Quadratic Method: Related Work

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.

John Savage

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$(x+p)(x+q)$and negating at the end. It is a nice piece of work, and involves both core components: factoring and the Babylonian trick!

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$x^2+x+1$, but they were unable to think of two numbers whose sum and product were both$1$. Starting from a few illustrative numerical examples, his article builds up to and proposes the following complete method of solving quadratic equations, which he advocates for using throughout sections on factoring and quadratic equations, and for developing the quadratic formula:

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

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$p$and$q$as two numbers waiting to be found, and proceeds with forward deductions based upon them to find out what their values actually are (although Step 2 is once written with the reverse direction of logic and the direction is not explicitly specified thereafter). As will be detailed later, this is still mathematically sound.

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$x^2+x+1$, and the article also says that students weren't sure the factorization$(x+\frac{1}{2}+\frac{\sqrt{3}}{2}i)(x+\frac{1}{2}-\frac{\sqrt{3}}{2}i)$produced by the method was correct until they multiplied the factors. Had the deductions been backward instead of forward, where previous lines were logical consequences of subsequent lines (as in the method I proposed), there would be no need to multiply the factors to check because the logic already went through.

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$u$such that$\frac{B^2}{4}-u^2=C$, then the product is$C$.") Reversing the logic would also achieve the benefit of removing the assumption in Step 5 that the complete set of square roots is known. In general, a logical framework starting from assuming existence cannot naively be reversed (another known derivation starts with$p+q=B\Rightarrow (p+q{)}^2=B^2$and then subtracts$4pq=4C$to get$(p-q{)}^2=B^2-4C$, but$(p+q{)}^2=B^2\nRightarrow p+q=B$, and further examples are provided in the previous link). However, in this case, by fortune it just so happens to be easy to rewrite the published presentation to correctly reverse the directions of implication, by using language matching the method that I shared. A generous interpretation of the writing style could even choose to read connectives like "thus" to mean "thus it suffices to find" (actually reversing the implication), or "therefore" to mean "therefore equivalently" (observing both directions of implication), and to read sequences of equations to mean that they are equivalent instead of taking the customary meaning of successive forward deduction that occurs in the context of solving systems of equations by substitution (e.g., the sum and product system of equations here). It is worth mentioning that the logic turns out to be so conveniently correctable that anyone who is familiar with mathematical deduction could rewrite each connective phrase in Savage's paper to ensure the desired backward logical flow.

There is a pedagogical difference in that Savage sought a factorization of$x^2+Bx+C$in the form$(x+p)(x+q)$, whereas my approach seeks a factorization in the form$(x-r)(x-s)$. Both can be used to teach this type of method effectively. Regarding the choice of sign in which version to teach, I would still advocate for the version I originally proposed, which uses the factorization$(x-r)(x-s)$. I think it has a pedagogical advantage because it conceptually represents a clean and complete reduction via factoring from the problem of solving a standard quadratic equation to a sum-product problem which is in turn solved by the Babylonian method (with no negation required as a final step). It also provides a proof of Viète's relations between a quadratic's coefficients and the sum and product of its roots, therefore connecting key points in mathematical history.

Completing the Square

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.

Babylonians and Greeks

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$x+y=A$and$xy=B$can be reduced to a quadratic of the form$x^2-Ax+B=0,$and expressed concern that therefore the ancient Babylonians and Greeks already had the solution to general quadratic 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$y=A-x,$and then plugging into the equation$xy=B$to deduce that$x(A-x)=B,$which rearranges to$x^2-Ax+B=0.$This shows that any$(x,y)$which satisfies the original system of equations$x+y=A$and$xy=B$must necessarily also satisfy$x^2-Ax+B=0.$In other words, the original sum-and-product system of equations can be reduced to the standard form quadratic$x^2-Ax+B=0.$

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$x^2-Ax+B=0$can be reduced to the system of equations$x+y=A$and$xy=B.$

In both Burton's book and Diophantus's work, the ancient mathematicians did not deduce the opposite reduction in general ($x^2-Ax+B=0$with general values of$A$and$B$implies exactly two solutions always exist, namely the two numbers with sum$A$and product$B$). A major reason was that ancient mathematicians did not work with negative numbers, not to mention irrational (or non-real complex) numbers. (Diophantus imposes the additional requirement that$(A/2{)}^2>B$is a positive perfect square in order to avoid non-real or irrational numbers.) Such a reduction needed to wait until complex numbers were invented, which was around 1500 AD.

So, although pages 67-68 of Burton and Book I Problem 27 of Diophantus both show complete solutions of the system$x+y=A$and$xy=B$with the key substitution, neither explicitly observes or shows that you can reduce the general standard form quadratic to this 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$x^2+Ax=B$and$x^2=Ax+B$for positive$A$and$B$, where their solution formulas are written without derivation. In contrast, on the same page the form$x^2+B=Ax$is said to have been "transformed by all sorts of ingenious devices" to the system$x+y=A$and$xy=B$, but no details are given of the transformation method, which is indeed difficult without the modern system of Algebra. The fact that Burton treats the first two forms ($x^2+Ax=B$and$x^2=Ax+B$) differently from the third form ($x^2+B=Ax$) means that there was no observation that every general standard form quadratic equation (possibly with non-positive coefficients) reduced to the sum-and-product system. Otherwise, there would have been no need to reduce only the third form and treat the first two forms differently.

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.

Historical Mathematical Manuscripts

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)