Why solving quadratic equations is a superpower

This installment is about motivating math and what happens when you don’t. A while back I was driving in to work and heard some guy on the radio ask why we were wasting time on quadratic equations in high school when we could be teaching financial math or letting the kids get work experience.

My initial reaction was that this gentleman had obviously been over-indulging recreational drugs or, possibly, had had parts of his brain removed with a melon baller.

On reflection, I realized that this was probably unfair. The man’s problem was not stupidity but ignorance. He only culpable action was shooting his mouth off when he had no idea what he was talking about– a sort of double-ignorance where you don’t even know what you don’t know. Looking back, I recalled that my teachers taught me how to solve quadratic equations but never once told me what they were good for.

One way to tell if something is significant is to see how many different people know about it. The first solution to quadratic equations we know of appears on Babylonian cuneiform tablets, one of the oldest human civilizations. The Berlin papyrus, from the Egyptian middle kingdom (before 1650 BCE), has a solution for quadratic equations. Book two of Euclid’s Elements  from 300 BCE, has a geometric solution to the quadratic equation, bringing classical Greek civilization on board. Brahmagupta explicitly described (without equations!) the quadratic formula in his treatise Brāhmasphuṭasiddhānta, published in 628 AD.

These ancient cultures are scattered through time and space and, for some reason, found quadratics to be so important that they not only solved them, but spread their solutions around so thoroughly that they survived into the modern era so I can write about it.

The Nova Scientia (new science), published by Niccolò Fontana Tartaglia in 1537, founded the science of ballistics. This work started a battle that pitted the Church’s support for Aristotle’s version of physics against the desires of generals and gunners to hit targets with their cannon. If you ignore air resistance, not a bad simplification for a large iron ball, then a cannonball follows a path described by a quadratic equation. Tartaglia showed that a cannonball goes farthest when the cannon is at an angle of 45 degrees. He did this with a series of careful experiments. This fact follows very directly from an understanding of quadratic equations as well, but at the time it was controversial.

Leaving the middle ages for the modern era of story problems, suppose you have a fixed amount of fence and want to make the largest possible rectangular pen or pens with it. If all the fence is in the horizontal or vertical direction then nothing is enclosed and the area is zero – pretty much as bad as possible. If we place H meters of fence in the horizontal direction and L-H meters of fence (everything that is left) in the vertical direction then the area enclosed in a pen is:

Area=H x (L-H)

This is a quadratic equation. The solution to that quadratic is that the area is biggest when the fence is evenly divided between the horizontal and vertical directions. If we are making a pen in the middle of the yard, this means a square pen encloses the most area. What if we build the pen against the side of the house or build three pens next to one another that share sides? Solving quadratic equations gives us these results:


The sides of the pens are labeled with the fraction of the fence used to make that side. Notice that the pen where one side of the pen is the house is twice as big while the three adjacent pens are pretty puny. In all three of these example the total amount of fence in each direction is the same. For the three adjacent pens, for example, there are two fourths in the horizontal direction and four eighths in the vertical direction giving us half the fence in each direction. This rule – always have the same amount of fence in each direction – solves all the rectangular pen problems. You don’t need a quadratic equation to apply this rule, but I used my ability to do quadratic equations to find the rule and, since I am a mathematician, to prove that it is correct in general (not just for my examples).

I’ll end with a silly example. I play role playing games. In one of these games where the players are role-playing super heroes you can get “levels” with your pistol, nuclear eye beams, sonic soliton scream, or whatever your good attack is. The levels either add to your roll to hit an opponent or they can make the accuracy of your attack drop off more slowly with the range to the opponent. I worked out the optimal assignment of my levels. For a close opponent, you put all the level on your chance of hitting. For a distant opponent, the correct choice is to split the levels evenly– something I figured out by solving a quadratic equation.  The person refereeing the game told me I was not allowed to enhance my character’s performance by using my own superpowers.

This is another place where someone found my application of math to whatever was sitting next to me to be eccentric. The fact I could do math was viewed as an unfair advantage. All I can say is, get your own unfair advantage.

I hope you found the post interesting and informative,

Dan Ashlock
Department of Mathematics and Statistics
University of Guelph, Ontario, Canada


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