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Topic: Quartic equation

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	Klein's Quartic Curve
		Klein's quartic curve is a surface of genus 3, which is to say that it is like a 3-holed torus.
		The 56 triangular faces of the dual tiling of Klein's quartic curve can be divided into 7 sets of 8, which have the same symmetries as the 8 corners of a truncated cube.
		Klein's quartic curve also has symmetries of order 7, which are not very clear in any of the pictures shown so far.
	gregegan.customer.netspace.net.au /SCIENCE/KleinQuartic/KleinQuartic.html (0 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.
		This is version 3 of quartic formula, born on 2002-01-22, modified 2005-07-07.
	planetmath.org /encyclopedia/QuarticFormula.html (236 words)

	Kids.Net.Au - Encyclopedia > Quartic equation (Site not responding. Last check: )
		A quartic equation is the result of setting a quartic function[?] to zero, an example quartic equation is the equation
		A quartic equation always has 4 solutions (or roots).They may be complex or there may be duplicate solutions.
		It is the highest degree of polynomial equation for which exact values of the roots can be found, by taking nth roots, and use of the normal algebraic operators.
	www.kids.net.au /encyclopedia-wiki/qu/Quartic_equation (135 words)

	quartic - Search Results - MSN Encarta
		In mathematics, a quartic equation is the result of setting a quartic function equal to zero.
		An example of a quartic equation is the equation the general form is where.
		A general quartic equation (also called a biquadratic equation) is a fourth-order polynomial...
	encarta.msn.com /quartic.html (145 words)

	Cubic Equation - Search Results - MSN Encarta
		In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		An example is the equation 2x 4x + 3x 4 = 0 and the general form may...
		A cubic equation is an equation involving a cubic polynomial, i.e., one of the form a_3x^3+a_2x^2+a...
	encarta.msn.com /Cubic_Equation.html (217 words)

	Q - Raytracing Reference
		Often, quartic refers to a shape described as a polynomial function of the coordinates (say x, y, z), such that the highest degree of any term is 4.
		A quartic equation is an equation of the form f(x) = 0, where f(x) is a quartic polynomial.
		It consists of adding the term (mx + b)^2 to both sides of the equation so as to convert them into perfect squares (m and b are calculated with some algebraic acrobatics followed by the solution of a cubic equation).
	fuzzyphoton.tripod.com /rtref/rtref_q.htm (391 words)

	Finding Real Roots of Quartics
		Quartic equations need to be solved when ray tracing 4th degree surfaces e.g., a torus.
		Quartics are the highest degree polynomials which can be solved analytically in general by the method of radicals i.e.: operating on the coefficients with a sequence of operators from the set: sum, difference, product, quotient, and the extraction of an integral order root.
		The signs of each of the quartic coefficients 'a', 'b', 'c', 'd' and the root of the cubic 'y', may be positive or negative, giving 32 possible combinations of signs.
	www-staff.it.uts.edu.au /~don/pubs/quartic.html (3291 words)

	The "Quartic Formula"
		Shortly after the discovery of a method to solve the cubic equation, Lodovico Ferraria (1522-1565), a student of Cardano, found a way to solve the quartic equation.
		First, the quartic equation is "depressed"; then one reduces the problem to solving a related cubic equation.
		One word of caution: A quartic equation may have four complex roots; so you should expect complex numbers to play a much bigger role in general than in my concrete example.
	www.sosmath.com /algebra/factor/fac12/fac12.html (0 words)

	Quartic Solutions LLC
		Quartic is a San Diego based Geographic Information System (GIS) Consulting firm which provides GIS consulting services to Southern California government and business organizations.
		A quartic equation is a fourth degree polynomial equation in the form:
		Thus the quartic (the 4th degree), and quintic (the 5th degree) equations were born.
	www.quarticsolutions.com /index.html (443 words)

