aa r X i v : . [ m a t h . HO ] D ec A Simple Proof of the Quadratic Formula
Po-Shen Loh ∗ December 16, 2019
Abstract
This article provides a simple proof of the quadratic formula, which also produces an efficientand natural method for solving general quadratic equations. The derivation is computationallylight and conceptually natural, and has the potential to demystify quadratic equations for stu-dents worldwide.
The quadratic formula was a remarkable triumph of early mathematicians, marking the completionof a long quest to solve quadratic equations, with a storied history stretching as far back as the OldBabylonian Period around 2000–1600 B.C. [18, 21]. Over four millennia, many recognized namesin mathematics left their mark on this topic, and the formula became a standard part of a firstcourse in Algebra.However, it is unfortunate that for billions of people worldwide, the quadratic formula is alsotheir first experience of a rather complicated formula which they memorize. Many typically learnit as the systematic alternative to a guess-and-check method that only factorizes certain contrivedquadratic polynomials. Countless mnemonic techniques abound, from stories of negative beesconsidering whether or not to go to a radical party, to songs set to the tune of
Pop Goes theWeasel.
A derivation by completing the square is usually included in the curriculum, but itsmotivation is often challenging for first-time Algebra learners to follow, and its written executioncan be cumbersome.This article introduces an independently discovered simple derivation of the quadratic formula,which also produces a computationally-efficient and natural method for solving general quadraticequations. The author would actually be very surprised if this pedagogical approach has eludedhuman discovery until the present day, given the 4,000 years of history on this topic, and the billionsof people who have encountered the formula and its proof. Yet this technique is certainly not widelytaught or known. After an earlier version of this arXiv preprint was publicized by formal andinformal media, the author was contacted by many people with potential references. However, theauthor still has not found a previously-existing publicly-shared reference detailing this pedagogicalapproach which is mathematically complete and formally correct. (Similar writings which comeclose, most notably Savage [22], are outlined in Section 3.3.) This article aims to provide a safelyreferenceable method and derivation which is logically sound. That said, it is entirely possible that ∗ Department of Mathematical Sciences, Carnegie Mellon University. Email: [email protected] . For a discussiondesigned for the general public, visit
The reader is encouraged to remember what it was like to be a first-time Algebra learner, whereit took some concentration to combine fractional expressions, especially those including multiplevariables and constants. This section is intentionally written at that level of simplicity, to emphasizehow straightforward all the algebraic manipulations and concepts are.
Throughout this article, we work over the complex numbers. The starting point is that if we canfind a factorization of the following form x + Bx + C = ( x − R )( x − S ) , (1)then a value of x makes the product equal zero precisely when at least one of the factors becomeszero, which happens precisely when x = R or x = S . By the distributive law, it suffices to find twonumbers R and S with sum − B and product C ; then, { R, S } will be the complete set of roots. Two numbers sum to − B precisely when their average is − B , and so it suffices to find twonumbers of the form − B ± z which multiply to C , where z is a single unknown quantity, becausethey will automatically have the desired average. (If z turns out to be 0, then we factor with R = S = − B .) The product ( − B + z )( − B − z ) conveniently matches the form of a difference ofsquares, and equals C precisely when (cid:18) − B (cid:19) − z = C, or equivalently, precisely when we have a z which satisfies z = B − C. Since a square root always exists (extending to complex numbers if necessary), arbitrarily select achoice of square root of B − C to serve as z , in order to satisfy the last equation. Tracing backthrough the logic, we conclude that the desired R and S exist in the form − B ± z , and so − B ± r B − C (2)are all the roots of the original quadratic. (cid:3) This is the standard method of factoring, which corresponds to the converse of relations that are typicallyattributed to Vi`ete [25]. Those relations provide inspiration for this method, but are not required for logical com-pleteness, because the converse is a straightforward consequence of the distributive property. By using the converse,this proof does not rely on the theorem that two roots (counting multiplicity) always exist. This substitution, and this entire solution to find two numbers given their sum and product, was known tothe Babylonians (see, e.g., Burton [4], Gandz [12], Irving [17], or Katz [18]). It also appeared in the first book ofDiophantus [9]. This approach therefore represents the fusion of these ancient techniques together with Renaissance-era mathematical sophistication. Further historical context follows later in this article. .2 Example of use as a method The computational and conceptual simplicity of this derivation actually renders it unnecessary tomemorize any formula at all, even for general coefficients of x . The proof naturally transforms intoa method, and students can execute its logical steps instead of plugging numbers into a formulathat they do not fully understand. Consider, for example, the following quadratic: x − x + 2 = 0 . Multiplying both sides by 2 to make the coefficient of x equal to 1, we obtain the equivalentequation x − x + 4 = 0 . If we find two numbers with sum 2 and product 4, then they are all the solutions. Two numbershave sum 2 precisely when they have average 1. So, it suffices to find some z such that two numbersof the form 1 ± z have product 4 (their average is automatically 1). The final condition is