With the launch of my new mathematics blog, I thought I would start with something of a nostalgia post: a note on the quadratic formula.

The proof of the quadratic formula (by completing the square) was one of the first that I learned. When I originally worked through it and finally arrived at a derivation of the formula, it was for me one of those early mathematical moments that continued the growth of an already deep-seated interest. It’s very much similar in memory to various other instances of mathematical experience, such as when I look back to the time when I first taught myself calculus. Sentimentality is not the right word here; it’s more a remembrance of an early moment of mathematical passion and discovery.

It is by no means on the list of my favourite proofs – all of which I will write about in the future – but it’s fitting to have this derivation (at the bottom) as part of the early development of this blog.

Furthermore, I think that one of the wonderful things about mathematics (even very basic arithmetic) is how the derivation of something as simple and basic as the quadratic formula is actually quite beautiful. Some of the deeper connections we might make are also, philosophically, very inspiring. And even further, the anthropological dimension of its history and discovery is incredibly interesting.

Often it seems that as one advances their early mathematical career, lessons and thought experiments on first principles and proofs of whatever formula are increasingly absent. So too is the why of maths. In the case of formulae, first principles are often sort of left implicit – you know, here’s a formula and here’s how and when to use it. The why of mathematics seems to be left out, at least especially early on in one’s career.

A lot of science is emerging that backs the idea that the best way to learn is not by attending lectures – though they are useful – but by exercising one’s critical thinking skills. Exploring first principles, working through the logic of whatever particular derivation, and in reflecting on proofs helps build a foundational sense of the properties in practice and strengthens key intuition of why.

History

Before I actually get into the derivation, what is interesting to note is that thoughts around the development of this basic formula possess an interesting history. (I summarized these notes some time ago and I cannot locate the original source, otherwise I would link to the historical record). In short: Math historians often cite that, although the first attempts to find a more general formula to solve quadratic equations can be tracked back to geometry (and trigonometry) of Pythagoras and Euclid, the history of thinking actually dates as far back as approximately 2000 or so BC.

Some denote the thinking at this time as the original problem, wherein Egyptian, Babylonian and Chinese engineers encountered a question of some urgency: namely, how certain shapes must be scaled to a total area. In other words, there needed to be a way to measure the lengths of the sides of walls.

Anthropologically, one has to remember that this problem emerged shortly after the first signs of civilization began to formally develop in Mesopotamia, and thus with it the Bronze Age. With this there was an increase in agricultural production and all the rest, taking off from the Neolithic Revolution many years before. Storage of excess materials, grain and resources was an ongoing problem in this early and important period of development.

But these early engineers were very intelligent for their time. They knew how to find the area of a square with the length of a side. They also knew how to utilize squared spaces. But the sides and area of more complex shapes posed a significant problem. But then something important happening. In Egypt around the time of 1500BC the concept of completing the square was formulated to help solve very basic problems concerning area. It also appears later in Chinese records.

Then in 700 AD, Baskhara, a famous Indian mathematician of whom many may already be familiar, was the first to recognise that any positive number has two square roots. This followed by another derivation of the quadratic formula performed by Mohammad bin Musa Al-Khwarismi, a famous Islamic mathematician. The historical account is that this particular derivation was then brought to Europe some time later by Jewish mathematician/astronomer Abraham bar Hiyya. Some time later it was then picked up in 1545 by Girolamo Cardano, a Renaissance scientist. Here Al-Khwarismi’s solution was integrated with Euclidean geometry, which helped pave the way for the modern formulation.

Indeed, it was François Viète in 16th century France who would introduce what we now would consider as more recognizable notation. Then, the big work. The famous enlightenment thinker, René Descartes, penned La Géométrie, within which modern Mathematics was born. From out of this the quadratic formula as we know it today would emerge and be adopted.

Deriving the Formula

With some of the history noted, here’s my effort at a derivation of the formula (by completing the square).