Algebraic equations

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Quadratic equation

A quadratic equation is a second-order polynomial equation in a single variable (1)with . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta the Pirates of Penzance impresses the pirates with his knowledge of quadratic equations in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse."The..

Biquadratic equation

The term "biquadratic equation" is sometimes used to as a synonym for quartic equation (Beyer 1987b, p. 34), but perhaps more commonly (e.g., Hazewinkel 1988; Gellert et al. 1989, p. 101) and more properly for a quartic equation having no odd powers, i.e.,Such equations are easy to solve, since they reduce to a quadratic equation in the variable and hence can be solved for using the quadratic formula and then in terms of the original variables by taking the square roots.

Principal quintic form

A general quintic equation(1)can be reduced to one of the form(2)called the principal quintic form.Vieta's formulas for the roots in terms of the s is a linear system in the , and solving for the s expresses them in terms of the power sums . These power sums can be expressed in terms of the s, so the s can be expressed in terms of the s. For a quintic to have no quartic or cubic term, the sums of the roots and the sums of the squares of the roots vanish, so(3)(4)Assume that the roots of the new quintic are related to the roots of the original quintic by(5)Substituting this into (1) then yields two equations for and which can be multiplied out, simplified by using Vieta's formulas for the power sums in the , and finally solved. Therefore, and can be expressed using radicals in terms of the coefficients . Again by substitution into (◇), we can calculate , and in terms of and and the . By the previous solution for and and again by using Vieta's formulas for the power..

Linear equation

A linear equation is an algebraic equation of the forminvolving only a constant and a first-order (linear) term, where is the slope and is the -intercept. The above form is aptly known as slope-intercept form; alternatively, linear equations can be written in a number of other forms including standard form, intercept form, etc.Occasionally, the above is called a "linear equation of two variables," where and are the variables. Equations such as and are linear equations of a single variable, and is an example of a linear equation with three variables.

Quintic equation

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois. However, certain classes of quintic equations can be solved in this manner.Irreducible quintic equations can be associated with a Galois group, which may be a symmetric group , metacyclic group , dihedral group , alternating group , or cyclic group , as illustrated above. Solvability of a quintic is then predicated by its corresponding group being a solvable group. An example of a quintic equation with solvable cyclic group is(1)which arises in the computation of .In the case of a solvable quintic, the roots can be found using the formulas found in 1771 by Malfatti, who was the first to "solve" the quintic using a resolvent of sixth degree (Pierpont..

Lambert's transcendental equation

An equation proposed by Lambert (1758) and studied by Euler in 1779.(1)When , the equation becomes(2)which has the solution(3)(4)(5)where is the Lambert W-function.

Algebraic equation

An algebraic equation in variables is an polynomial equation of the formwhere the coefficients are integers (where the exponents are nonnegative integers and the sum is finite).Examples of algebraic equations are given in the following table.curveequationCayley's sexticeight curveline through and plane through , , and unit circleunit sphereThe roots of an algebraic equation in one variable are known as algebraicnumbers.

Cubic formula

The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial. A general cubic equation is of the form(1)(the coefficient of may be taken as 1 without loss of generality by dividing the entire equation through by ). The Wolfram Language can solve cubic equations exactly using the built-in command Solve[a3 x^3 + a2 x^2 + a1 x + a0 == 0, x]. The solution can also be expressed in terms of the Wolfram Language algebraic root objects by first issuing SetOptions[Roots, Cubics -> False].The solution to the cubic (as well as the quartic) was published by Gerolamo Cardano (1501-1576) in his treatise Ars Magna. However, Cardano was not the original discoverer of either of these results. The hint for the cubic had been provided by Niccolò Tartaglia, while the quartic had been solved by Ludovico Ferrari. However, Tartaglia himself had probably caught wind of the solution from another source. The solution..

Quartic equation

A quartic equation is a fourth-order polynomialequation of the form(1)While some authors (Beyer 1987b, p. 34) use the term "biquadratic equation" as a synonym for quartic equation, others (Hazewinkel 1988, Gellert et al. 1989) reserve the term for a quartic equation having no cubic term, i.e., a quadratic equation in .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. 283). The Wolfram Language can solve quartic equations exactly using the built-in command Solve[a4 x^4 + a3 x^3 + a2 x^2 + a1 x + a0 == 0, x]. The solution can also be expressed in terms of Wolfram Language algebraic root objects by first issuing SetOptions[Roots, Quartics -> False].The roots of this equation satisfy Vieta'sformulas: (2)(3)(4)(5)where the denominators on the right side are all . Writing the quartic in the standard..

Cubic equation

A cubic equation is an equation involving a cubicpolynomial, i.e., one of the formSince (or else the polynomial would be quadratic and not cubic), this can without loss of generality be divided through by , givingA closed-form formula known as the cubic formulaexists for the solutions of a cubic equation.

Quadratic formula

The formula giving the roots of a quadraticequation(1)as(2)An alternate form is given by(3)

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