Quadratic formula (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Quadratic formula" in English language version.

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7books.google.com

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archive.org (Global: 6^th place; English: 6^th place)

Starting from a quadratic equation of the form ⁠ $\textstyle ax^{2}+bx=c$ ⁠, Śrīdhara's method, as quoted by Bhāskara II (c. 1150): "Multiply both sides of the equation by a number equal to four times the [coefficient of the] square, and add to them a number equal to the square of the original [coefficient of the] unknown quantity. [Then extract the root.]".
Smith 1923, p. 446 Smith, David Eugene (1923), History of Mathematics, vol. 2, Boston: Ginn
Forsythe, George E. (1969), "Solving a Quadratic Equation on a Computer", The Mathematical Sciences: A Collection of Essays, MIT Press, pp. 138–152, ISBN 0-262-03026-8
Smith 1923, p. 380. Smith, David Eugene (1923), History of Mathematics, vol. 2, Boston: Ginn
Smith 1923, p. 134. Smith, David Eugene (1923), History of Mathematics, vol. 2, Boston: Ginn
Rene Descartes, The Geometry

archives-ouvertes.fr (Global: 1,031^st place; English: 879^th place)

hal.archives-ouvertes.fr

Goualard, Frédéric (2023), The Ins and Outs of Solving Quadratic Equations with Floating-Point Arithmetic (Technical report), University of Nantes, HAL hal-04116310

arxiv.org (Global: 69^th place; English: 59^th place)

Mastronardi, Nicola; Van Dooren, Paul (2015), "Revisiting the stability of computing the roots of a quadratic polynomial", Electronic Transactions on Numerical Analysis, 44: 73–83, arXiv:1409.8072, archived from the original on 2025-03-03, retrieved 2024-03-15

books.google.com (Global: 3^rd place; English: 3^rd place)

Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2
Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw–Hill Companies, Chapter 13 §4.4, p. 291, ISBN 0-07-141083-X
Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, p. 134, ISBN 978-0-8218-0587-9
Irving 2013, p. 34. Irving, Ron (2013), Beyond the Quadratic Formula, MAA, ISBN 978-0-88385-783-0
The Cambridge Ancient History Part 2 Early History of the Middle East, Cambridge University Press, 1971, p. 530, ISBN 978-0-521-07791-0
Irving 2013, p. 39. Irving, Ron (2013), Beyond the Quadratic Formula, MAA, ISBN 978-0-88385-783-0
Irving 2013, p. 42. Irving, Ron (2013), Beyond the Quadratic Formula, MAA, ISBN 978-0-88385-783-0

csusm.edu (Global: low place; English: low place)

public.csusm.edu

Aitken, Wayne, "A Chinese Classic: The Nine Chapters" (PDF), Mathematics Department, California State University, retrieved 28 April 2013

doi.org (Global: 2^nd place; English: 2^nd place)

Specifically, Fagnano began with the equation ⁠ $xx+bb=ax$ ⁠ and found the solutions to be $x={\frac {2bb}{a\mp a{\sqrt {1-{\dfrac {4bb}{aa}}}}}}.$ (In the 18th century, the square ⁠ $\textstyle x^{2}$ ⁠ was conventionally written as ⁠ $xx$ ⁠.)

Fagnano, Giulio Carlo (1750), "Applicazione dell' algoritmo nuovo Alla resoluzione analitica dell' equazioni del secondo, del terzo, e del quarto grado" [Application of a new algorithm to the analytical resolution of equations of the second, third, and fourth degree], Produzioni matematiche del conte Giulio Carlo di Fagnano, Marchese de' Toschi, e DiSant' Ononio (in Italian), vol. 1, Pesaro: Gavelliana, Appendice seconda, eq. 6, p. 467, doi:10.3931/e-rara-8663
Goff, Gerald K. (1976), "The Citardauq Formula", The Mathematics Teacher, 69 (7): 550–551, doi:10.5951/MT.69.7.0550, JSTOR 27960584
Hoehn, Larry (1975), "A More Elegant Method of Deriving the Quadratic Formula", The Mathematics Teacher, 68 (5): 442–443, doi:10.5951/MT.68.5.0442, JSTOR 27960212
Debnath, Lokenath (2009), "The legacy of Leonhard Euler – a tricentennial tribute", International Journal of Mathematical Education in Science and Technology, 40 (3): 353–388, doi:10.1080/00207390802642237, S2CID 123048345
Thompson, H. Bradford (1987), "Good numerical technique in chemistry: The quadratic equation", Journal of Chemical Education, 64 (12): 1009, Bibcode:1987JChEd..64.1009T, doi:10.1021/ed064p1009
Baker, Henry G. (1998), "You Could Learn a Lot from a Quadratic: Overloading Considered Harmful", SIGPLAN Notices, 33 (1): 30–38, doi:10.1145/609742.609746