equivalentto each of these equivalent equations: 1 − z = 4 z = − . We can satisfy the last equation by choosing i √ z . Tracing back through the logic, we concludethat 1 ± i √ x coefficient If one specifically wishes to derive the commonly memorized quadratic formula using this method,one only needs to divide the equation ax + bx + c = 0 by a (assume nonzero) to obtain an equivalentequation which matches the form of (1): x + (cid:18) ba (cid:19) x + (cid:16) ca (cid:17) = 0 . Plugging ba and ca for B and C in (2), the roots are: − b a ± r b a − ca = − b a ± r b − ac a = − b ± √ b − ac a . (cid:3) Observe that with this approach, all of the useful and interesting conceptual insights are fullyisolated in a computationally light derivation of an explicit formula, while also producing an efficientand understandable algorithm. The routine but laborious computational portion is required only ifa general formula is sought for memorization purposes. In light of the efficient algorithm, however,it becomes questionable whether there is merit to memorizing a formula without understanding.For example, although the solution to a general linear equation ax + b = 0 is x = − ba (assume a = 0), the equation is typically solved via manipulation instead of plugging into a memorizedformula. 3 Discussion
Before learning the quadratic formula, students learn how to multiply binomials, and they see usefulexpansions such as ( u + v ) = u + 2 uv + v and ( u + v )( u − v ) = u − v . Indeed, the first ofthese expansions is the cornerstone of the traditional proof of the quadratic formula by completingthe square. The second of these expansions is also of wide importance: among other things, it iseventually used to rationalize the denominator of expressions such as √ −√ by multiplying thenumerator and denominator by √ √ ×
37 = (40 + 3)(40 −
3) = 40 − = 1591 . This is an ancient trick. Some historians believe the Babylonians used it thousands of yearsago, multiplying in their base-60 number system by subtracting from tables of squares (see, e.g.,Derbyshire [7]). It was then natural for them to develop the same parameterization for finding twonumbers, given their sum and product.
The most common proof of the quadratic formula is via completing the square, and that was alsothe method used by al-Khwarizmi [1] in his systematic solutions to abstract quadratic equations.Compared to our approach, the motivation is less direct, as the step of completing the square (forthe simple situation of x + Bx + C = 0) simultaneously combines three insights: (i) The x and Bx can be entirely absorbed into a square of the form ( x + D ) by using only partof the expansion ( u + v ) = u + 2 uv + v “backwards,” to attempt to factor an expressionthat begins with u + 2 uv . (ii) This perfect square can be created by adding and subtracting the appropriate constant, whichis ( B ) . (iii) After these manipulations are complete, the equation will have ( x + B ) and some constants,and any such equation can be solved by moving constants around and taking a square root. The full combination of these insights is required to understand the motivation for why one shouldeven write down the specific offsetting quantities + B − B in the first line of the completing thesquare: x + Bx + B − B C = 0 . It is interesting to note that this step of completing the square uses the fact that the complete set of solutions to x = K is {±√ K } , which is not obvious to a first-time student of quadratics. In contrast, our approach only requiresthat there exists an explicit choice of x which satisfies x = K , which is often explicitly constructable.
4n contrast, our approach starts from students’ existing experience searching for a pair of num-bers with given sum and product, which naturally arises during the factoring method. It showsthem that the (sometimes frustrating) guess-and-check process can be replaced by one idea: toparameterize the pair by its average plus or minus a common unknown offset. No particular for-mula needs to be written for the offset itself (unlike the case of carefully selecting B ), and we cansimply call it an unknown z . Instantly, their previous experience of trial and error is replaced bya “forward” expansion of the form ( u + v )( u − v ) = u − v , which produces an exciting lone z ,revealing the pair of numbers with all guesswork eliminated. Could such a simple proof and pedagogical method possibly be new? The author researched theEnglish-language literature on the history of mathematics, and consulted English translations ofold manuscripts, from mathematical traditions ranging from Diophantus [9] to Brahmagupta [3],Yanghui [26], and al-Khwarizmi [1]. This section is too brief to do full justice to the history,and mainly serves to point the interested reader to relevant resources with much richer detail. Inparticular, several books have surveyed the topic of the quadratic formula, such as Chapter 2 ofIrving [17], and mathematical history books such as Burton [4], Derbyshire [7], and Katz [18].As preserved in their cuneiform tablets, the Babylonians had evidence of formulas for a widevariety of problems of quadratic nature, dating back to the Old Babylonian Period around 2000–1600 B.C. Although today we can easily use substitution to reduce them to standard one-variablequadratic equations, the Babylonians did not have a way to solve those standard quadratics.However, they did consider the problem of finding the dimensions of a rectangular field givenits semiperimeter and area, and had the key substitution used in our solution method. This isdiscussed in Gandz’s extensive 150-page study of quadratic equations [12], as well as in Berriman[2], Burton [4], Gandz [13], Katz [18], and Robson [21]. The ancient Egyptians also had evidence ofwork with a two-term quadratic equation, preserved on scraps of a Middle Kingdom papyrus [10].Ancient Chinese mathematicians had solutions to practical problems of quadratic nature, suchas Problem 20 in Chapter 9 of Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art),which was written over several centuries and completed around 100 