CHATTING

Online chat may refer to any kind of communication over the Internet, that offers a real-time direct transmission of text-based messages from sender to receiver, hence the delay for visual access to the sent message shall not hamper the flow of communications in any of the directions. Online chat may address point-to-point communications as well as multicast communications from one sender to many receivers and voice and video chat or may be a feature of a Web conferencing service. Contents [hide] 1 Instrumentation 2 History 3 Chatiquette 4 Cultural impact 5 Social criticism 6 Software and protocols 7 See also 8 References [edit]Instrumentation Online chat in a lesser stringent definition may be primarily any direct text-based or video-based ( webcams ), one-on-one chat or one-to-many group chat (formally also known as synchronous conferencing), using tools such as instant messengers, Internet Relay Chat, talkers and possibly MUDs. The expression online chat comes from the word chat which means "informal conversation". Online chat includes web-based applications that allow communication - often directly addressed, but anonymous - between users in a multi-user environment. Web conferencing is a more specific online service, that is often sold as a service, hosted on a web server controlled by the vendor. [edit]History The first online chat system was called Talkomatic, created by Doug Brown and David R. Woolley in 1974 on the PLATO System at the University of Illinois. It offered several channels, each of which could accommodate up to five people, with messages appearing on all users' screens character-by-character as they were typed. Talkomatic was very popular among PLATO users into the mid-1980s. The first[1] dedicated online chat service that was widely available to the public was the CompuServe CB Simulator in 1980,[2] created by CompuServe executive Alexander "Sandy" Trevor in Columbus, Ohio. Ancestors include network chat software such as UNIX "talk" used in the 1970s. [edit]Chatiquette The term chatiquette (chat etiquette) is a variation of netiquette (internet etiquette) and describes basic rules of online communication.[3][4][5][6] To avoid misunderstandings and to simplify the communication between users in a chat these conventions or guidelines have been created. Chatiquette varies from community to community, generally describing basic courtesy; it introduces new user into the community and the associated network culture. As an example, it is considered rude to write only in upper case, because it looks as if the user is shouting. The word chatiquette has been used in connection with various chat systems (e.g. IRC) since 1995.[7][8] [edit]Cultural impact Despite being virtual, chat can spill into the outside world.[9] There can also be a strong sense of online identity leading to impression of subculture.[10] Compare Internet sociology. Chats are valuable sources of various types of information, the automatic processing of which is the object of chat/text mining technologies.[11] [edit]Social criticism This section may contain original research. Please improve it by verifying the claims made and adding references. Statements consisting only of original research may be removed. (May 2010) There has been much criticism about what online chatting has done in today's society. Many people[who?] are accusing it of replacing proper English with short hand with an almost completely new hybrid language.[citation needed] Writing is changing as it takes on some of the functions and features of speech. Internet chatrooms and rapid real-time conferencing allow users to interact with whoever happens to coexist in cyberspace. These virtual interactions involve us in `talking' more freely and more widely than ever before (Merchant, 2001).[12] With chatrooms replacing many face-to-face conversations it is necessary to be able to have quick conversation as if the person were present, so many people learn to type as quickly as they would normally speak. Critics[who?] are wary that this casual form of speech is being used so much that it will slowly take over common grammar; however, such a change has yet to be seen. With the increasing population of online chatrooms there has been a massive growth[13] of new words created or slang words, many of them documented on the website Urban Dictionary Sven Birkerts says "as new electronic modes of communication provoke similar anxieties amongst critics who express concern that young people are at risk, endangered by a rising tide of information over which the traditional controls of print media and the guardians of knowledge have no control on it".[14] This person is arguing that the youth of the world may have too much freedom with what they can do or say with the almost endless possibilities that the Internet gives them, and without proper controlling it could very easily get out of hand and change the norm of literacy of the world. In Guy Merchant's journal article Teenagers in Cyberspace: An Investigation of Language Use and Language Change in Internet Chatrooms; he says "that teenagers and young people are in the leading the movement of change as they take advantage of the possibilities of digital technology, drastically changing the face of literacy in a variety of media through their uses of mobile phone text messages, e-mails, web-pages and on-line chatrooms. This new literacy develops skills that may well be important to the labor market but are currently viewed with suspicion in the media and by educationalists.[who?] [15] Merchant also says "Younger people tend to be more adaptable than other sectors of society and, in general, quicker to adapt to new technology. To some extent they are the innovators, the forces of change in the new communication landscape. (Merchant, 2001)."[16] In this article he is saying that young people are merely adapting to what they were given.

