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Topic: Cubic equation

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Quartic equation

Quintic equation

Quadratic equation

Polynomial

Cubic function

Linear equation

Algebra

Exterior covariant derivative

Logarithm

Elliptic curve

Linear function

Irrational number

Real number

Complex number

Partial differential equation

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	Math Forum: Ask Dr. Math FAQ: Cubic and Quartic Equations
		To try to go backward and come up with a closed form for the Cubic Formula in terms of the original a, b, c, d would be a real pain.
		The roots of the original equation are then x = -a/4 and the roots of that cubic with a/4 subtracted from each.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /dr.math/faq/faq.cubic.equations.html (1200 words)

	cubic equation
		A polynomial equation of the third degree, the general form of which is
		Early studies of cubics helped legitimize negative numbers, give a deeper insight into equations in general, and stimulate work that eventually led to the discovery and acceptance of complex numbers.
		He also noted an important fact connecting solutions of a cubic equation to its coefficients, namely, that the sum of the solutions is the negation of b, the coefficient of the x
	www.daviddarling.info /encyclopedia/C/cubic_equation.html (264 words)

	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.
		Pure cubic equations are therefore of the form x 3 =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.
	www.1911encyclopedia.org /Equation (10531 words)

	The "Cubic Formula"
		Solving a cubic equation, on the other hand, was the first major success story of Renaissance mathematics in Italy.
		First, the cubic equation is "depressed"; then one solves the depressed cubic.
		Shortly after the discovery of a method to solve the cubic equation, Lodovico Ferraria (1522-1565), a student of Cardano, found a similar method to solve the quartic equation.
	www.sosmath.com /algebra/factor/fac11/fac11.html (436 words)

	Cubic equation Summary
		In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		is non-zero (otherwise it is a quadratic equation).
		Cubic equations were first discovered by Jaina mathematicians in ancient India sometime between 400 BC and 200 CE.
	www.bookrags.com /Cubic_equation (2260 words)

	Three Problems Of Antiquity
		Cubic equations possess a pertinent property which constitutes the contents of a lemma below.
		After proving the lemma, I shall derive cubic equations for the three problems and show that they satisfy the conditions of the lemma.
		The number of prime factors on the left side of the latter equation is divisible by 3.
	www.cut-the-knot.org /arithmetic/cubic.shtml (641 words)

	Property Modeling
		The parameter a of equation 2-3 represents a rough measure of the attraction forces between the molecules, and the parameter b is related to the size of the molecule.
		To calculate the liquid fugacity in equation 2-29 the liquid's compressibility is input to equation 2-24, and to calculate the vapor fugacity, the vapor's compressibility is used.
		Equating the fugacity for each component in the liquid to its respective fugacity in the vapor provide the two additional equations that are needed to solve for the vapor phase composition and the liquid phase composition.
	www.me.gatech.edu /energy/andy_phd/two.htm (5641 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.
		Equation (4) is a cubic equation nested within the quartic equation.
	www.bookrags.com /Quartic_equation (3541 words)

	Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths, based on Lill's Method.
		In forming the right-angle path for a quadratic equation, the directions for the a, b, and c lines are individually absolute in the sense that a negative coefficient would be represented by a line in the opposite direction, without affecting the directions for the other coefficients.
		For a cubic equation, once one solution is found, it is easy to know whether or not the remaining two are real or complex by constructing the solution circle and seeing if it intersects with the b-line of the subpath quadratic.
		The point to consider is that a cubic equation can be solved using the expanded closed-path quintic and extracting one real root by finding a quartic sub-path and sequentially extracting other real roots (if there are any) from that closed-path quartic, or by extracting 3 real roots by finding 3 quartic subpaths.
	www.concentric.net /~pvb/ALG/rightpaths.html (3802 words)

	Cubic Equation - Search Results - MSN Encarta
		Cubic Equation - Search Results - MSN Encarta
		Cubic Equation, any equation in which the largest power of x is x
		Later scholars, including 12th-century Persian mathematician Omar Khayyam, solved certain cubic equations geometrically by using conic sections.
	encarta.msn.com /Cubic_Equation.html (120 words)

	Allegany College
		This equation is not good for extrapolation because after the year 1995, there is a steady increase in the life expectancy, and since we are alive today, it must be invalid.
		This equation is not good for extrapolation because it starts with the life expectancy at infinity, and coming down at the year 1862 then increasing again, only to finally rise into infinity.
		This equation reacts similar to the logarithmic equation, but in the long run the numbers are not as realistic as the logarithmic equation, making it not the best for extrapolation.
	www.ac.cc.md.us /Department/math/life.htm (1090 words)

	A Catalog of Cubic Plane Curves
		This cubic plane curve is asymptotic to the semicubical parabola.
		This cubic curve is asymptotic to a line, and crosses itself at a node located at the origin.
		This cubic curve is asymptotic to a line, and has a cusp at the origin.
	staff.jccc.net /swilson/planecurves/cubics.htm (905 words)

