## What use is a quadratic equation?

A quadratic equation is an equation *where at least one term is squared*. It looks something like this:

ax² + bx + c

or

c = ax² + bx

.. and students are asked to solve, factorise or draw it.

Quadratics are usually the most dreaded of homework assignments. I’m sure many students have wondered about their use.

Will they ever be useful – or is a quadratic just useless knowledge?

Although there has been some debate in recent years, history and more modern developments have shown that quadratic equations **are** an essential part of mathematics.

Quadratic equations have played a significant role in everything from understanding the Earth’s orbit, the invention of telescopes, and the development of mobile phones.

Originally formed in Ancient Babylon to calculate payments to the taxman (!), quadratic equations have been around quite a long time. The Ancient Greek mathematicians studied them and the quadratic curves (circle, ellipse, hyperbola, and parabola) that could be created.

But, it wasn’t until the 16^{th} century that quadratic equations really stepped into the spotlight. Using a quadratic, Copernicus proposed that the Earth orbited in a circle around the Sun. He wasn’t quite right with his idea. Although he did provide the basis that Johannes Kepler used to calculate that the Earth’s orbit was actually an ellipse.

*Ellipse Equation*

ax² + by² = 1

This produced the modern understanding of our planet’s orbits, as well as the orbits of other bodies in the Solar System, and quadratic equations played a starring role in the production.

A quadratic equation not only gave us a tool to calculate how planets moved, but it also helped us get the ability to *see them*. Galileo’s telescope contained two intersecting mirrors, each in the shape of a hyperbola, and was able to magnify strong enough to see the satellites of Jupiter. Also using a ‘quadratic curve,’ Newton designed his reflecting telescope with parabola shaped mirrors. Without these inventions, we’d have a very hard time trying to see the solar system in any detail.

Modern communications also have a basis in quadratic equations. It was a quadratic equation that led to the usage of the ‘imaginary number (*i*)’. Within quantum theory, the imaginary number is fundamental to Schrӧdinger’s equation.

*i* (∂u / ∂t) + v²+ v(x)u = 0

This equation is very important in the design of circuits that perform tasks within computers, MP3 players, and mobile phones.

Interestingly, the formula used to figure out how to carry our voices from one mobile phone to another also involves the imaginary number.

Without a quadratic equation, none of those technologies would have been possible.

Although it may be hard to see when looking at your homework, quadratic equations have played a large part in the development of inventions throughout history. Many technologies were able to be designed because of a quadratic equation in one form or another. It has been essential in building both our understanding of how things work and the production of things we use every day.

A huge thank you to Cassandra Hoffman for all her support in preparing this post.

Please add your comments below – thanks!

Tim Bartrum says

Wow, this is amazing! Thanks for this interesting article 😀