Posted Oct 26, 2019; updated Dec 16, 2020

It is surprising that something so retrospectively simple was not widely known, and so I am documenting the history of discovery around this particular approach. 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. If you know of another publicly verifiable dated reference with the same method of solving general quadratic equations, please contact me, and I would be happy to learn more. I am recording the most similar approaches on this page.

After sharing an earlier version of my article online, I was helpfully pointed to the nice article Factoring Quadratics, by multiple award-winning teacher John Savage in *The Mathematics Teacher* Volume 82 (January 1989), pages 35–36, by Benjamin Dickman, who found this reference via Google Scholar keyword search. This provided a coherent method of solving quadratic equations that overlapped in almost all calculations, with an apparent logical difference in assuming that every quadratic can be factored (requiring a significantly more advanced fact which would make it not suitable for first-time learners, unlike the method I proposed). It also has a pedagogical difference in choice of sign, factoring in the form

Perhaps due to a friendly writing style which occasionally reverses the direction of logic, and leaves some logic up for interpretation, the published article as-written appears to use the additional (true but advanced) fact that every quadratic can be factored into two linear factors, which is beyond first-time learners, or it has some reversed directions of implication that are not technically correct. The directional reversals prevented the observation that an extra assumption in Completing the Square could be avoided. That said, they can be fixed, and Savage's contribution contains very nice insights.

Since Savage's article is not easily viewable, I summarize it here. If anyone believes that I have misrepresented Savage's writing, please contact me and I will happily update this page as appropriate. The article is quite fun to read, and describes from the point of view of a teacher how Savage invented a method of factoring trinomials while teaching radicals and imaginary numbers to his tenth-grade enrichment class. The article opens with an anecdote where he asked his students to factor

- To factor
as${x}^{2}+Bx+C$ , we want to find two numbers$(x+p)(x+q)$ and$p$ with sum$q$ and product$B$ .$C$ - Since the sum of
and$p$ is$q$ , the average is$B$ . The numbers$B/2$ and$p$ can then be represented by$q$ .$\frac{B}{2}\pm u$ - Since the product is
, then$C$ .$\frac{{B}^{2}}{4}-{u}^{2}=C$ - Since
, then$\frac{{B}^{2}}{4}-{u}^{2}=C$ .${u}^{2}=\frac{{B}^{2}}{4}-C$ - Since the complete set of square roots of any number
is$D$ and$\{\pm \sqrt{D}\}$ , then${u}^{2}=\frac{{B}^{2}}{4}-C$ .$u=\pm \sqrt{\frac{{B}^{2}}{4}-C}$ - Thus
and$p$ are$q$ .$\frac{B}{2}\pm \sqrt{\frac{{B}^{2}}{4}-C}$ - By the zero-product property, the solutions are
and$-p$ , which are$-q$ .$-\frac{B}{2}\mp \sqrt{\frac{{B}^{2}}{4}-C}$

Perhaps due to the friendly writing style, the directions of implication in Steps 2–5 appear to be written in a way that presumes the definite existence of

Specifically, successive rows of equations are written with single connective words such as "therefore", "thus", and "so", or with no connective words. These all represent forward deductions where the next statement is a logical consequence of all previous statements. The logical issues with approaches based on forward deduction are detailed on this page, which explains that those approaches require the initial knowledge that a factorization definitely exists. That matches the framing of the opening anecdote, which was asked in a way that presumed the existence of a factorization of

As detailed on the above page, forward deduction is still mathematically correct, as it is a consequence of the Fundamental Theorem of Algebra (or the quadratic formula) that every quadratic can be factorized into two linear factors. Indeed, perhaps that was already known to tenth-grade enrichment students. However, for first-time Algebra learners, it has not yet been proven that every quadratic has two roots (counting multiplicity), and so although forward deduction works, in order for this to become a self-contained elementary technique like the method I proposed, that existence assumption would need to be removed.

It turns out to be relatively easy to edit Savage's published presentation to explicitly remove any reliance on the initial assumption that a factorization exists, by reversing the directions of logic in Steps 2–5. (For example, the logic in Step 3 would need to become "If there exists a

There is a pedagogical difference in that Savage sought a factorization of

The arithmetic calculations in the method I proposed are almost exactly the same as those in the standard method of completing the square. That should be expected, because both methods result in the same quadratic formula. However, the logic is quite different, as the usual way of completing the square actually requires an additional assumption. That dedicated page contains a full comparison.

The ancient Babylonians and Greeks already knew how to solve the problem of finding two numbers with given sum and product, using the same technique outlined in the corresponding step of Savage's article, and in my article. This is explicitly stated in Book I Problem 27 of Diophantus's Arithmetica, from around 250 AD. A detailed discussion with some similar-looking calculations appears in The History of Mathematics: An Introduction by David Burton. Several readers have pointed out that the system of equations

They are indeed correct that the sum-and-product system can be reduced to a standard form quadratic, as that follows from using substitution to deduce that

However, the mathematical logic required is the opposite direction of reduction: in order to solve general quadratic equations, one needs the statement that a quadratic of the form

In both Burton's book and Diophantus's work, the ancient mathematicians did not deduce the opposite reduction in general (

So, although pages 67-68 of Burton and Book I Problem 27 of Diophantus both show complete solutions of the system

This is particularly clear from page 70 of Burton, which separates quadratic equations into several forms based on the signs of their coefficients. The first two forms discussed on page 70 are

In conclusion, the sum-and-product system was solved in the same way by the ancient Babylonians and Greeks, but they did not know how to reduce general standard form quadratic equations to that system.

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)