2009
The Quadratic Formula
A few weeks ago my eighth-grade daughter learned about the quadratic formula for solving ax2+bx+c=0. She's impressed I can still rattle it off like a song "Negative bee plus or minus the square root of bee squared minus for-ay-see all over two-ay". She then asked the question I love to hear, "How did anyone come up with that formula?" So I decided to show her how to derive it but couldn't figure it out on the spot. How embarrassing! I did manage to figure it out later and showed her the next day.
Her textbook gives conditions to use the quadratic formula.
The first because they haven't learned complex numbers in the eighth grade. How about the second? After all the solution for a=0 is well defined at x=-c/b (assuming b≠0). What happens when you take the limit as a goes to zero? For the square root with sign opposite of b the limit doesn't exist going to plus or minus infinity depending on whether you approach zero from above or below. But for the other square root after applying l'Hôpital's rule and greatly simplifying you do get -c/b. Possibly the most complicated way to solve bx+c=0.