 CIRCLE
A circle is a simple shape of Euclidean geometry that is the set of all points in the plane that are equidistant from a given point, the centre. The distance between any of the points and the centre is called the radius. A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk. A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant. A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone. Contents [hide] 1 Terminology 2 History 3 Analytic results 3.1 Length of circumference 3.2 Area enclosed 3.3 Equations 3.4 Tangent lines 4 Properties 4.1 Chord 4.2 Sagitta 4.3 Tangent 4.4 Theorems 4.5 Inscribed angles 5 Circle of Apollonius 5.1 Cross-ratios 5.2 Generalised circles 6 Circles inscribed in or circumscribed about other figures 7 Circle as limiting case of other figures 8 See also 9 References 10 Further reading 11 External links [edit]Terminology Chord: a line segment whose endpoints lie on the circle. Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle. Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter. Circumference: the length of one circuit along the circle itself. Tangent: a straight line that touches the circle at a single point. Secant: an extended chord, a straight line cutting the circle at two points. Arc: any connected part of the circle's circumference. Sector: a region bounded by two radii and an arc lying between the radii. Segment: a region bounded by a chord and an arc lying between the chord's endpoints. Chord, secant, tangent, radius, and diameter Arc, sector, and segment [edit]History The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origins of the words "circus" and "circuit" are closely related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilisation possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[citation needed] The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo Tughrul Tower from inside Circles on an old Arabic astronomical drawing Some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π.[1] 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. 1880 CE– Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.[2] [edit]Analytic results [edit]Length of circumference Further information: Pi The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: [edit]Area enclosed Area enclosed by a circle = π × area of the shaded square Main article: Area of a disk As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,[3] which comes to π multiplied by the radius squared: Equivalently, denoting diameter by d, that is, approximately 79 percent of the circumscribing square (whose side is of length d). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality. [edit]Equations [edit]Cartesian coordinates Circle of radius r = 1, centre (a, b) = (1.2, −0.5) In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to The equation can be written in parametric form using the trigonometric functions sine and cosine as where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. An alternative parametrisation of the circle is: In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis. In homogeneous coordinates each conic section with equation of a circle is of the form It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity. [edit]Polar coordinates In polar coordinates the equation of a circle is: where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes In the general case, the equation can be solved for r, giving the solution with a minus sign in front of the square root giving the same curve. [edit]Complex plane In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written . The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line. [edit]Tangent lines Main article: Tangent lines to circles The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is or If y1 ≠ b then the slope of this line is This can also be found using implicit differentiation. When the centre of the circle is at the origin then the equation of the tangent line becomes and its slope is [edit]Properties The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.) The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T. All circles are similar. A circle's circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2π and π, respectively. The circle which is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle. [edit]Chord Chords are equidistant from the centre of a circle if and only if they are equal in length. The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector: A perpendicular line from the centre of a circle bisects the chord. The line segment (circular segment) through the centre bisecting a chord is perpendicular to the chord. If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle. If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental. For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle. An inscribed angle subtended by a diameter is a right angle (see Thales' theorem). The diameter is the longest chord of the circle. If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd. If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.[4] The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).[5] The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.