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Topic: Biquadratic equation

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Quartic equation

Solution point

Quadratic equation

Functional equation

Cubic equation

Ordinary differential equation

List of equations

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	Equation - LoveToKnow 1911
		Equations of the first degree are called simple or linear; of the second, quadratic; of the third, cubic; of the fourth, biquadratic; of the fifth, quintic, and so on.
		Quadratic equations arose in the Greek investigations in the doctrine of proportion, and although they were presented and solved in a geometrical form, the methods employed have no relation to the generalized conception of algebraic geometry which represents a curve by an equation and vice versa.
		This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois's theory.
	www.1911encyclopedia.org /Equation (10531 words)

	biquadratic - definition by dict.die.net (Site not responding. Last check: )
		biquadratic adj : of or relating to the fourth power n 1: an algebraic equation of the fourth degree [syn: biquadrate, quartic, fourth power] 2: and equation of the fourth degree [syn: biquadratic equation] 3: a polynomial of the fourth degree [syn: biquadratic polynomial, quartic polynomial]
		Biquadratic equation (Alg.), an equation of the fourth degree, or an equation in some term of which the unknown quantity is raised to the fourth power.
		Biquadratic root of a number, the square root of the square root of that number.
	dict.die.net /biquadratic (121 words)

	[No title]
		Pure cubic equations are therefore of the form x3=r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube root of each side of the equation.
		Thus the equation 23+222+p2164rz 64=0 becomes, after reduction, v3+2pv2+(p2—4r)v—q2=o; it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are 4a, ;b, ;c, so that our rule may now be expressed thus: Let y4+py2+qy+r=o be any biquadratic equation wanting its second term.
		Thus we have the biquadratic equation y4+2Py22—84 R.y+P2-4Q=o, one of the roots of which is y= J a+ J b+ A) c, while a, b, c are the roots of the cubic equation z3+Pz2+Qz—R=o.
	encyclopedia.jrank.org /correction/edit?content_id=23314&locale=en (10461 words)

	EQUATION - Definition
		{Equation of the center} (Astron.), the difference between the place of a planet as supposed to move uniformly in a circle, and its place as moving in an ellipse.
		{Equations of condition} (Math.), equations formed for deducing the true values of certain quantities from others on which they depend, when different sets of the latter, as given by observation, would yield different values of the quantities sought, and the number of equations that may be found is greater than the number of unknown quantities.
		{Equation of time} (Astron.), the difference between mean and apparent time, or between the time of day indicated by the sun, and that by a perfect clock going uniformly all the year round.
	www.hyperdictionary.com /dictionary/equation (388 words)

	BIQUADRATIC - Definition
		{Biquadratic equation} (Alg.), an equation of the fourth degree, or an equation in some term of which the unknown quantity is raised to the fourth power.
		{Biquadratic root of a number}, the square root of the square root of that number.
		Thus the square root of 81 is 9, and the square root of 9 is 3, which is the biquadratic root of 81.
	www.hyperdictionary.com /dictionary/biquadratic (101 words)

	Archimedes Project
		Biquadratic Equation, is that which rises to 4 dimensions, or in which the unknown quantity rises to the 4th power; as.
		Any biquadratic equation may be conceived to be generated or produced from the continual multiplication of four simple equations, as ; or from that of two quadratic equations, as ; or, lastly, from that of a cubic and a simple equation, as : which was the invention of Harriot.
		And, on the contrary, a biquadratic equation may be resolved into four simple equations, or into two quadratics, or into a cubic and a simple equation, having all the same roots with it.
	archimedes.mpiwg-berlin.mpg.de /cgi-bin/toc/toc.cgi?dir=hutto_dicti_078_en_1795;step=textonly;corpus=&page=226&n=226 (657 words)

	English - English dict. (Wordnet) - biquadratic (Site not responding. Last check: )
		n 1: an algebraic equation of the fourth degree [syn:
		2: and equation of the fourth degree [syn: {biquadratic
		3: a polynomial of the fourth degree [syn: {biquadratic
	vdict.com /i/7/simple.php?dictionary=7&word=biquadratic (37 words)

	Ecuación de 2º grado (22) (Site not responding. Last check: )
		Biquadratic equations are those equations which relate to the fourth power and do not contain any terms of the third or first power.
		These equations can be solved graphically in the same way as the quadratic equations by drawing the graph of the LHS of the equation once the RHS is equal to 0.
		Write the biquadratic equation in the box on the left and the quadratic equation which also needs solving in the other box, on the right.
	descartes.cnice.mecd.es /ingles/4th_year_secondary_educ_B/Quadratic_equations/Ecu_seg22.htm (627 words)

