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I am looking for good motivational examples of how quadratic equations can naturally arise in real life for someone starting high school. The high school book my child is using just jumps into factoring and solving them without any clear purpose. The few motivations seem to be contrived, such as the flight of a ball.

A related question I found is the following: Everyday Example Problems for Solving Linear and Quadratic Equations. My question is different because I am looking for motivations.

A possible starter example:

A farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? [1]

The only concern I have here is that the quadratic function does come up but the solutions to the obvious quadratic equation are not immediately intuitive.

Ideally an example of how they arose in antiquity would be great.

  1. https://studentsuccess.asu.edu/sites/default/files/optimization.pdf
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  • $\begingroup$ I'm not sure I understand what you want. Are you looking for simple optimization problems where you want to minimize/maximize $f(x)=ax^2+bx+c$? $\endgroup$
    – user5402
    Aug 30, 2018 at 14:41
  • $\begingroup$ I am not looking for optimization problems. The example I gave happened to have optimization, but my point is that this is a problem which can arise naturally, giving rise to quadratic functions. Linear functions and equations arise very naturally in real life: one egg is $1, how much to feed a family of 8? $\endgroup$
    – John Smith
    Aug 30, 2018 at 14:48
  • $\begingroup$ There's many in college algebra or pre-calculus books like this one. I don't know if you consider it natural enough. $\endgroup$
    – user5402
    Aug 30, 2018 at 14:56
  • $\begingroup$ There are hundreds (maybe thousands) of school-level algebra books from the 1800s that are freely available on the internet. In these books where quadratic equations appear, look at the sets of exercises labeled "problems" (or "applications", or some other similar word). A few hours of this one Saturday afternoon should provide you with more examples than you could ever use in a class. $\endgroup$
    – Dave L Renfro
    Aug 30, 2018 at 18:12
  • $\begingroup$ I've enjoyed thinking about fireworks, e.g., "Parabolic envelope of fireworks." $\endgroup$
    – Joseph O'Rourke
    Aug 30, 2018 at 22:12

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The few motivations seem to be contrived, such as the flight of a ball.

...

Ideally an example of how they arose in antiquity would be great.

The latter question might be better addressed on History of Science and Mathematics, but it seems quite likely that they arose in antiquity largely to tackle such issues as... the flight of a ball. War is a powerful driver of technology, and understanding the trajectory of missiles launched from a catapult would improve the effectiveness of your artillery.

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  • $\begingroup$ The popularity of trebuchets in our high school shop class show that artillery is indeed motivational for many high school students. $\endgroup$
    – Rory Daulton
    Sep 1, 2018 at 11:00
  • $\begingroup$ flight of a ball (ballistics parabola) is definitely not contrived. Is one of the most common applications (artillery). How objects move in a gravity field is a normal common physical problem $\endgroup$
    – guest
    Sep 1, 2018 at 12:28
  • $\begingroup$ @guest, yes, that was my point. The stuff on a yellow background is quoted from the question. $\endgroup$
    – Peter Taylor
    Sep 1, 2018 at 13:19
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    $\begingroup$ Yeah we are on the same side. I just wouldn't even say "from antiquity". As a served naval officer. All the new cruisers and destroyers still have 5" 54 guns on them. Sometimes I think people on this site think everyone designs i-phones and that all the meat in the supermarket was born in plastic packaging. Lots of people laying pipe, dropping bombs, drilling for oil, building chemical plants, yada yada. $\endgroup$
    – guest
    Sep 1, 2018 at 13:24
  • $\begingroup$ @guest, the quadratic equation hasn't been used for artillery in my lifetime or yours. One of the earliest uses of electronic computers was calculating range tables, and before that they were done by hand. To get decent accuracy at the muzzle velocities achievable with gunpowder you need to take into account air resistance. Going on the values from Wikipedia, without air resistance your 5" 54 would have a range of about 59km rather than its actual effective range of about 24km. $\endgroup$
    – Peter Taylor
    Sep 2, 2018 at 20:09
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Note mathematics are models of the real world.

What makes quadratic equations nice is they either have a maximum or minimum. So for many problems we are trying to solve; our systems have maxima and minima. a quadratic representation lets us model such a system's behavior close to the maximum or minimum value.

In trying to help my students to understand the motivation for various models, I will sometimes show curve fitting (done via software) to real data. This illustrates real data can be noisy and that our models may not be exact but can be good enough to predict from the data.

As an aside MAA has a nice textbook - Functions, Data, and Models An Applied Approach to College Algebra, that explains from this perspective, motivating the algebra from how we have to apply it to real world data.

So when you look back at the flight of the ball the quadratic form happens naturally, a ball will rise to a maximum point fighting gravity until gravity stops its upward motion and then fall towards the ground with gravity. Since gravity works in the up and down direction (y or h) in the equation this motion will be symmetric around that point. This language is also how we would describe a graph of a quadratic equation. If you progress to calculus you can derive the quadratic nature of this system by knowing that the acceleration due to gravity is constant. But that is another discussion for a later time.

Here are some resources about the thought process of choosing the model. Once a model is selected you use the tools of algebra to bring out all the predictive power of that model.

CK-12 Linear, Exponential, and Quadratic Models

Monterey Institute Algebra Course

Math is Fun Examples

Note many of the resources you find will start with the flight of a projectile because this is a real life experience that is common to many people so it is used to bridge conceptually from the formula to application.

To Specifically answer your questions about the historical motivation:

University of Virginia Math History

Mathnasium a nice summary timeline

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I think something a little different that what's normally done in most high school classes is finding equilibria in second order difference equations (but it certainly can be done if one discusses recursive functions as models).

Let me give an example: suppose you have a population of fish that grow logistically (this can be explained to a high school student without too much effort) with a carrying capacity of 20000 and a growth rate of .02. Then the population $P(t)$ at time $t$ years can be given by $$P(t) = P(t-1) +(.02)\cdot(1- P(t-1)/20000)\cdot P(t-1).$$ Find the positive populations $s$ at which the population doesn't change (i.e., find the equilibria). What if we allow fisherpeople to fish 80 fish a year from the population? How do the equilibria change?

From there you could talk about stable vs unstable equilibria, draw a phase diagram, modify the model and see what happens, etc.

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