[6]:p.71 [edit]Sagitta The sagitta is the vertical segment. The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines: Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y / 2)2. Solving for r, we find the required result. [edit]Tangent The line perpendicular drawn to a radius through the end point of the radius is a tangent to the circle. A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length. If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary. If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ). [edit]Theorems Secant-secant theorem See also: Power of a point The chord theorem states that if two chords, CD and EB, intersect at A, then CA × DA = EA × BA. If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC2 = DG × DE. (Tangent-secant theorem.) If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE. (Corollary of the tangent-secant theorem.) The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle). If the angle subtended by the chord at the centre is 90 degrees then l = r√2, where l is the length of the chord and r is the radius of the circle. If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem. [edit]Inscribed angles See also: Inscribed angle theorem Inscribed angle theorem An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees). [edit]Circle of Apollonius Main article: Apollonian circles Apollonius' definition of a circle: d1 / d2 constant Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B.[7][8] (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points. The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB, since the segments are similar: Analogously, a line segment PD through some point D on AB extended bisects the corresponding exterior angle BPQ where Q is on AP extended. Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90 degrees, i.e., a right angle. The set of points P such that angle CPD is a right angle forms a circle, of which CD is a diameter. Second, see[9]:p.15 for a proof that every point on the indicated circle satisfies the given ratio. [edit]Cross-ratios A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one: Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane. [edit] Generalised circles See also: Generalised circle If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition is not a circle, but rather a line. Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius. [edit]Circles inscribed in or circumscribed about other figures In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.[10] About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.[11] A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.[12] A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle. [edit]Circle as limiting case of other figures The circle can be viewed as a limiting case of each of various other figures: A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero. A superellipse has an equation of the form for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which n = 2. A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results. A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.

This article is about quadratic equations and solutions. For more general information about quadratic functions, see Quadratic function. For more information about quadratic polynomials, see Quadratic polynomial. In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.) The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below). Plots of real-valued quadratic function ax2 + bx + c, varying each coefficient separately Contents [hide] 1 Quadratic formula 1.1 Discriminant 1.2 Monic form 2 History 3 Examples of use 3.1 Geometry 3.2 Quadratic factorization 3.3 Application to higher-degree equations 4 Derivations of the quadratic formula 4.1 By completing the square 4.2 By shifting ax2 4.3 By Lagrange resolvents 5 Other methods of root calculation 5.1 Alternative quadratic formula 5.2 Floating-point implementation 5.3 Vieta's formulas 5.4 Trigonometric solution for complex roots 5.5 Geometric solution 6 Generalization of quadratic equation 6.1 Characteristic 2 7 See also 8 References 9 External links [edit]Quadratic formula A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real. Having the roots are given by the quadratic formula[1] where the symbol "±" indicates that both are solutions of the quadratic equation.[2] [edit]Discriminant Example discriminant signs ■ <0: data-blogger-escaped-x2="x2" data-blogger-escaped-x="x">0: 3⁄2x2+1⁄2x−4⁄3 In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta, the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant: A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases: If the discriminant is positive, then there are two distinct roots, both of which are real numbers: For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one distinct real root, sometimes called a double root: If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:[3] where i is the imaginary unit. Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. [edit]Monic form Dividing the quadratic equation by the quadratic coefficient a gives the simplified monic form of where p = b/a and q = c/a. This in turn simplifies the root and discriminant equations somewhat to and [edit]History This section may contain inappropriate or misinterpreted citations that do not verify the text. Please help improve this article by checking for inaccuracies. (help, talk, get involved!) (September 2010) Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form: which are equivalent to the equation:[4] The original pair of equations were solved as follows: Form Form Form Form (where x ≥ y is assumed) Find x and y by inspection of the values in (1) and (4).[5] There is evidence pushing this back as far as the Ur III dynasty.[6] In the Sulba Sutras in ancient India circa 8th century BC quadratic equations of the form ax2 = c and ax2 + bx = c were explored using geometric methods. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula.