Euler’s derivation of the quartic formula

Let us consider the quartic equation

y4+py2+qy+r=0, (1)

where p,q,r are arbitrary known complex numbersMathworldPlanetmathPlanetmath.  We substitute in the equation

y:=u+v+w. (2)

We get firstly

Thus (1) attains the form


When u,v,w are determined so that

u2+v2+w2=-p2, (3)
uvw=-q8, (4)

the expressions in the brackets vanish and our equation shrinks to the form

v2w2+w2u2+u2v2=p2-4r16. (5)

Squaring (4) gives

u2v2w2=q264. (6)

The left hand sides of (3), (5) and (6) are the elementary symmetric polynomials of u2, v2, w2, whence these three squares are the roots z1, z2, z3 of the so-called cubic resolvent equation

z3+p2z2+p2-4r16z-q264=0. (7)

Therefore we may write


All 8 sign combinationsMathworldPlanetmathPlanetmath of those square roots satisfy the equations (3), (5), (6). In order to satisfy also (4) the signs must be chosen suitably.  If  u0,v0,w0 is some suitable combination of the values of the square roots, then all possible combinations are


Accordingly, we have the

Theorem (Euler 1739).  The roots of the equation (1) are

{y1=u0+v0+w0,y2=u0-v0-w0,y3=-u0+v0-w0,y4=-u0-v0+w0, (8)

where u0,v0,w0 are square roots of the roots of the cubic resolvent (7).  The signs of the square roots must be chosen such that


The equations (8) imply an important formulaMathworldPlanetmathPlanetmath

(y1-y2)(y1-y3)(y1-y4)(y2-y3)(y2-y4)(y3-y4)= -26(v02-w02)(w02-u02)(u02-v02)
= -64(z2-z3)(z3-z1)(z1-z2),

which yields the

Corollary.  A quartic equation has a multiple root always and only when its cubic resolvent has such one.


  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
  • 2 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
Title Euler’s derivation of the quartic formula
Canonical name EulersDerivationOfTheQuarticFormula
Date of creation 2013-03-22 17:51:58
Last modified on 2013-03-22 17:51:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 12D10
Synonym quartic formula by Euler
Related topic TchirnhausTransformations
Related topic CasusIrreducibilis
Related topic ZeroRuleOfProduct
Related topic ErnstLindelof
Related topic KalleVaisala
Related topic BiquadraticEquation2
Related topic SymmetricQuarticEquation