Quadratic nightmares

[cjsmith]

How does one solve a quadratic equation quickly? (Without sacrificing accuracy, of course, and without making it more likely that an arithmetic goof will creep in.)

I know how to use the quadratic formula, but I'm slow and methodical. On a test, minutes count. I've been warned that on the upcoming chemistry exam, seconds may count. This one has been announced as "hard" and "if you haven't done all the practice problems three times over, you won't have time to complete the test". It's going to be FULL of this stuff (for equilibrium calculations). I don't have time between now and the test to do all the practice problems three times over, not if I want sleep too.

How do you solve these dang things FAST?

Oooo. Are there calculators that just do this for you, once you've set up the equation? If so, I'd better go get one. If that's what other students have, I'd be a fool to handicap myself.

Comments

[hitchhiker.livejournal.com]

what in chemistry needs quadratic equations to be solved?

[cjsmith.livejournal.com]

Equilibrium calculations. Like so:

A <-> B + C

Kc = [B][C]/[A] = (some number they give you in the problem, say, 3.673 x 10-5)

Start with 0.1M A, no B, and no C. At equilibrium, how much C do you have?
Final concentrations are A = 0.1 - x, B = x, and C = x. Plug that into Kc and solve for x.

Pain in the ASS.

[hitchhiker.livejournal.com]

ah yes. never did like physical chemistry!

[cjsmith.livejournal.com]

I can't imagine people who do chemistry-related stuff for a living are sitting there plugging the quadratic formula into their calculators all day, seeing how fast they can solve these. O'course, that may be where I'm wrong. If that's really what life as a chemist is like, then I'm being tested in the right way.

[brooksmoses]

Hah. Haha. No. (At least, not as far as I am aware, having done some bits of this sort of thing.) For one thing, you'll note that your A <-> B + C equation is a trivially simple system. In reality, you've probably got A <-> B + C, C <-> D + 2E, E + A <-> B + 3G, and so on and so forth. And so you plug all these things into a computer program that solves the whole set simultaneously. There are lots of programs specifically for this purpose.

Actually, it's probably more complicated than that, because likely you're just using the equilibrium concentrations as input into some rate-based model or something like that. And so what you've got is this model that's calculating the behavior of a flame in a car-engine cylinder, and so it's doing equilibrium calculations like that for several tens of thousands of points in the cylinder, over and over again for each timestep as it steps computationally through a few cycles of the engine's rotation.

Also, there are (when you get to more complicated things) rather easier ways to look at this bit of math than trying to solve the rate equations directly. The most annoying point in my Ph.D. qualifying oral exams was realizing that I should have studied them and hadn't (because of not realizing they were in scope), and having to say, "I know there's this easier way, but I've forgotten it, so I'll have to do this the hard way" -- to the fellow who had invented the easier way.