[citation needed] Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. Pythagoras and Euclid used a strictly geometric approach, and found a general procedure to solve the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[7] In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation as follows: To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. (Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)[5] This is equivalent to: The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.[8] He also described the method of completing the square and recognized that the discriminant must be positive,[9] which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.[10] While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,[11] as well as irrational numbers as solutions.[12] Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.[13] The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.[14] His solution was largely based on Al-Khwarizmi's work.[15] The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[16] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in a 1896 paper by Henry Heaton.[17] [edit]Examples of use [edit]Geometry For the quadratic function: f (x) = x2 − x − 2 = (x + 1)(x − 2) of a real variable x, the x-coordinates of the points where the graph intersects the x-axis, x = −1 and x = 2, are the solutions of the quadratic equation: x2 − x − 2 = 0. The solutions of the quadratic equation are also the roots of the quadratic function:[18] since they are the values of x for which If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. It follows from the above that, if the discriminant is positive, the graph touches the x-axis at two points, if zero, the graph touches at one point, and if negative, the graph does not touch the x-axis. [edit]Quadratic factorization The term is a factor of the polynomial if and only if r is a root of the quadratic equation It follows from the quadratic formula that In the special case () where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as [edit]Application to higher-degree equations Certain higher-degree equations can be brought into quadratic form and solved that way. For example, the 6th-degree equation in x: can be rewritten as: or, equivalently, as a quadratic equation in a new variable u: where Solving the quadratic equation for u results in the two solutions: Thus Concentrating on finding the three cube roots of 2 + 2i – the other three solutions for x (the three cube roots of 2 - 2i ) will be their complex conjugates – rewriting the right-hand side using Euler's formula: (since e2kπi = 1), gives the three solutions: Using Eulers' formula again together with trigonometric identities such as cos(π/12) = (√2 + √6) / 4, and adding the complex conjugates, gives the complete collection of solutions as: and [edit]Derivations of the quadratic formula [edit]By completing the square The quadratic formula can be derived by the method of completing the square,[19] so as to make use of the algebraic identity: Dividing the quadratic equation by a (which is allowed because a is non-zero), gives: or The quadratic equation is now in a form to which the method of completing the square can be applied. To "complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a complete square: which produces The right side can be written as a single fraction, with common denominator 4a2. This gives Taking the square root of both sides yields Isolating x, gives [edit]By shifting ax2 ax2 with vertex shifted from the origin to (xV, yV). a=-1 in this example. The quadratic formula can be derived by starting with equation which describes the parabola as ax2 with the vertex shifted from the origin to (xV, yV). Solving this equation for x is straightforward and results in Using Vieta's formulas for the vertex coordinates the values of x can be written as Note. The formulas for xV and yV can be derived by comparing the coefficients in and Rewriting the latter equation as and comparing with the former results in from which Vieta's expressions for xV and yV can be derived. [edit]By Lagrange resolvents For more details on this topic, see Lagrange resolvents. An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, which is an early part of Galois theory.[20] This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group. This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial assume that it factors as Expanding yields where and Since the order of multiplication does not matter, one can switch α and β and the values of p and q will not change: one says that p and q are symmetric polynomials in α and β. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can be expressed in terms of α + β and αβ. The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree n is related to the ways of rearranging ("permuting") n terms, which is called the symmetric group on n letters, and denoted For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple. To find the roots α and β, consider their sum and difference: These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations: Thus, solving for the resolvents gives the original roots. Formally, the resolvents are called the discrete Fourier transform (DFT) of order 2, and the transform can be expressed by the matrix with inverse matrix The transform matrix is also called the DFT matrix or Vandermonde matrix. Now is a symmetric function in α and β, so it can be expressed in terms of p and q, and in fact as noted above. But is not symmetric, since switching α and β yields (formally, this is termed a group action of the symmetric group of the roots). Since is not symmetric, it cannot be expressed in terms of the polynomials p and q, as these are symmetric in the roots and thus so is any polynomial expression involving them. However, changing the order of the roots only changes by a factor of and thus the square is symmetric in the