	EQUATION (from Lat. ae... - Online Information article about EQUATION (from Lat. ae...
		equation states an equality existing between two classes of quantities, distinguished as known and unknown; these correspond to the data of a problem and the thing sought.
		hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown, and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth.
		Vandermonde's remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c.
	encyclopedia.jrank.org /EMS_EUD/EQUATION_from_Lat_aequatio_aequ.html (10856 words)

	yPass.net - Online Dictionary
		n 1: an algebraic equation of the fourth degree [syn: \{biquadrate\},
		2: and equation of the fourth degree [syn: {biquadratic
		3: a polynomial of the fourth degree [syn: {biquadratic
	www.ypass.net /dictionary/index.html?word=biquadratic (39 words)

	ON-LINE MATHEMATICS DICTIONARY
		An equation of the form f(x)=0 where f is a polynomial.
		The curve whose equation in polar coordinates is r*theta=a.
		An equation of the form f(x)=0 where f(x) is a second degree polynomial.
	pax.st.usm.edu /cmi/inform_html/glossary.html (4061 words)

	Biquadratic equation
		Quartic equations were first considered by Jaina Mathematicians in ancient India between 400 BC and 200 AD.
		Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately.
		Equation (4) is a cubic equation nested within the quartic equation.
	www.algebra.com /algebra/homework/equations/Biquadratic_equation.wikipedia (2059 words)

	Education
		The relations between the roots and coefficients of a quadratic equation, a cubic equation and a biquadratic - equation.
		Proofs of (i) irrational roots of a polynomial equation occur in conjugate pairs, (ii) complex roots of a polynomial equation occur in conjugate pairs - Problems of solving equations given an irrational root and given a complex root - problems.
		Equation of the tangent to a circle - Derivation.
	www.onlinebangalore.com /educ/cet/cetmat.html (1911 words)

	PlanetMath: quartic formula
		The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer.
		A practical algorithm for solving quartic equations by radicals is given in the concluding paragraph of the Galois-theoretic derivation of the quartic formula.
		My (not ease) job was to extract the Ts and Rs from your set of equations...
	planetmath.org /encyclopedia/BiquadraticEquation.html (235 words)

	Quartic equation Summary
		The extreme competitiveness of the mathematicians involved in solving the cubic and quartic equations is consistent with the aggressive individualism of the Renaissance.
		With the solutions of the cubic and quartic equations known, it might seem that the solution of the quintic equation, which included the fifth power of the unknown, and even higher order equations would be achieved eventually.
		Quartic equations were first discovered by Jaina Mathematicians in ancient India between 400 BC and 200 CE.
	www.bookrags.com /Quartic_equation (3541 words)

	Ecuacion_seg_2 (Site not responding. Last check: )
		Therefore, the "discriminant of the equation is 0".
		We have seen that the number of roots of a quadratic equation depends on the sign of the number which we get inside the square root of the quadratic equation formula.
		Biquadratic equations are those equations which relate to the fourth power and do
	descartes.cnice.mecd.es /ingles/Bach_CNST_1/Simultaneous_equations_inequations/Ecuacion_seg_2.htm (1164 words)

	Fast Track Communication
		In [1] Adler derived a quadrilateral lattice equation as the nonlinear superposition principle for Bäcklund transformations (BTs) of the Krichever–Novikov (KN) equation [2, 3].
		The main integrability property of this equation is that of multidimensional consistency, cf [5, 6], which implies that solutions of (1.1) can be embedded in a multidimensional lattice such that they obey simultaneously equations of a similar form (albeit with different choices of the lattice parameters) in all two-dimensional sublattices.
		Equation (1.1) emerged as the most general equation in a classification of scalar quadrilateral lattice equations integrable in this sense, [7], which includes the previously known cases of lattice equations of Korteweg-de Vries type, cf [8, 9].
	www.iop.org /EJ/article/1751-8121/40/1/F01/a7_1_f01.html (2800 words)

	RIGEP Mathematics - Definitions
		An equation of the form f(x)=0 where f is a polynomial.
		The curve whose equation in polar coordinates is r*theta=a.
		An equation of the form f(x)=0 where f(x) is a second degree polynomial.
	www.wireplastik.com /folio/riil/definitions.html (4012 words)