	Equation Summary
		Equations are statements that use numbers and symbols to demonstrate that two groups of mathematical data are equal.
		Equations are often used to state the equality of two expressions containing one or more variables.
		However, if the equation were based on the natural numbers for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed.
	www.bookrags.com /Equation (2213 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.
		Quartic equations are polynomial equations with one unknown variable (usually denoted by x), which is never raised to a power greater than 4.
		Quartic equations were first discovered by Jaina Mathematicians in ancient India between 400 BC and 200 CE.
	www.bookrags.com /Quartic_equation (3541 words)

	History of Algebra
		Dardi used completing the cube to solve cubic equations, and a similar method for quartic equations.
		Equations of fifth and sixth degrees were further explored by Piero della Francesca.
		The idea of logarithms probably came from astronomers, where they had to multiply and divide complicated trigonometric functions, which may have as many as eight digits; and thus felt that if these could be reduced to addition or multiplication, it would be easier.
	library.thinkquest.org /C0110248/algebra/history2.htm (654 words)

	math lessons - Quartic equation
		As with other polynomials, it is sometimes possible to factor a quartic equation directly; but more often such a feat is Herculean, especially when the roots are irrational or complex.
		After much effort, such a formula was indeed found for quartics — but since then it has been proven (by Evariste Galois) that such an approach dead-ends with quartics; they are the highest-degree polynomial equations whose roots can be expressed in a formula using a finite number of arithmetic operators and n-th roots.
		Equation (4) is a cubic equation nested within the quartic equation.
	www.mathdaily.com /lessons/Quartic_equation (1534 words)

	Quadratic etc equations
		In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation.
		Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.
		Ferrari managed to solve the quartic with perhaps the most elegant of all the methods that were found to solve this type of problem.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Quadratic_etc_equations.html (1458 words)

	Math Forum: Ask Dr. Math FAQ: Cubic and Quartic Equations
		The roots of the original equation are then x = -a/4 and the roots of that cubic with a/4 subtracted from each.
		Once you have factored the quartic into two quadratics, finishing the finding of the roots is simple, using the Quadratic Formula.
		For some historical background, see Quadratic, cubic and quartic equations from the MacTutor Math History archives.
	mathforum.org /dr.math/faq/faq.cubic.equations.html (1200 words)

	The Quartic - Page One
		Ferrari was the first to develop an algebraic technique for solving the general quartic.
		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.
		The equation that must be solved to make it factorable is called the resolvent cubic.
	www.vimagic.de /hope/4/index.html (151 words)

	Qaurtic Proof
		We shall assume that as in the case of the cubic equation, that the root will consists of two independent terms.We also assume that each term will be a root of the fourth order.
		We thus have an equation in v that we can solve and this equation nearly looks the same as the one for case one..
		So we can again solve for x We have seen therefore that the path we follow for case 2 is exactly the same as for case one and this should not come as a surprise.
	www.geocities.com /dirkie6/page6.html (916 words)

	Amazon.com: quartic: Books (Site not responding. Last check: )
		An elementary treatise on cubic and quartic curves, by A. Basset.
		On the in-and-circumscribed triangles of the plane rational quartic curve, by Joseph Nelson Rice.
		The rational quartic curve in space of three and four dimensions, being an introduction to rational curves.
	www.amazon.com /s?ie=UTF8&rh=i:books,k:quartic&page=1 (863 words)

	Solutions to Polynomial Equations (Site not responding. Last check: )
		In the case of the cubic equation below, we will need to extract cube roots, so our notation will be more like that of (2).
		is significantly more difficult to solve than the quadratic equation, and its general solution was not found until the sixteenth century.
		The cubic formula is more complicated than the quadratic formula and cannot reasonably be written without a change of variables.
	www.math.rutgers.edu /~erowland/polynomialequations.html (363 words)