roots, and thus expressible in terms of p and q. Using the equation yields and thus . If one takes the positive root, breaking symmetry, one obtains: and thus Thus the roots are which is the quadratic formula. Substituting yields the usual form for when a quadratic is not monic. The resolvents can be recognized as being the vertex, and is the discriminant (of a monic polynomial). A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating and which one can solve by the quadratic equation, and similarly for a quartic (degree 4) equation, whose resolving polynomial is a cubic, which can in turn be solved. However, the same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots. [edit]Other methods of root calculation [edit]Alternative quadratic formula In some situations it is preferable to express the roots in an alternative form. This alternative requires c to be nonzero; for, if c is zero, the formula correctly gives zero as one root, but fails to give any second, non-zero root. Instead, one of the two choices for ∓ produces the indeterminate form 0/0, which is undefined. However, the alternative form works when a is zero (giving the unique solution as one root and division by zero again for the other), which the normal form does not (instead producing division by zero both times). The roots are the same regardless of which expression we use; the alternative form is merely an algebraic variation of the common form: The alternative formula can reduce loss of precision in the numerical evaluation of the roots, which may be a problem if one of the roots is much smaller than the other in absolute magnitude. In this case, b is very close to , and the subtraction in the numerator causes loss of significance. A mixed approach avoids both all cancellation problems (only numbers of the same sign are added), and the problem of c being zero: Here sgn denotes the sign function. [edit]Floating-point implementation A careful floating point computer implementation differs a little from both forms to produce a robust result. Assuming the discriminant, b2 − 4ac, is positive and b is nonzero, the code will be something like the following:[21] Here sgn(b) is the sign function, where sgn(b) is 1 if b is positive and −1 if b is negative; its use ensures that the quantities added are of the same sign, avoiding catastrophic cancellation. The computation of x2 uses the fact that the product of the roots is c/a. Note that while the above formulation avoids catastrophic cancellation between b and , there remains a form of cancellation between the terms b2 and −4ac of the discriminant, which can still lead to loss of up to half of correct significant figures.[22][23] The discriminant b2−4ac needs to be computed in arithmetic of twice the precision of the result to avoid this (e.g. quad precision if the final result is to be accurate to full double precision).[24] [edit]Vieta's formulas Main article: Vieta's formulas Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form: and These results follow immediately from the relation: which can be compared term by term with: The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex’s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression: The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving Graph of two evaluations of the smallest root of a quadratic: direct evaluation using the quadratic formula (accurate at smaller b) and an approximation for widely spaced roots (accurate for larger b). The difference reaches a minimum at the large dots, and rounding causes squiggles in the curves beyond this minimum. As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other. If |x 2| << |x 1|, then x 1 + x 2 ≈ x 1, and we have the estimate: The second Vieta's formula then provides: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. The figure shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently the difference between the methods begins to increase as the quadratic formula becomes worse and worse. This situation arises commonly in amplifier design, where widely separated roots are desired to insure a stable operation (see step response). [edit]Trigonometric solution for complex roots This section requires expansion. (July 2012) In the case of complex roots the roots can also be found trigonometrically.[25] [edit]Geometric solution Geometric solution of ax2+bx+c using Lill's method. Solutions are −AX1/SA, −AX2/SA The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in the accompanying diagram. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.[26] [edit]Generalization of quadratic equation 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. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.) The symbol in the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Note that even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field. [edit]Characteristic 2 In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is and note that there is only one root since In summary, See quadratic residue for more information about extracting square roots in finite fields. In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax2 + bx + c are and For example, let a denote a multiplicative generator of the group of units of F4, the Galois field of order four (thus a and a + 1 are roots of x2 + x + 1 over F4). Because (a + 1)2 = a, a + 1 is the unique solution of the quadratic equation x2 + a = 0. On the other hand, the polynomial x2 + ax + 1 is irreducible over F4, but it splits over F16, where it has the two roots ab and ab + a, where b is a root of x2 + x + a in F16. This is a special case of Artin-Schreier theory.

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