A.D. Practical problems ofquadratic nature continued to be considered by other Chinese mathematicians, such as the 13th-century Yang Hui. See, e.g., the book in Chinese by Zeng [27] or the book in English by Lam[19].The Greeks had several methods of approaching certain types of quadratic problems, bothalgebraic and geometric, as surveyed in Eells [11]. Heath’s translation [15] of Diophantus [9] fromaround 250 A.D. clearly shows the solution of the core problem of finding two numbers with givensum and product (Book I Problem 27), using the key parameterization in terms of the average.Indian mathematicians also had derived a formula for quadratics. Although Brahmagupta [3]did not discover it himself, one root of the quadratic formula (without derivation) appears in hiswritings circa 628 A.D. See, e.g., the translation by Colebrooke [6] or the commentary by Sharmaet al. [23]. A derivation due to Sridhara from around 900 A.D. appears in Puttaswamy [20].The Persian mathematician al-Khwarizmi published his influential work [1] around 825 A.D.,where he abstractly considered and solved the general form of quadratic equations, without startingfrom practical applications. His work split into several cases, because he did not allow numbersto be negative or zero. Consequently, his formulas did not produce all roots, although they didproduce all roots according to the standards of what a number was at the time.5s mathematics in Western Europe flourished during the Renaissance, successive formulationsand proofs appeared, from Stevin [24] to Vi`ete [25] and Descartes [8], ultimately taking on themodern form that we know today. In the years since then, new proofs have occasionally appeared,such as two in
The American Mathematical Monthly:
Heaton [16] in 1896 and Cirul [5] in 1937.After an earlier version of this arXiv preprint circulated across the Internet, references of morerecent similar work were identified. The most similar is Savage [22]. His approach essentiallyoverlapped in almost all calculations, but had a pedagogical difference in choice of sign, factoringin the form ( x + p )( x + q ) and negating at the end. Perhaps due to a friendly writing style, thatpublished article has some reversed directions of implication that are not formally correct. Thedirectional reversals brought in the same extra assumption as in Footnote 3 when completing thesquare, creating another pedagogical difference. That said, those oversights can easily be correctedby using language similar to our presentation. Gowers [14] also had happened upon a similarapproach, while informally presenting a natural way to deduce the cubic formula. As he waswriting for a different purpose, his version as written uses Vi`ete’s sum and product relations atthe outset, requiring initial knowledge that there always exist two roots (a pedagogical differencefor first-time Algebra learners), and deducing forward. It can easily be converted to avoid thisexistence assumption by using factoring as in our approach.In summary, the author has not yet found a previously-existing book or paper which states thesame pedagogical method as this present article and precisely justifies the steps, but there existindependent references that contain the key ideas and can be adapted to achieve this. That said, itis entirely possible that the method in this present article was previously observed by people whodid not share their findings. The two main components of our derivation have existed for hundreds of years (polynomial factoringconverse of Vi`ete’s relations) and for thousands of years (Babylonian solution to the sum-productproblem). Furthermore, the reduction from the Babylonian problem to a standard quadratic equa-tion has been well-known for an extremely long time. Even al-Khwarizmi [1], after abstractlyanalyzing general quadratic equations, showed how to use his formula to find two numbers withsum 10 and product 21. Like many students in the modern era, he used substitution to reduce theproblem to a single-variable quadratic equation, and solved it with the quadratic formula. Why,then, didn’t early mathematicians just reverse their steps and find our simple derivation?Perhaps the reason is because it is actually mathematically nontrivial to attempt to factor x + Bx + C = ( x − R )( x − S ) over complex numbers. Even if the original quadratic polynomialhas real coefficients, it is sometimes impossible to find two real numbers with sum − B and product C . Early mathematicians did not know how to reason with a full (algebraically closed) systemof numbers. Indeed, al-Khwarizmi did not even use negative numbers, nor did Vi`ete, not tomention the complex numbers that might arise in general. Perhaps, by the time our mathematicalsophistication had advanced to a sufficient stage, the Babylonian trick had faded out of recentmemory, and we already found the method of completing the square to be sufficiently elementaryfor integration into mainstream curriculum.It is worth noting that the author discovered the solution method in this paper in the course offilming mathematical explanations, to explain advanced concepts to particularly young students.Given his audience, he was systematically going through the middle school math curriculum, cre-ating alternative explanations in elementary language. To prepare students for the mindset of6actoring, he posed a standalone sum-product problem, designed to be solved via guess-and-check.While teaching it one evening, his background in coaching math competition students led him toindependently reinvent the Babylonian parameterization in terms of the average, and to recog-nize the difference of squares. Later, when teaching factoring, he suddenly realized that the sametechnique worked in general, leading to a simple proof of the quadratic formula! May this storyencourage the reader to think afresh about old things; seeing as how progress was made on this4,000 year old topic, more surprises certainly await the light of discovery. References [1] Al-Khwarizmi, Muhammad ibn Musa. (circa 825).
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