dtic.mil (Global: 833^rd place; English: 567^th place)

apps.dtic.mil

Forsythe, George E. (1966), How Do You Solve a Quadratic Equation (PDF) (Tech report), Stanford University, STAN-CS-66-40 (AD639052)

ethz.ch (Global: 2,224^th place; English: 1,900^th place)

era-prod11.ethz.ch

Specifically, Fagnano began with the equation ⁠ $xx+bb=ax$ ⁠ and found the solutions to be $x={\frac {2bb}{a\mp a{\sqrt {1-{\dfrac {4bb}{aa}}}}}}.$ (In the 18th century, the square ⁠ $\textstyle x^{2}$ ⁠ was conventionally written as ⁠ $xx$ ⁠.)

Fagnano, Giulio Carlo (1750), "Applicazione dell' algoritmo nuovo Alla resoluzione analitica dell' equazioni del secondo, del terzo, e del quarto grado" [Application of a new algorithm to the analytical resolution of equations of the second, third, and fourth degree], Produzioni matematiche del conte Giulio Carlo di Fagnano, Marchese de' Toschi, e DiSant' Ononio (in Italian), vol. 1, Pesaro: Gavelliana, Appendice seconda, eq. 6, p. 467, doi:10.3931/e-rara-8663

handle.net (Global: 102^nd place; English: 76^th place)

hdl.handle.net

Rocha, Rodrigo Luis da (2023). O uso da expressão 'fórmula de bhaskara' em livros didáticos brasileiros e sua relação com o método resolutivo da equação do 2º grau [The use of the expression 'bhaskara formula' in Brazilian textbooks and its relationship with the method for solving quadratic equations] (master's thesis) (in Portuguese). Universidade Federal do Paraná. hdl:1884/82597.
Guedes, Eduardo Gomes (2019). A equação quadrática e as contribuições de Bhaskara [The quadratic equation and Bhaskara's contributions] (master's thesis) (in Portuguese). Universidade Federal do Paraná. hdl:1884/66582.
Banerjee, Isha (July 2, 2024). "India Molded Math. Then Europe Claimed It". The Juggernaut. For instance, some Indian schools call the quadratic formula Sridharacharya's formula and some Brazilian schools call it Bhaskara's formula.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

Thompson, H. Bradford (1987), "Good numerical technique in chemistry: The quadratic equation", Journal of Chemical Education, 64 (12): 1009, Bibcode:1987JChEd..64.1009T, doi:10.1021/ed064p1009

jstor.org (Global: 26^th place; English: 20^th place)

Goff, Gerald K. (1976), "The Citardauq Formula", The Mathematics Teacher, 69 (7): 550–551, doi:10.5951/MT.69.7.0550, JSTOR 27960584
Hoehn, Larry (1975), "A More Elegant Method of Deriving the Quadratic Formula", The Mathematics Teacher, 68 (5): 442–443, doi:10.5951/MT.68.5.0442, JSTOR 27960212

kent.edu (Global: 9,450^th place; English: 5,700^th place)

etna.math.kent.edu

Mastronardi, Nicola; Van Dooren, Paul (2015), "Revisiting the stability of computing the roots of a quadratic polynomial", Electronic Transactions on Numerical Analysis, 44: 73–83, arXiv:1409.8072, archived from the original on 2025-03-03, retrieved 2024-03-15

khanacademy.org (Global: 3,659^th place; English: 2,881^st place)

"Discriminant review", Khan Academy, retrieved 2019-11-10
"Understanding the quadratic formula", Khan Academy, retrieved 2019-11-10

knaw.nl (Global: 2,678^th place; English: 2,423^rd place)

dwc.knaw.nl

Struik, D. J.; Stevin, Simon (1958), The Principal Works of Simon Stevin, Mathematics (PDF), vol. II–B, C. V. Swets & Zeitlinger, p. 470

mathwarehouse.com (Global: low place; English: low place)

