Earliest method used to solve quadratic equation

Quadratic, cubic and quartic equations. It is often claimed that the Babylonians about BC were the first to solve quadratic equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation.

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Earliest method used to solve quadratic equation

In elementary algebra , the quadratic formula is a formula that provides the solutions to a quadratic equation. Other ways of solving a quadratic equation, such as completing the square , yield the same solutions. This version of the quadratic formula is used in Muller's method for finding the roots of general functions. Many different methods to derive the quadratic formula are available in the literature. The standard one is a simple application of the completing the square technique. The left-hand side is now ready for the method of completing the square , i. Because the left-hand side is now a perfect square, we can easily take the square root of both sides:. Then the steps of the derivation are: [10]. The following method was used by many historical mathematicians: [12]. An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents , [13] which is an early part of Galois theory. This approach focuses on the roots themselves rather than algebraically rearranging the original equation. So the polynomial factors as. The Galois theory approach to analyzing and solving polynomials is to ask whether, given coefficients of a polynomial each of which is a symmetric function in the roots, one can "break" the symmetry and thereby recover the roots. The quadratic formula is exactly correct when performed using the idealized arithmetic of real numbers , but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available on computers, the limited precision of intermediate calculations can lead to substantially inaccurate results. Unfortunately, introductory algebra textbooks typically do not address this problem, even though it causes students to obtain inaccurate results in other school subjects such as introductory chemistry.

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The numbers a , b , and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient , the linear coefficient and the constant coefficient or free term. The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers , there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. Completing the square is one of several ways for deriving the formula.

There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square. Setting each factor to zero,. Then to check,. Setting each factor to 0,. A quadratic with a term missing is called an incomplete quadratic as long as the ax 2 term isn't missing. Many quadratic equations cannot be solved by factoring. This is generally true when the roots, or answers, are not rational numbers. A second method of solving quadratic equations involves the use of the following formula:. When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant.

Earliest method used to solve quadratic equation

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

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Hidden categories: CS1 errors: generic name CS1 maint: multiple names: authors list Articles with short description Short description is different from Wikidata Wikipedia articles needing clarification from October All articles that may contain original research Articles that may contain original research from October Wikipedia articles needing clarification from September Commons category link from Wikidata Webarchive template wayback links Articles with BNF identifiers Articles with BNFdata identifiers Articles with J9U identifiers Articles with LCCN identifiers. In respect of content there is scarcely any difference between the two groups of texts. Euclid had no notion of equation, coefficients etc. The standard one is a simple application of the completing the square technique. Historical events in geometry Srishti Garg. His importance lies in his discoveries of mathematical knowledge which was later transferred to Arab and European scholars. The Equation that Couldn't Be Solved. A lesser known quadratic formula, as used in Muller's method , provides the same roots via the equation. Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis , where real numbers are approximated by floating point numbers called "reals" in many programming languages. Unit 5th topic Drugs used in congestive Heart failure and shock. For the formula used to find solutions to such equations, see Quadratic formula. Main article: Completing the square. This article is about algebraic equations of degree two and their solutions. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. McKeague

The term "quadratic" comes from the Latin word "quadratus" meaning square, which refers to the fact that the variable x is squared in the equation. Did you know that when a rocket is launched, its path is described by a quadratic equation?

Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. Squares equal to roots. The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. If linear problems are found in their texts then the answers are simply given without any working; these problems were obviously thought too. Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. However, because legend and obfuscation cloud his work even more than that of other pre-Socratic philosophers, one can give only a tentative account of his teachings, and some have questioned whether he contributed much to mathematics or natural philosophy. The formula and its derivation remain correct if the coefficients a , b and c are complex numbers , or more generally members of any field whose characteristic is not 2. The quadratic equation may be solved geometrically in a number of ways. Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 including those that are themselves composite , facilitating calculations with fractions. B Hughes, The earliest correct algebraic solutions of cubic equations, Vita mathematica Washington, DC, , - Let h and k be respectively the x -coordinate and the y -coordinate of the vertex of the parabola that is the point with maximal or minimal y -coordinate. History of mathematics - Pedagogy of Mathematics.