Editing Quadratic formula (section)

==Historical development==
The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations.{{sfn|Irving|2013|p=34}} 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>{{citation |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>

The Greek mathematician [[Euclid]] (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his ''[[Euclid's Elements|Elements]]'', an influential mathematical treatise{{sfn|Irving|2013|p=39}} Rules for quadratic equations appear in the Chinese ''[[The Nine Chapters on the Mathematical Art]]'' circa 200 BC.<ref name=Aitken>{{cite web |mode=cs2 |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>{{sfn|Smith|1923|loc=[https://archive.org/details/historyofmathema0001davi/page/380 {{pgs|380}}] }} In his work ''[[Arithmetica]]'', the Greek mathematician [[Diophantus]] (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid.{{sfn|Irving|2013|p=39}} His solution gives only one root, even when both roots are positive.{{sfn|Smith|1923|loc=[https://archive.org/details/historyofmathema0001davi/page/134 {{pgs|134}}] }}

The [[Indian mathematics|Indian mathematician]] [[Brahmagupta]] included a generic method for finding one root of a quadratic equation in his treatise ''[[Brāhmasphuṭasiddhānta]]'' (circa 628 AD), written out in words in the style of that time.<ref name=Bradley>Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006).</ref><ref>Mackenzie, Dana. ''The Universe in Zero Words: The Story of Mathematics as Told through Equations'', p. 61 (Princeton University Press, 2012).</ref> His solution of the quadratic equation {{tmath|1=\textstyle ax^2 + bx = c}} was 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 name=Stillwell2004>{{citation |last=Stillwell |first=John |title=Mathematics and Its History |edition=2nd |year=2004 |publisher=Springer |isbn=0-387-95336-1|page=87}}</ref>
In modern notation, this can be written {{tmath|1=\textstyle x = \bigl(\sqrt{c \cdot 4a + b^2} - b\bigr) \big/ 2a }}. The Indian mathematician [[Śrīdhara]] (8th–9th century) came up with a similar algorithm for solving quadratic equations in a now-lost work on algebra quoted by [[Bhāskara II]].<ref>{{MacTutor |id=Sridhara |title=Sridhara |year=2000}}</ref> The modern quadratic formula is sometimes called ''Sridharacharya's formula'' in India and ''Bhaskara's formula'' in Brazil.<ref>{{cite thesis |last=Rocha |first=Rodrigo Luis da |title=O uso da expressão 'fórmula de bhaskara' em livros didáticos brasileiros e sua relação com o método resolutivo da equação do 2º grau |trans-title=The use of the expression 'bhaskara formula' in Brazilian textbooks and its relationship with the method for solving quadratic equations |language=pt |hdl=1884/82597 |year=2023 |type=master's thesis |publisher=Universidade Federal do Paraná }} {{pb}} {{cite thesis |year=2019 |last=Guedes |first=Eduardo Gomes |type=master's thesis |hdl=1884/66582 |publisher=Universidade Federal do Paraná |title=A equação quadrática e as contribuições de Bhaskara |trans-title=The quadratic equation and Bhaskara's contributions |language=pt }} {{pb}} {{cite magazine |url=https://www.thejuggernaut.com/fibonacci-sequence-indian-math-history-quadratic-formula |title=India Molded Math. Then Europe Claimed It. |quote=For instance, some Indian schools call the quadratic formula Sridharacharya's formula and some Brazilian schools call it Bhaskara's formula. |magazine=The Juggernaut |last=Banerjee |first=Isha |date=July 2, 2024 |url-access=subscription }}</ref>

The 9th-century Persian mathematician [[Muḥammad ibn Mūsā al-Khwārizmī]] solved quadratic equations algebraically.{{sfn|Irving|2013|p=42}} 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 special cases of the quadratic formula in the form we know today.<ref>{{citation |url=http://archive.org/details/TheGeometry|title=The Geometry|last=Rene Descartes|language=en}}</ref>