"Axis of Symmetry of a Parabola. How to find axis from equation or from a graph. To find the axis of symmetry ...", www.mathwarehouse.com, retrieved 2019-11-10

semanticscholar.org (Global: 11^th place; English: 8^th place)

api.semanticscholar.org

Debnath, Lokenath (2009), "The legacy of Leonhard Euler – a tricentennial tribute", International Journal of Mathematical Education in Science and Technology, 40 (3): 353–388, doi:10.1080/00207390802642237, S2CID 123048345

st-andrews.ac.uk (Global: 1,547^th place; English: 1,410^th place)

mathshistory.st-andrews.ac.uk

O'Connor, John J.; Robertson, Edmund F. (2000), "Sridhara", MacTutor History of Mathematics Archive, University of St Andrews

thejuggernaut.com (Global: low place; English: low place)

Rocha, Rodrigo Luis da (2023). O uso da expressão 'fórmula de bhaskara' em livros didáticos brasileiros e sua relação com o método resolutivo da equação do 2º grau [The use of the expression 'bhaskara formula' in Brazilian textbooks and its relationship with the method for solving quadratic equations] (master's thesis) (in Portuguese). Universidade Federal do Paraná. hdl:1884/82597.
Guedes, Eduardo Gomes (2019). A equação quadrática e as contribuições de Bhaskara [The quadratic equation and Bhaskara's contributions] (master's thesis) (in Portuguese). Universidade Federal do Paraná. hdl:1884/66582.
Banerjee, Isha (July 2, 2024). "India Molded Math. Then Europe Claimed It". The Juggernaut. For instance, some Indian schools call the quadratic formula Sridharacharya's formula and some Brazilian schools call it Bhaskara's formula.

web.archive.org (Global: 1^st place; English: 1^st place)

Mastronardi, Nicola; Van Dooren, Paul (2015), "Revisiting the stability of computing the roots of a quadratic polynomial", Electronic Transactions on Numerical Analysis, 44: 73–83, arXiv:1409.8072, archived from the original on 2025-03-03, retrieved 2024-03-15

Quadratic formula (English Wikipedia)

archive.org (Global: 6th place; English: 6th place)

archives-ouvertes.fr (Global: 1,031st place; English: 879th place)

hal.archives-ouvertes.fr

arxiv.org (Global: 69th place; English: 59th place)

books.google.com (Global: 3rd place; English: 3rd place)

csusm.edu (Global: low place; English: low place)

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doi.org (Global: 2nd place; English: 2nd place)

dtic.mil (Global: 833rd place; English: 567th place)

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ethz.ch (Global: 2,224th place; English: 1,900th place)

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handle.net (Global: 102nd place; English: 76th place)

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harvard.edu (Global: 18th place; English: 17th place)

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jstor.org (Global: 26th place; English: 20th place)

kent.edu (Global: 9,450th place; English: 5,700th place)

etna.math.kent.edu

khanacademy.org (Global: 3,659th place; English: 2,881st place)

knaw.nl (Global: 2,678th place; English: 2,423rd place)

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mathwarehouse.com (Global: low place; English: low place)

semanticscholar.org (Global: 11th place; English: 8th place)

api.semanticscholar.org

st-andrews.ac.uk (Global: 1,547th place; English: 1,410th place)

mathshistory.st-andrews.ac.uk

thejuggernaut.com (Global: low place; English: low place)

web.archive.org (Global: 1st place; English: 1st place)

archive.org (Global: 6^th place; English: 6^th place)

archives-ouvertes.fr (Global: 1,031^st place; English: 879^th place)

arxiv.org (Global: 69^th place; English: 59^th place)

books.google.com (Global: 3^rd place; English: 3^rd place)

doi.org (Global: 2^nd place; English: 2^nd place)

dtic.mil (Global: 833^rd place; English: 567^th place)

ethz.ch (Global: 2,224^th place; English: 1,900^th place)

handle.net (Global: 102^nd place; English: 76^th place)

harvard.edu (Global: 18^th place; English: 17^th place)

jstor.org (Global: 26^th place; English: 20^th place)

kent.edu (Global: 9,450^th place; English: 5,700^th place)

khanacademy.org (Global: 3,659^th place; English: 2,881^st place)

knaw.nl (Global: 2,678^th place; English: 2,423^rd place)

semanticscholar.org (Global: 11^th place; English: 8^th place)

st-andrews.ac.uk (Global: 1,547^th place; English: 1,410^th place)

web.archive.org (Global: 1^st place; English: 1^st place)