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E30 in the Enestrom index. Translated from the Latin original "De formisradicum aequationum cuiusque ordinis coniectatio" (1733). For an equation ofdegree n, Euler wants to define a "resolvent equation" of degree n-1 whoseroots are related to the roots of the original equation. Thus by solving theresolvent we can solve the original equation. In sections 2 to 7 he works thisout for quadratic, cubic and biquadratic equations. Apparently he gives a newmethod for solving the quartic in section 5. Then in section 8 Euler says thathe wants to try the same approach for solving the quintic equation and generalnth degree equations. In the rest of the paper Euler tries to figure out inwhat cases resolvents will work.  Two references I found useful were Chapter 14, p.p. 106-113 of C. EdwardSandifer, "The Early Mathematics of Leonhard Euler", published 2007 by TheMathematical Association of America and Olaf Neumann, "Cyclotomy: from Eulerthrough Vandermonde to Gauss", p.p. 323-362 in the collection "Leonhard Euler:Life, Work and Legacy" edited by Bradley and Sandifer, 2007. Stacy Langton hasgiven a lot of details about Euler's work on the theory of equations, and alsosome advice on the translation; of course any mistakes are my own. If Langtonends up writing anything about Euler' and the theory of equations I wouldhighly recommend reading it.

A conjecture on the forms of the roots of equations