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[parent] biquadratic equation (Topic)

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

$\displaystyle ax^4+bx^2+c = 0$ (1)

Here, $a$ , $b$ , $c$ are known real or complex numbers and $a \neq 0$ .

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 $x^2$ , i.e. $$a(x^2)^2+bx^2+c = 0,$$ whence $$x^2 = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ (see quadratic formula or quadratic equation in $\mathbb{C}$ ). Taking square roots of the values of $x^2$ (see taking square root algebraically), one obtains the four roots of (1).

Example. Solve the biquadratic equation

$\displaystyle x^4+x^2-20 = 0.$ (2)

We have
$\displaystyle x^2 = \frac{-1\pm\sqrt{1^2-4\cdot1\cdot(-20)}}{2\cdot1} = \frac{-1\pm9}{2},$ (3)

i.e. $x^2 = 4$ or $x^2 = -5$ . The solution is
$\displaystyle x = \pm2 \quad \lor \quad x = \pm i\sqrt{5}.$ (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) $x = 1+\sqrt2+\sqrt3$
$(x-1)^2 = 2+2\sqrt{6}+3$
$y^2-5 = 2\sqrt{6}$
$y^4-10y^2+1 = 0$ (one has substituted $x-1 := y$ )

2) $x = \sqrt{\sqrt{2}-1}$
$x^2 = \sqrt{2}-1$
$(x^2+1)^2 = 2$
$x^4+2x^2-1 = 0$




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See Also: biquadratic extension, biquadratic field, Euler's derivation of the quartic formula, irreducible polynomials obtained from biquadratic fields, logical or

Also defines:  biquadratic equation

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Cross-references: operations, coefficients, rational, polynomial, rational numbers, solution, taking square root algebraically, square roots, quadratic formula, quadratic equation, equation, complex numbers, real, terms, degree, odd
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This is version 8 of biquadratic equation, born on 2008-03-03, modified 2008-12-29.
Object id is 10360, canonical name is BiquadraticEquation2.
Accessed 1975 times total.

Classification:
AMS MSC12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 30-00 (Functions of a complex variable :: General reference works )

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