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From Andrew Ellinor & Satyajit Mohanty's article on a technique for solving cubics:

Using Lagrange's resolvents, to solve the cubic, one has to first solve a quadratic. Given the general cubic, $$x^3 + ax^2 + bx + c = 0$$ It's [sic] resolvent equation is given by $$z^2 + z \left( 2 a^3 - 9 a b + 27 c \right) + \left( a^2 - 3b \right)^3 = 0$$ such that that the solution is given by $$x= \frac{ -a+z_{1}^{\frac{1}{3}} + z_{2}^{\frac{1}{3}} }{3}$$ where $z_{1,2}$ are roots of the resolvent


Does anyone know where, in which work, Lagrange did actually develop this technique?

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The relevant publication seems to have been published across two annual volumes (for the years 1770 and 1771, but likely printed in 1773) of the Proceedings of the Royal Academy of Science and Literature in Berlin. At present, I have not been able to find a scan of the actual publication, but it is reproduced in the third volume of the collected works of Lagrange published in 1869:

J. Lagrange, "Réflexions sur la résolution algébrique des équations", Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, années 1770 et 1771. In: J.-A. Serret (ed.), Oeuvres de Lagrange, vol. 3, Paris: Gauthier-Villars 1869, pp. 205-421

The Gallica digital repository of the Bibliothèque nationale de France (BNF) provides a scan of the volume online. Continuing my search for the original source, I found the first part of the publication here:

J. Lagrange, "Réflexions sur la résolution algébrique des équations", Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, 1770 (1772), pp. 134-215 (Google scan online).

At long last, I also managed to locate the second part of the original publication:

J. Lagrange, "Suite des Réflexions sur la Résolution algébrique des Équations", Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, 1771 (1773), pp. 138-253 (Google scan online).

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    $\begingroup$ I note that @Conifold provided largely identical information in a comment while I was in the process of writing this answer. $\endgroup$
    – njuffa
    Jun 6, 2023 at 20:57

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