Editing Quadratic equation (section)

==History==
[[Babylonian mathematics|Babylonian mathematicians]], as early as 2000 BC (displayed on [[First Babylonian dynasty|Old Babylonian]] [[clay tablet]]s) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the [[Third Dynasty of Ur]].<ref name=Friberg2009>{{cite journal|last=Friberg|first=Jöran|title=A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma|journal=Cuneiform Digital Library Journal|year=2009|volume=3|url=http://cdli.ucla.edu/pubs/cdlj/2009/cdlj2009_003.html}}</ref> In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
<math display="block"> x+y=p,\ \ xy=q, </math>
which is equivalent to the statement that {{mvar|x}} and {{mvar|y}} are the roots of the equation:<ref name=Stillwell2004>{{cite book |last=Stillwell |first=John |title=Mathematics and Its History (2nd ed.) |year=2004 |publisher=Springer |isbn=978-0-387-95336-6}}</ref>{{rp|86}}
<math display="block">z^2+q=pz.</math>

The steps given by Babylonian scribes for solving the above rectangle problem, in terms of {{mvar|x}} and {{mvar|y}}, were as follows:
#Compute half of ''p''.
#Square the result.
#Subtract ''q''.
#Find the (positive) square root using a table of squares.
#Add together the results of steps (1) and (4) to give {{math|''x''}}.
In modern notation this means calculating <math>x = \frac{p}{2} + \sqrt{\left(\frac{p}{2}\right)^2 - q}</math>, which is equivalent to the modern day [[quadratic formula]] for the larger real root (if any) <math>x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}</math> with {{math|1=''a'' = 1}}, {{math|1=''b'' = −''p''}}, and {{math|1=''c'' = ''q''}}.

Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian [[Berlin Papyrus 6619|Berlin Papyrus]], dating back to the [[Middle Kingdom of Egypt|Middle Kingdom]] (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.<ref>{{cite book|title=The Cambridge Ancient History Part 2 Early History of the Middle East|url=https://books.google.com/books?id=slR7SFScEnwC&pg=PA530|year=1971|publisher=Cambridge University Press|isbn=978-0-521-07791-0|page=530}}</ref> Babylonian mathematicians from circa 400 BC and [[Chinese mathematics|Chinese mathematicians]] from circa 200 BC used [[Dissection problem|geometric methods of dissection]] to solve quadratic equations with positive roots.<ref name=Henderson>{{cite web|last=Henderson|first=David W.|title=Geometric Solutions of Quadratic and Cubic Equations |publisher=Mathematics Department, Cornell University |url=http://www.math.cornell.edu/~dwh/papers/geomsolu/geomsolu.html|access-date=28 April 2013}}</ref><ref name=Aitken>{{cite web|last=Aitken|first=Wayne|title=A Chinese Classic: The Nine Chapters|url=http://public.csusm.edu/aitken_html/m330/china/ninechapters.pdf|publisher=Mathematics Department, California State University|access-date=28 April 2013}}</ref> Rules for quadratic equations were given in ''[[The Nine Chapters on the Mathematical Art]]'', a Chinese treatise on mathematics.<ref name=Aitken/><ref>{{cite book|last=Smith|first=David Eugene|title=History of Mathematics|url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA380|year=1958|publisher=Courier Dover Publications|isbn=978-0-486-20430-7|page=380}}</ref> These early geometric methods do not appear to have had a general formula. [[Euclid]], the [[Greek mathematics|Greek mathematician]], produced a more abstract geometrical method around 300 BC. With a purely geometric approach [[Pythagoras]] and Euclid created a general procedure to find solutions of the quadratic equation. In his work ''[[Arithmetica]]'', the Greek mathematician [[Diophantus]] solved the quadratic equation, but giving only one root, even when both roots were positive.<ref>{{cite book |title=History of Mathematics, Volume 1 |first1=David Eugene |last1=Smith |publisher=Courier Dover Publications |year=1958 |isbn=978-0-486-20429-1 |page=134 |url=https://books.google.com/books?id=12qdOZ0gsWoC}} [https://books.google.com/books?id=12qdOZ0gsWoC&pg=PA134 Extract of page 134]</ref>

