Posted Oct 13, 2019; last updated Aug 6, 2021

I've recently been systematically thinking about how to explain school math concepts in more thoughtful and interesting ways, while creating my *Daily Challenge* lessons. One night in September 2019, while brainstorming different ways to think about the quadratic formula, I was surprised to come up with a simple method of eliminating guess-and-check from factoring that I had never seen before.

- If you find
and$r$ with sum$s$ and product$-B$ , then$C$ , and they are all the roots${x}^{2}+Bx+C=(x-r)(x-s)$ - Two numbers sum to
when they are$-B$ $-\frac{B}{2}\pm u$ - Their product is
when$C$ $\frac{{B}^{2}}{4}-{u}^{2}=C$ - Square root always gives valid
$u$ - Thus
work as$-\frac{B}{2}\pm u$ and$r$ , and are all the roots$s$

*Known hundreds of years ago (Viète)**Known thousands of years ago (Babylonians, Greeks)*

The individual steps of this method had been separately discovered by ancient mathematicians. The combination of these steps is something that anyone could have come up with, but after releasing this webpage to the wild, the only previous reference that surfaced, of a similar coherent method for solving quadratic equations, was a nice article by mathematics teacher John Savage, published in *The Mathematics Teacher* in 1989. His approach overlapped in almost all calculations, with a pedagogical difference in choice of sign, but had a difference in logic, as (possibly due to a friendly writing style which leaves some logic up for interpretation) it appears to use the additional (true but significantly more advanced) fact that every quadratic can be factored into two linear factors, or has some reversed directions of implication that are not technically correct. In particular, my approach's avoidance of an extra assumption in Completing the Square was not achieved by Savage's method. The related work page compares the method described by Savage, with the method that I proposed. Since I still have not seen any previously-existing book or paper which states this type of method in a way that is suitable for first-time learners (avoiding advanced knowledge) and justifies all steps clearly and consistently, I chose to share it to provide a safely referenceable version.

The presentation below is based on the approach in my originally posted article, but goes further. It uses my sign convention and my own logical steps (as opposed to using Savage's version) in order to be logically sound, and also because I think my choice is helpful for understanding the deeper relationship between a quadratic and its solutions. It also shows a clean reduction of the problem from solving a standard quadratic, to a classical problem solved by the Babylonians and Greeks. This video is a self-contained practical lesson that walks through many examples with each logical step explained. The text discussion below goes a bit deeper and includes commentary which may be useful for teachers.

Let's start by reviewing the facts that are usually taught to introduce quadratic equations. First, we use the distributive rule to multiply (also called FOIL):

The key takeaway is that the

Here's another example:

Since we had both

The reason it is useful to know what happens when multiplying is because if we can do this in reverse, we can solve quadratic equations. For example, suppose we want to find all*zero-product property* uses the basic axiom that you can divide by any nonzero number: suppose for contradiction that

So, the

Let's try the reverse process for the example

As summarized in the related work, Savage's version has the similar calculations except that he seeks a factorization into the mathematically equivalent form

To make this even more natural for a first-time learner, I would advocate introducing the concept of factoring with an initial example that has a negative

Here's a way to pinpoint numbers that work without any guessing at all! The sum of two numbers is

By looking for two numbers of the form

Note that in this approach, we only need the existence of one particular number whose square equals another particular number. In this example, it is obvious that

As I noted in my complete article, although I (like many others) independently came up with the trick of how to find two numbers given their sum and product, the Babylonians and Greeks already knew that particular trick thousands of years prior. However, mathematics had not been sufficiently developed for them to be able to use that trick on its own to solve general quadratic equations. Specifically, they did not work with polynomial factoring or negative numbers (not to mention non-real complex numbers). For an in-depth discussion, please visit the related work page.

Now that guessing has been eliminated, we can actually solve any quadratic with this method. Consider this example:**if** we can find two numbers with sum**This method works for every quadratic equation, without needing any memorization, and every step has a simple mathematical justification.**

If one wishes to derive the quadratic formula, this method also provides an alternative simple proof of it.

For a general quadratic equation

The above formula is already enough to solve any quadratic equation, because you can multiply or divide both sides by a number so that nothing is in front of the

This method consists of two main steps, starting from a general quadratic equation in standard form

- Because of polynomial factoring, if we can find two numbers with sum
and product$-B$ then those are the complete set of solutions.$C,$ - Use the ancient Babylonian/Greek trick (extended to complex numbers) to find those two numbers in every circumstance.

In order for these steps to be mathematically sound as a complete method, it is essential that under all circumstances, Step 2 finds two numbers to use in Step 1, even if they are non-real complex numbers. It is therefore unlikely that mathematicians before Cardano (~1500 AD) could have done this completely.

Both steps are individually well-known. In retrospect, their combination to form a complete and coherent method for solving general quadratic equations is simple and obvious. Therefore, the main contribution of this method is to point out something useful that has been hiding in plain sight.

While researching the novelty of this approach, I came across several ancient mathematical works. Thanks to the Internet, it is now possible for everyone to view and appreciate the creativity of early mathematicians.

- Arithmetica, by Diophantus (circa 250)
- Brahma-Sphuta Siddhanta, by Brahmagupta (circa 628)
- al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala , by al-Khwarizmi (circa 825)
- Ars Magna, by Cardano (1545)
- Opera Mathematica, by Viète (1579)
- Geschichte der Elementar-Mathematik in systematischer Darstellung, by Tropfke (1902—1903)