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

[leback.livejournal.com]

I'm pretty sure that the graphing calculator I had in high school (TI-82?) could be programmed to do this.

I fully expect that you will ace this test, with minutes to spare. :-)

[cjsmith.livejournal.com]

That is awesome. I may need one of those.

I turned in the last test with a few seconds to spare. That one was announced as "easy" and "if you've done the practice problems you should have no trouble". Conceptually, I'm fine with all of this material, but it's clear I'm dead slow.

[leback.livejournal.com]

At the time, my calculator was pretty fancy stuff, but I suspect there are cheaper programmable calculators (or ones that have the quadratic formula built in) out there now. Does the instructor have any rules about what kinds of calculators you're allowed to use for the exam? I assume they don't want students carrying things in which they can store all the information they're meant to have memorized, but maybe this isn't that sort of test...

To some extent, having a solid conceptual handle on the stuff can make up for slowness -- if you only finish half of something, but you still get as many problems right as anyone who handed it in early, you probably still get an A. :-)

[cjsmith.livejournal.com]

There are no rules at all limiting the kinds of calculators one can use. This instructor is the kind who gives us constants, too, so storing those in memory isn't an issue. :-)

Sadly, there's no curve on this; if I get 85% on the test, no matter how many concepts I have down cold, I get a B. It's time to get myself one of those calculators, right bloody now.

[leback.livejournal.com]

Okay, the no-curve thing is nuts. I'm sorry, but after the stories you told about your classmates last quarter, I'm pretty confident that, slow and methodical though you may be, you're going to be at or near the top of the class on this exam. So if you get a B, everybody gets a B (or worse) -- which I know doesn't help you any, but says to me the instructor's doing something wrong.

Anyway, good luck finding an appropriate calculator. I see that there are plenty of 99-cent iPhone apps that will solve quadratic equations, so if you don't come across any other reasonably easy and inexpensive option, you're welcome to borrow my phone for the exam. :-)

[cjsmith.livejournal.com]

I honestly don't know why he's making a test he knows we can't complete in the time given. I do know that he's frustrated that many of the students don't seem to be doing the "homework" (practice problems); maybe he's trying to make the point that doing them -- aka Listening To The Teacher, Thank You -- will help. If that's it, then I really wish he'd chosen another way to make his point. I am doing the work and I'm not pleased to have to face this kind of hurdle just because others aren't listening.

[lizzie-omalley.livejournal.com]

One of the people at the high school I teach at was a doctor in a "former life" and is now teaching a night class at the local Jr. college as well as high school. He was lamenting the other day that the students don't read so they can't keep up with his lectures. I am thinking it's a universal problem in the US these days.

[cjsmith.livejournal.com]

Given what I see around me, I suspect that's right. There are students in my classes who listen, who do the work, and who are doing very well. They're rare. If I were a teacher I'd be at my wits' end with most of the class.

[midendian.livejournal.com]

Graphing calculators are amazing little things. Even a decade ago I had a TI-89 that did symbolic manipulation (ie algebra, without resorting to numbers). It'll solve them faster than you can type them in...

[cjsmith.livejournal.com]

Awesome. Half the students in class have big fat fancy calculators, so I'd wager half the students in class are not punching in "b squared minus 4 a c" etc by hand and potentially goofing it up. I clearly need to upgrade.

[mighty-sam.livejournal.com]

When is the exam? I think I have a graphing calculator I could send you.

Definitely get one if they're allowed. As you say, you're handicapped without it. And make sure you have time to practice with it before the exam. Solving a quadratic equation on one just requires plugging in a few values. Takes seconds.

You can also store all other kinds of useful stuff on them: constants, formulae, whatnot (all of course, within the bounds allowed by the instructor).

[cjsmith.livejournal.com]

Monday at 11:30. Might be a bit late for this one. Maybe someone local has one I could borrow, though.

[mighty-sam.livejournal.com]

I'm sure someone does. Get it and start practicing with it as soon as you can. You'll be glad you did.

[cjsmith.livejournal.com]

Thanks! I bet this is a major clue I've been missing so far. :-)

[lizzie-omalley.livejournal.com]

Of course, I would recomend an HP. I am so used to RPN that I forget to use the = key.

[cjsmith.livejournal.com]

I know what you mean. I went through all my undergraduate years with a HP-15c (well, my first set of undergraduate years). Amazingly, when I pulled it out of the desk drawer about eight months ago, it still worked! I'm using a TI now, but am perfectly comfortable with either flavor.

[lizzie-omalley.livejournal.com]

Yeah. I have a little HP 42s that I keep in my purse. I have long since forgotten how to use it to it's full capacity but love being able to do simple chain problems on it. Somewhere I have a larger graphing calculator that I bought for calculus.

[cjsmith.livejournal.com]

My HP-15c can solve for zeros of an arbitrary function, with the generalized SOLVE capability it has, but entering the information is not at all simple and it can take up to two minutes to iterate among estimates. I *may* be able to program it to do the quadratic formula given three pre-loaded registers. Not sure whether my time is better spent investigating how to write that program or in doing fifty chem problems over and over and getting them wrong if I rush.

[ame-chan.livejournal.com]

Oh, this is good to know. We were not allowed the use of calculators for tests in the class I just finished and I don't think we'll get them for the next one, but it would probably be handy to have one and learn to use it anyway. Chem and ochem and other courses are coming where I might want one.

Having just spent today's final wrestling with #$@#!@ quadratics, I really hear you!!

[cjsmith.livejournal.com]

I guess I'm not too surprised that you're not allowed to use calculators in an algebra or pre-calc class -- the calculators are simply too powerful, so the students wouldn't have to learn algebra. But man, doing that crap by hand is a pain. And it's frustrating as hell to have my grade hinge on being lightning fast with these stupid things when what I'm supposed to be learning is chemistry.

[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.