Solving quadratic equations with continued fractions information
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
where a ≠ 0.
The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.
If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.
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example to illustrate the solution of a quadraticequation using continuedfractions. We begin with the equation x 2 = 2 {\displaystyle x^{2}=2} and manipulate...
Artin–Schreier theory. Solvingquadraticequationswithcontinuedfractions Linear equation Cubic function Quartic equation Quintic equation Fundamental theorem...
Poitou and George Szekeres. Gauss's continuedfraction Padé table Solvingquadraticequationswithcontinuedfractions Convergence problem Infinite compositions...
solutions to quadraticequations or as coefficients in an equation. He was also the first to solve three non-linear simultaneous equationswith three unknown...
to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so...
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field...
1624). In Europe, it was likewise used to solve Diophantine equations and in developing continuedfractions. The extended Euclidean algorithm was published...
equation, a first order nonlinear ordinary differential equation Euler conservation equations, a set of quasilinear first-order hyperbolic equations used...
§ Brouncker's formula. Some approximations of pi include: Integers: 3 Fractions: Approximate fractions include (in order of increasing accuracy) 22/7, 333/106, 355/113...
false position method and quadraticequations. Written evidence of the use of mathematics dates back to at least 3200 BC with the ivory labels found in...
computing square roots, and general methods of solving linear and some quadraticequations, solution to Pell's equation. Muhammad ibn Mūsā al-Khwārizmī (820 CE)...
\varphi } satisfies the quadraticequation φ 2 = φ + 1 {\displaystyle \varphi ^{2}=\varphi +1} and is an irrational number with a value of φ = 1 + 5 2...
requires solving an indeterminate quadraticequation (which reduces to what would later be misnamed Pell's equation). As far as we know, such equations were...
describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadraticequation) in his main work, the...
titled with two such types of manipulation. However, even for solvingquadraticequations, the factoring method was not used before Harriot's work published...
include multiplication tables and methods for solving linear, quadraticequations and cubic equations, a remarkable achievement for the time. Tablets...
linear and quadraticequations and the elementary arithmetic of binomials and trinomials. This approach, which involved solvingequations using radicals...
solvingquadraticequations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with...
quadratic irrationals and their Pierce expansions", Fibonacci Quarterly, 36 (2): 146–153 Kraaikamp, Cor; Wu, Jun (2004), "On a new continuedfraction...
the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or...
trigonometry. Arithmetic: Continuedfractions. Algebra: Solutions of simultaneous quadraticequations. Whole number solutions of linear equations by a method equivalent...
trigonometry, and spherical trigonometry. It also contains continuedfractions, quadraticequations, sums-of-power series, and a table of sines. The Arya-siddhanta...
their equations. Once Euler had solved this equation, he considered the case a = b {\displaystyle a=b} . Taking limits, he derived the equation ln x...
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