	Information on Cubic equation
		Cubic foot, a volume equivalent to a cubical solid which measures a foot in each of its dimensions.
		Cubic number, a number produced by multiplying a number into itself, and that product again by the same number.
		Cubical parabola (Geom.), two curves of the third degree, one plane, and one on space of three dimensions.
	www.wkonline.com /d/Cubic_equation.html (122 words)

	math lessons - Cubic equation
		A cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known at that time.
		The cube root function is in some respects not a well-behaved function, or one convenient for the purposes of finding the roots of a cubic equation.
	www.mathdaily.com /lessons/Cubic_equation (1367 words)

	cardano
		Pacioli ponders the cubic and decides the problem is too difficult for the mathematics of the day [2 p 134].
		The solution of the cubic equation is kept secret by del Ferro so that he has it to use in case his position is ever challenged.
		The solution of the cubic is told to Antonio Fior a student of del Ferro on del Ferro's death bed [2 pp 134-136].
	muskingum.edu /~rdaquila/m370/cardano.html (1095 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.
		The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Quadratic_etc_equations.html (1458 words)

	Cubic Equations
		This WorkBook is a gentle introduction to abstract algebra, and theory of equations, written in homage to Gerolimo Cardano, who, with Tartaglia, established procedures for solving cubic equations.
		The aim is to give a self-contained exposition of the basic facts about cubic equations within the context of abstract algebra, and at the same time to provide many opportunities to explore.
		For those of you familiar with matrix representations, the symbolic cubic numbers that we introduce are simply the elements of the algebra of 3x3 circulant matrices.
	www.mathwright.com /library/oldcardanox3.html (838 words)

	PlanetMath: cubic formula
		This is version 6 of cubic formula, born on 2002-01-06, modified 2005-03-05.
		I am a high school math teacher and designed a special project for my students that used the formula I found on your website for solutions to cubic equations.
		We looked at the graph of the cubic equation and found that it should have 1 real and 2 complex solutions.
	planetmath.org /encyclopedia/CubicFormula.html (245 words)

	Solutions to Polynomial Equations
		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)

	Convergence \| How Tartaglia Solved the Cubic Equation
		And imagine moreover that negative numbers, and also negative solutions of equations, were rejected - called false or fictitious - because then one thought geometrically, and the side of a square, or the edge of a cube cannot be negative.
		That means, instead of the single cubic equation of today there were not less than 13 equations, 7 with all four terms (cubic, quadratic, linear, and absolute term), 3 without the linear term, and 3 without the quadratic term:
		But the ten cubic equations containing the quadratic term were too difficult to be solved.
	mathdl.maa.org /convergence/1/convergence/1/?pa=content&sa=viewDocument&nodeId=1345 (278 words)

	Math Forums @ Math Goodies - What is the curve for a cubic equation named?
		For a quadratic equation, it is a "parabola".
		By the way, the graph of the cubic is for my Algebra2 class, not the Internet class.
		"cubic" is special, and descriptive enough as it is, being other than just another polynomial, not otherwise having a special name, as with a polynomial of degree 27, or whatever.
	mathgoodies.com /forums/topic.asp?TOPIC_ID=30847⏛ (1077 words)

	Karl's Calculus Tutor - Box 5.3a The Cubic Formula
		Generations of mathematicians searched for a cubic solution before Niccolo Fontana Tartaglia and Girolamo Cardano hit on it in the 16th century (Cardano published the solution in Ars Magna in 1545).
		The trick is to convert the cubic, by a circuitous route, to a quadratic and then apply the quadratic formula.
		In order for this equation to hold we must be able to cancel all the terms on the left of the equal.
	www.karlscalculus.org /cubic.html (595 words)

	Miscellaneous Mathematical Utilities
		Because these utilities are written in Javascript, make sure Javascript is enabled in your Internet browser.
		Numerical Solver for Kepler's Equation for Elliptical Motion
		A Downloadable Console Utility, N Equations in N Unknowns
	www.akiti.ca /Mathfxns.html (193 words)

	Cubic Equation Solver
		Problem Statement: Write a FORTRAN program to read the coefficient of a cubic equation, find the roots of the polynomial based on the cubic formula, and output the answers.
		Algebraic theory on polynomials says that there should be three roots for a cubic equation; complex roots, when exist, will appear in complex conjugate pairs.
		Problem Statement: Re-do the cubic equation solver with MATLAB by following the same approach as in FORTRAN, including the use of the function structure.
	www.glue.umd.edu /~nsw/ench250/cubiceq.htm (389 words)

	cubic
		It explains how Tartaglia solved the cubic equation, and how it led to the first computation with complex numbers.
		However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations.
		Remarkably, dal Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.
	www.math.wisc.edu /~angenent/276/cubic.html (587 words)

	Roots of a cubic equation (Site not responding. Last check: )
		The calculator solves for the roots of a quintic equation.
		Enter values into the fields to form equation of the type
		Press "see graphical function" to display the graph for the function you input (requires Java).
	www.hull.ac.uk /php/chsajb/general/cubic.html (65 words)

	Complex numbers: quadratic and cubic equations
		Instead, quadratic equations were classified into four different kinds depending on the signs of the coefficients a, b, and c.
		Equations of the third degree are called cubic equations.
		As with the quadratic equation, there are several forms for the cubic when negative terms are moved to the other side of the equation and zero terms dropped.
	www.clarku.edu /~djoyce/complex/cubic.html (682 words)

	Solving cubic equations
		This page is intended to be read after two others: one on what it means to solve an equation and the other on algebraic numbers, field extensions and related ideas.
		Of course, once we have solved the equation for y, it is easy to obtain a solution for x, since x is a very simple linear function of y.
		Once we have solved this equation for y, it is easy to obtain a solution of the original equation for x, since x is a very simple linear function of y.
	www.dpmms.cam.ac.uk /~wtg10/cubic.html (2889 words)

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