	RAND \| Papers \| On the Derivation of Booker's Quartic from Appleton-Hartree Equation.
		On the Derivation of Booker's Quartic from Appleton-Hartree Equation.
		Presentation of a simple method for the derivation of the Booker's quartic equation for oblique incidence of electromagnetic waves in the magneto-ionic theory of ionospheric propagation.
		Using those results the Booker's quartic equation is derived directly in a simpler manner from the Appleton-Hartree equation, using rotation transformation of the rectangular coordinate system.
	www.rand.org /pubs/papers/P3222 (291 words)

	Ray tracing primitives
		To intersect a ray with this, substitute Equation 24 in Equation 36.
		To intersect a ray with this, substitute Equation 24 in Equation 47.
		Substituting these three equations into Equation 54 will show that they are correct and is a useful exercise in algebraic manipulation.
	www.cl.cam.ac.uk /teaching/2000/AGraphHCI/SMEG/node2.html (2109 words)

	Hitting a Moving Target: The Missile Guidance System - The Code Project - C# Libraries
		Now all that the caller will have to do is use these vectors to appropriately configure the actual missile object's state and introduce it into their runtime environment (physics engine etc.); the specifics of this environmental configuration can be found in the penultimate paragraph of the algebraic section.
		equation that results when an intercept is attempted in more than 1D space (in 1D the square of a square root can be cancelled out, as the value for that dimension is also the magnitude of the whole vector, but a 1D intercept occurring on a line is a bit trivial, don't you think?).
		A quintic, or an equation of any nonzero coefficient of a higher degree polynomial, does not have a formula that, given the coefficients, produces all possible values for the variable in question, namely time in the case of the guided missile.
	www.codeproject.com /cs/library/Missile_Guidance_System.asp?msg=2116001 (3714 words)

	¥³.The Sixteenth-Century Mathematics of Italy: Commercial Mathematics
		Probably the most spectacular mathematical achievement of the sixteenth century was the dicovery, by Italian mathematicians, of the algebraic solution of cubic and quartic equations.
		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)

	Algebra - Numericana
		Quartic equation involved in the classic "Ladders in an Alley" problem.
		There are popular implementations (on some handheld calculators and elsewhere) which provide pointwise solutions to quadratic equations, but they don't qualify as proper mathematical generalizations of the square root function.
		Note that this quartic equation may include roots which do not correspond to solutions of the original problem...
	home.att.net /~numericana/answer/algebra.htm (4774 words)

	Solving Cubics and Quartics
		Many algorithms use the idea of first solving a particular cubic equation, the coefficients of which are derived from those of the quartic.
		There have been many algorithms proposed for solving quartic and cubic equations, but most have been proposed with aims of elegance, generality or simplicity rather than error minimisation or overflow avoidance.
		A computer program was written to perform a comparison of the stabilities of the five algorithms for the solution of quartic equations.
	linus.socs.uts.edu.au /~don/pubs/solving.html (3623 words)

	Steps to the Quintic
		The quartic can also be solved, but now the answer is quite long, so in order to avoid choking your link, it is not displayed here.
		Ruffini (1799) and Abel (1826) proved that it is not possible to give an explicit solution for the general quintic equation with symbolic coefficients in terms of square roots, cube roots, and so on.
		Again, the output is a pair whose first element is the transformation in form of a pure function and whose second element is the new quintic, called the canonical quintic.
	library.wolfram.com /examples/quintic/steps.html (615 words)

	[No title] (Site not responding. Last check: )
		The dynamic solution of the system matrices is in the form of a quartic equation.
		The "Solve Quartic" button solves the quartic for the four roots.
		Evaluation of the solution to the quartic equation can then be performed.
	www.aerologic.com /stab/dyn.html (129 words)

	Center for Technology and Teacher Education: Mathematics Activities
		Graph the regression equation on the same axis as the scatterplot.
		Graph the quartic regression equation on the same axis as the linear regression equation.
		Graph your model along with the quartic regression equation and the linear regression equation.
	www.teacherlink.org /content/math/activities/gc-athletes/guide.html (810 words)

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