symmetric quartic equation

0.1 Symmetric quartic

Besides the biquadratic equationPlanetmathPlanetmath, there are other of quartic equations

a0z4+a1z3+a2z2+a3z+a4= 0  (a0 0), (1)

which can be reduced to quadratic equations.  If the left hand side of (1) is P(z), one may write the identityPlanetmathPlanetmath

P(z)z2=(a0z2+a4z2)+(a1z+a3z)+a2. (2)

If we assume first that  a4=a0  and  a3=a1,  the identity is


We set  z+1z:=x,  whence  z2+1z2=x2-2;  hence the identity is simplified to


Accordingly, we obtain the roots ( of the so-called symmetric quartic equation

a0z4+a1z3+a2z2+a1z+a0= 0  (a0 0), (3)

if we first determine the roots x1 and x2 of the quadratic

a0x2+a1x+a2-2a0= 0

and then solve the equations  z+1z=x1 and z+1z=x2 which can be written

z2-x1z+1= 0,z2-x2z+1= 0. (4)

Note, that the roots of either equations (4) are inverse numbers of each other (see properties of quadratic equations).  Therefore, as well the inverse number of any root of the symmetric quartic (3) is a root of (3); this fact is, by the way, clear also because of the identity


Example.  The equation

2z4-5z3+4z2-5z+2= 0

is symmetric.  Thus we solve first

2x2-5x+4-22= 0,

which yields  x1=0,  x2=52.  Secondly we solve

z2+1= 0,z2-52z+1= 0

which yield all the four roots  z=±i,  z=12,  z=2  of the quartic.

0.2 Almost symmetric quartic

There is still the quartic equation

a0z4+a1z3+a2z2-a1z+a0= 0  (a00), (5)

which reduces to quadratics — the identity (2) for it reads


The substitution  z-1z:=x  converts it to


Thus, (5) can be solved by determining first the roots x1 and x2 of

a0x2+a1x+a2+2a0= 0,

then the roots z of  z-1z=x1 and z-1z=x2  which may written

z2-x1z-1= 0,z2-x2z-1= 0.

Hence one infers, that if z is a root of (5), so is also its opposite inverseMathworldPlanetmath -1z; this is apparent also due to the identity



  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
  • 2 E. Lindelöf: Einführung in die höhere Analysis. Nach der ersten schwedischen und zweiten finnischen Auflage auf deutsch herausgegeben von E. Ullrich. Teubner, Leipzig (1934).
Title symmetric quartic equation
Canonical name SymmetricQuarticEquation
Date of creation 2013-03-22 18:05:28
Last modified on 2013-03-22 18:05:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Topic
Classification msc 30-00
Classification msc 12D99
Synonym symmetric quartic
Related topic AlgebraicEquation
Related topic EulersDerivationOfTheQuarticFormula
Related topic ErnstLindelof