# biquadratic equation

A biquadratic equation^{} (in a narrower sense) is the special case of the quartic equation (http://planetmath.org/QuarticFormula) containing no odd degree terms:

$a{x}^{4}+b{x}^{2}+c=0$ | (1) |

Here, $a$, $b$, $c$ are known real or complex numbers^{} and $a\ne 0$.

For solving a biquadratic equation (1) one does not need the quartic formula (http://planetmath.org/QuarticFormula) since the equation may be thought a quadratic equation with respect to ${x}^{2}$, i.e.

$$a{({x}^{2})}^{2}+b{x}^{2}+c=0,$$ |

whence

$${x}^{2}=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$$ |

(see quadratic formula or quadratic equation in $\u2102$ (http://planetmath.org/QuadraticEquationInMathbbC)). Taking square roots of the values of ${x}^{2}$ (see taking square root algebraically), one obtains the four roots (http://planetmath.org/Equation) of (1).

Example. Solve the biquadratic equation

${x}^{4}+{x}^{2}-20=0.$ | (2) |

We have

${x}^{2}={\displaystyle \frac{-1\pm \sqrt{{1}^{2}-4\cdot 1\cdot (-20)}}{2\cdot 1}}={\displaystyle \frac{-1\pm 9}{2}},$ | (3) |

i.e. ${x}^{2}=4$ or ${x}^{2}=-5$. The solution is

$x=\pm 2\mathit{\hspace{1em}}\vee \mathit{\hspace{1em}}x=\pm i\sqrt{5}.$ | (4) |

Remark. In one wants to form of rational numbers^{} 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) $x=1+\sqrt{2}+\sqrt{3}$

${(x-1)}^{2}=2+2\sqrt{6}+3$

${y}^{2}-5=2\sqrt{6}$

${y}^{4}-10{y}^{2}+1=0$ (one has substituted (http://planetmath.org/TchirnhausTransformations) $x-1:=y$)

2) $x=\sqrt{\sqrt{2}-1}$

${x}^{2}=\sqrt{2}-1$

${({x}^{2}+1)}^{2}=2$

${x}^{4}+2{x}^{2}-1=0$

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 |