In 628 AD, [[Brahmagupta]], an [[Indian mathematics|Indian mathematician]], gave in his book ''[[Brāhmasphuṭasiddhānta]]'' the first explicit (although still not completely general) solution of the quadratic equation {{math|''ax''<sup>2</sup> + ''bx'' {{=}} ''c''}} as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."<ref>Brāhmasphuṭasiddhānta, Colebrook translation, 1817, page 346; cited by {{cite book |last=Stillwell |first=John |title=Mathematics and Its History (3rd ed.) |series=Undergraduate Texts in Mathematics |year=2010 |publisher=Springer |isbn=978-0-387-95336-6 |page=93 |doi=10.1007/978-1-4419-6053-5}}</ref> This is equivalent to
<math display="block">x = \frac{\sqrt{4ac+b^2}-b}{2a}.</math>
The ''[[Bakhshali Manuscript]]'' written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear [[indeterminate equation]]s (originally of type {{math|''ax''/''c'' {{=}} ''y''}}). [[Muhammad ibn Musa al-Khwarizmi]] (9th century) developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric [[Mathematical proof|proofs]] in the process.<ref name=Katz2007>{{Cite journal | last1 = Katz | first1 = V. J. | last2 = Barton | first2 = B. | doi = 10.1007/s10649-006-9023-7 | title = Stages in the History of Algebra with Implications for Teaching | journal = Educational Studies in Mathematics | volume = 66 | issue = 2 | pages = 185&ndash;201 | year = 2006 | s2cid = 120363574 }}</ref> He also described the method of completing the square and recognized that the [[discriminant]] must be positive,<ref name=Katz2007/><ref name=Boyer1991/>{{rp|230}} which was proven by his contemporary [['Abd al-Hamīd ibn Turk]] (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.<ref name=Boyer1991>{{cite book|last=Boyer|first=Carl B.|editor-link=Uta Merzbach|editor-first=Uta C.|editor-last=Merzbach|title=A History of Mathematics|year=1991|publisher=John Wiley & Sons, Inc.|isbn=978-0-471-54397-8|url=https://archive.org/details/historyofmathema00boye}}</ref>{{rp|234}} While al-Khwarizmi himself did not accept negative solutions, later [[Mathematics in medieval Islam|Islamic mathematicians]] that succeeded him accepted negative solutions,<ref name=Katz2007/>{{rp|191}} as well as [[irrational number]]s as solutions.<ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."</ref> [[Abū Kāmil Shujā ibn Aslam]] (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a [[square root]], [[cube root]] or [[Nth root|fourth root]]) as solutions to quadratic equations or as [[coefficient]]s in an equation.<ref>Jacques Sesiano, "Islamic mathematics", p. 148, in {{citation|title=Mathematics Across Cultures: The History of Non-Western Mathematics|editor1-first=Helaine|editor1-last=Selin|editor1-link=Helaine Selin|editor2-first=Ubiratan|editor2-last=D'Ambrosio|editor2-link=Ubiratan D'Ambrosio|year=2000|publisher=[[Springer Science+Business Media|Springer]]|isbn=978-1-4020-0260-1}}</ref> The 9th century Indian mathematician [[Sridhara]] wrote down rules for solving quadratic equations.<ref>{{cite book|last=Smith|first=David Eugene|title=History of Mathematics|url=https://books.google.com/books?id=12qdOZ0gsWoC&pg=PA280|year=1958|publisher=Courier Dover Publications|isbn=978-0-486-20429-1|page=280}}</ref>

The Jewish mathematician [[Abraham bar Hiyya|Abraham bar Hiyya Ha-Nasi]] (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.<ref name=Livio2006>{{cite book |last=Livio |first=Mario |title=The Equation that Couldn't Be Solved |year=2006 |publisher=Simon & Schuster |isbn=978-0743258210 |url=https://books.google.com/books?id=veQ9a3nixDUC&q=Abraham+bar+Hiyya+Ha-Nasi+quadratic&pg=PA62}}</ref> His solution was largely based on Al-Khwarizmi's work.<ref name=Katz2007/> The writing of the Chinese mathematician [[Yang Hui]] (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier [[Liu Yi (mathematician)|Liu Yi]].<ref name=Ron>{{cite book|last=Ronan|first=Colin|title=The Shorter Science and Civilisation in China|url=https://books.google.com/books?id=XsMxmS7NyukC&pg=PA15|year=1985|publisher=Cambridge University Press|isbn=978-0-521-31536-4|page=15}}</ref> By 1545 [[Gerolamo Cardano]] compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by [[Simon Stevin]] in 1594.<ref>{{Citation |title=The Principal Works of Simon Stevin, Mathematics |volume=II-B |first1=D. J. |last1=Struik |first2=Simon |last2=Stevin |publisher=C. V. Swets & Zeitlinger |year=1958 |page=470 |url=http://www.dwc.knaw.nl/pub/bronnen/Simon_Stevin-%5bII_B%5d_The_Principal_Works_of_Simon_Stevin,_Mathematics.pdf}}</ref> In 1637 [[René Descartes]] published ''[[La Géométrie]]'' containing the quadratic formula in the form we know today.