h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Quadratic Formula - A44008887
Author: Jhawkesby - U13402970

Here is my guide entry on the quadratic formula. It is also my first Flea Market rescue.

Original entry - A2697311
Original author - U556438
Flea Market conversation - F74125?thread=572835
Peer Review conversation - F1827865?thread=455553

It is suppose to help teach someone how to use the quadratic formula and it gives an example. Typing the maths was quite time consuming but I hope it is readable. For anyone who is just quickly reviewing it might seem a bit confusing but when you look at it again it isn't. The / symbol represents division and the brackets help show what is grouped together. If there is anything else to add or change I don't mind.

Well done, JH.

I think you've taken a lot of trouble to write this clearly, and it should be understandable to most people who are interested in this sort of thing.

I would suggest a couple of changes and additions, though...

Before you give the formula, or maybe immediately after, explain what a quadratic equation is in real terms. Give a couple of practical examples (like the geometry one, but I'd use real objects: perhaps you're building an enclosure out of fencing, for example).

Some more excellent practical examples here (projectiles, stopping distances, etc):

http://plus.maths.org/issue30/features/quadratic/index-gifd.html

Oh, you'll love this, too (well I did) - the Hansard record of a House of Commons debate on quadratic equations:

http://www.publications.parliament.uk/pa/cm200203/cmhansrd/vo030626/debtext/30626-21.htm


OK, the next thing is that you don't really mention where the formula does and doesn't work. I would urge you to cover this, as the other entries on the subject don't, for some reason.

1) If b² > 4ac then all is fine and dandy and you get two distinct, real solutions (or 'roots' as the mathematician would call them)

2) If b² = 4ac then you get a repeated root, and there's only one real distinct solution.

3) But if b² < 4ac then your formula tries to calculate the square root of a negative number. Your calculator won't like it. There will be no real solutions, but there will be complex number solutions (see my entry on Complex Numbers for discussion of this, including an example involving a fence)

You can see these three situations if you plot the graph of ax²+bx+c=0

If the graph cuts the x-axis in two places, then it's situation 1. (It cuts it at the solutions to the equation of course)

If the graph curve just touches the x-axis at one place (a tangent) then it's situation 2.

If the graph doesn't touch the x-axis at all, then it's situation 3.

Icy

A25746249 Complex Numbers - An Introduction

Thankyou for reviewing Icy.
I will definitely sort out when it does and doesn't work but first I will sort out giving the examples. Do you mind giving me an example of a sentence I could do because I want it to seem easy to follow and not sound complicated.

I sorted out what you mentioned apart from the graph bit because I am not sure what to do about that at all. I feel as though if I did mention it then it will either be to short for a paragraph if I was going to keep it simple or it will need a few diagrams if I was going to explain it a bit more but I am sure there is a way around this so if anyone can help please. Doing this guide entry is a bit complicated for me because it is my first maths guide entry so I need a bit of help.

You left a 'postive' somewhere instead of 'positive'. Just next to the formula you give in the 'negative' section.

By the way, in the formula from that section, shouldn't it still be b^2-4ac under the square root (instead of -b^2-4ac)?



Shouldn't you include more examples? You tempt the reader with applications in Physics or Biology and then you don't mention them anymore. This gives in the end, to my mind, too much algebraic manipulation and not enough applications.

Sorted out the spelling mistake.

It is suppose to be negative I am sure but if it isn't then it doesn't make sense.

What kind of examples should I put for the Physics and Biology bit.

Oh, but just because you have a 'minus' sign doesn't make a number negative (although it certainly makes things a bit confusing).

I'm positive it should be "+ b^2" under the square root. Indeed you can retrieve the formula from the very first one with (a,-b,c) instead of (a,b,c). Write B = -b, you obtain:

x = -B +/- sqrt(B^2 - 4ac) / 2a

and then you replace -B by -(-b), that is b, and B^2 by (-b)^2, which gives again b^2.



I will try to think of entertaining examples (although I'm not sure I can come up with physics or biology inspired ones).

Ah yes, one from physics: you can compute the trajectory of a golf ball (or a cannon ball if you're feeling warlike). Using Newton's laws of gravity (and assuming negligible air friction), you naturally obtain a quadratic equation.

OK I will take your word for it.
I have changed it. I also noticed it really doesn't matter anyway because whichever it is the answer will be positive.

I wouldn't be so trusting I can't remember a time when I didn't make a computational mistake when posting this sort of things on hootoo. Thank you anyway

Another (a bit more far-fetched) application of the quadratic formula, I think, would be the determination of explicit formulas for recursive sequences defined by u(n+2) = b.u(n+1) + c.u(n), where u(0) and u(1) are given. Like, for example, Fibonacci's sequence.

It might be worth noting that, while the quadratic formula is usually written with a plus-or-minus symbol, this is technically unnecessary, as the square root will have positive and negative solutions anyway.

Root 4 is plus or minus 2.

TRiG.

I prefer the plus/minus symbol here, as the square root is often conventionally taken to mean the positive value in mathematics.

I see what you mean TRiG but here as Icy explains most people do take the positve answer. Anyway it is suppose to be written like that.

As a reply to post 6 can anybody help on what kind of examples shall I put for Physics and Biology.

Also anything else that needs to be done.

Sorry that smiley was suppose to be

Sorry I didn't have much time to look for examples. (Maybe it is worth asking for applications of the formula in <./>askh2g2</.>?)

As I hinted, one application is the study of the trajectory of a little object subject to gravity only, when you know the initial velocity and direction of thrust. (I'm sure there are more official ways of describing the problem - a native English speaker should help, probably).

Here it is a bit too difficult to explain without a picture, so this would just have to be mentioned without actually writing out an example.

Let's see if I can come up with better examples.

Mmh, I found this link to expand on what I said above: http://tinyurl.com/5jzdxg

* for lunch*

Thankyou Toy Box for the link and I will try Ask h2g2 now.
As you say it is a bit hard to give examples without a diagram but I am sure there must be a way.

This is in partial response to a biological example of the quadratic formula.

Suppose you have the seeds from a certain plant, and you'd like to figure out how much water this plant needs to produce its best and healthiest growth. So you go out to your greenhouse and plant the seeds.

One seedling gets 100 micro-liters per day.
One seedling gets one milliliter per day.
One seedling gets 10 milliliters per day.
One seedling gets 100 milliliters per day.
One seedling gets one liter per day.
One seedling gets 10 liters per day.

Keep all other variables (sunlight, soil type, plant food, singing folk ballads to the plants) exactly the same. And track the response of each seedling. Unsurprisingly the first seedling sprouts but almost immediately dies from insufficient water. The second seedling is quite small, the third larger, the fourth larger still, the fifth grows even more quickly but soon dies from root rot, and the sixth is washed away.

After three weeks you graph the size of each plant at 3 weeks, and are happy to see that the dots form some sort of curve. Perhaps you can find an equation that fits the curve. (Hint, hint.) If you do, you have probably found a quadratic equation that models your data.

You can draw a graph with the size of the plant along the vertical axis and the amount of water along the horizontal axis.

Your quadratic equation does more than connect the dots on a graph. It gives a (very rough) prediction of the amount of water producing optimal growth - even if you were not lucky enough to try that amount of water. What is that prediction? It is the amount of water corresponding to the peak of your curve. Further experimentation, that is, trying amounts of water near that predicted optimal amount of water. can help you refine your model and improve your gardening results.



Happy Nerd