	ALGEBRA FOR STATICS
		A system of linear equations is a set of two or more equations that are linear in the designated variables.
		This involves elimination of one equation by taking one equation and solving for one variable in terms of the other variables and then substituting this expression into the remaining equations to reduce the number of equations and unknown variables.
		Once the system is reduced to one equation and one variable, back substitution is used to solve for the previously eliminated unknown variables.
	em-ntserver.unl.edu /Math/mathweb/algebra/algesb97.html (1126 words)

	quartic - Search Results - MSN Encarta
		The general form of a quartic equation is
		Such a function is sometimes called a biquadratic...
		A quartic equation is a fourth-order polynomial equation of the form z^4+a_3z^3+a_2z^2+a_1z+a_0=0.
	ca.encarta.msn.com /quartic.html (117 words)

	Equation - Article from FactBug.org - the fast Wikipedia mirror site (Site not responding. Last check: )
		In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables.
		In equations, the values of the variables for which the equation is true are called solutions.
		Thus to solve the equation, one must find what values fulfill the condition stated as an equation.
	www.factbug.org /cgi-bin/a.cgi?a=9284 (198 words)

	[No title]
		Solving first degree polynomial equations in one variable is trivial; the solutions to second-degree polynomial equations can involve irrational (square root) and complex numbers; cubic and biquadratic equation solutions are algorithmetic.
		A single linear equation in a single variable can be manipulated into the form $ax~=~b$, where all the factors of the variable have been collected together and the constant terms have likewise been collected together via arithmetic equivalence operations, i.e.
		A system of linear algebraic equations can be expressed efficiently in matrix notation:.EQ (6.1) bold A bold x~=~bold b,.EN where $bold x$ is a column array (vector) of n unknown variables, $bold A$ is a square $n times n$ matrix of coefficient parameters, and $bold b$ is a column array of n "target" values.
	www.unm.edu /~dmclaugh/Computation/6_Numerical (3293 words)

	¥³.The Sixteenth-Century Mathematics of Italy: Commercial Mathematics
		The only thing focused was a solving of an equation of the third and fourth degree and symbolizing algebra of France.
		Believing this claim was a bluff, Fior challinged Targaglia to a public contest of solving cubic equations, whereupon the latter exerted himself and only a few days before the contest found an algebraic solution for cubics lacking a quadratic term.
		It was not long after the cubic had been solved that an algebraic solution was discovered for the general quartic (or biquadratic) equation.
	library.thinkquest.org /22584/emh1400.htm (880 words)

	OEDILF
		Biquadratic describes a polynomial or algebraic equation of the fourth degree (meaning that the highest exponent of any term is to the fourth power, such as x
		A quartic equation is a fourth degree function set to zero.
		) are zero, then we have a biquadratic equation, which can be solved in a manner similar to the quadratic equation, well known to all algebra students.
	www.oedilf.com /db/Lim.php?VerseId=61597 (169 words)

	Abstract FPE
		Abiquadratic equation of state for the attractive hard sphere chain fluid with dipolar interactions is developed.
		The background of the development is the mapping of molecular model equations onto a simplified mathematical form.
		For the hard sphere reference model a simple theoretical equation is developed, which is almost as accurate as the Carnahan-Starling (CS) equation and includes the nearly correct high density limit.
	van-der-waals.pc.uni-koeln.de /papers/FPE182.html (177 words)

	[No title]
		Cayley, On a new auxiliary equation in theory of equations of the 5th degree, Philos.
		Harley, A contribution to the history of the problem of the reduction of the general equation of the fifth degree to a trinomial form, Quart.
		G.B. Jerrard, Reflections on the resolution of algebraic equations of the fifth degree, Phil.
	www1.elsevier.com /homepage/sac/cam/mcnamee/19.htm (2975 words)

	All Elementary Mathematics - Study Guide - Algebra - Equations of higher degrees...
		Some kinds of the higher degrees equations may be solved using a quadratic equation.
		Sometimes one can resolve the left-hand side of equation to factors, each of them is a polynomial of the degree not higher than second.
		The known Cardano’s formulas for solution of this kind equations are very difficult and almost aren’t used in practice.
	www.bymath.com /studyguide/alg/sec/alg24.html (412 words)

	四次方程[Quartic Equation] - 揭示大自然的规律 - 5dblog.com (Site not responding. Last check: )
		A general quartic equation (also called a biquadratic equation) is a fourth-order polynomial equation of the form
		Ferrari was the first to develop an algebraic technique for solving the general quartic, which was stolen and published in Cardano's Ars Magna in 1545 (Boyer and Merzbach 1991, p.
		The quartic can be solved by writing it in a general form that would allow it to be algebraically factorable and then finding the condition to put it in this form.
	blog.5d.cn /user2/iamet/200409/17989.html (461 words)

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