biquadratic equation

A biquadratic equationPlanetmathPlanetmath (in a narrower sense) is the special case of the quartic equation ( containing no odd degree terms:

ax4+bx2+c=0 (1)

Here, a, b, c are known real or complex numbersMathworldPlanetmathPlanetmath and  a0.

For solving a biquadratic equation (1) one does not need the quartic formula ( since the equation may be thought a quadratic equation with respect to x2, i.e.




(see quadratic formula or quadratic equation in (  Taking square roots of the values of x2 (see taking square root algebraically), one obtains the four roots ( of (1).

Example.  Solve the biquadratic equation

x4+x2-20=0. (2)

We have

x2=-1±12-41(-20)21=-1±92, (3)

i.e.  x2=4  or  x2=-5.  The solution is

x=±2x=±i5. (4)

Remark.  In one wants to form of rational numbersPlanetmathPlanetmathPlanetmath a polynomial equation with rational coefficients and most possibly low degree by using two square root operations, then one gets always a biquadratic equation.  A couple of examples:

1) x=1+2+3
y4-10y2+1=0  (one has substituted (  x-1:=y)

2) x=2-1

Title biquadratic equation
Canonical name BiquadraticEquation
Date of creation 2013-03-22 17:52:45
Last modified on 2013-03-22 17:52:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Topic
Classification msc 30-00
Classification msc 12D99
Related topic BiquadraticExtension
Related topic BiquadraticField
Related topic EulersDerivationOfTheQuarticFormula
Related topic IrreduciblePolynomialsObtainedFromBiquadraticFields
Related topic LogicalOr
Defines biquadratic equation