In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ...

where a ≠ 0. (For a = 0, the equation becomes a linear equation.) Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x², the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term. For other senses of this word, see coefficient (disambiguation). ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

Quadratic equations are called quadratic because quadratus is Latin for "square"; in the leading term the variable is squared. For other uses, see Latin (disambiguation). ... In algebra, the square of a number is that number multiplied by itself. ...

Plots of real-valued quadratic function ax² + bx + c, varying each coefficient separately

Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1276 × 877 pixel, file size: 114 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1276 × 877 pixel, file size: 114 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Quadratic formula

A quadratic equation with real or complex coefficients has two (not necessarily distinct) solutions, called roots, which may be real or complex, given by the quadratic formula: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

where the symbol "±" indicates that both

and

are solutions.

Discriminant

Example discriminant signs
■ <0: x²+¹⁄₂
■ =0: −⁴⁄₃x²+⁴⁄₃x−¹⁄₃
■ >0: ³⁄₂x²+¹⁄₂x−⁴⁄₃

In the above formula, the expression underneath the square root sign: Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (900 × 900 pixel, file size: 44 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (900 × 900 pixel, file size: 44 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

is called the discriminant of the quadratic equation. In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

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, 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, there is exactly one distinct root, and that root is a real number. Sometimes called a double root, its value is:

  

• If the discriminant is negative, there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:

  

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. The integers are commonly denoted by the above symbol. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a quadratic irrational, also known as a quadratic surd or quadratic irrationality, is an irrational number that is the solution to some quadratic equation with rational coefficients. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

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 roots of the quadratic equation: x² − x − 2 = 0.

The roots of the quadratic equation Image File history File links Polynomial of degree 2 : f(x) = x2 - x - 2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Polynomial of degree 2 : f(x) = x2 - x - 2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... For other uses, see Root (disambiguation). ...

are also the zeros of the quadratic function: This article is about the zeroes of a function. ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ...

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 zeros of f are exactly the x-coordinates of the points where the graph touches the x-axis. Please refer to Real vs. ... In mathematics, the domain of a function is the set of all input values to the function. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

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. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Quadratic factorization

The term

is a factor of the polynomial

if and only if r is a root of the quadratic equation In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

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 ...

Application to higher-degree equations

Certain higher-degree equations may be quadratic in form, such as:

which can be written as

where

.

Note that the highest exponent is twice the value of the exponent of the middle term. This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring, the quadratic formula, or completing the square. ... Completing the square is an algebra technique, also used in many types of calculus. ...

Generally speaking, if the polynomial is quadratic in some variable u where

then the quadratic equation can be used to help find solutions.

History

The Babylonians, as early as 1800 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form: (Redirected from 1800 BC) (19th century BC - 18th century BC - 17th century BC - other centuries) (3rd millennium BC - 2nd millennium BC - 1st millennium BC) Events 1787 - 1784 BC -- Amorite conquests of Uruk and Isin 1786 BC -- Egypt: End of Twelfth Dynasty, start of Thirteenth Dynasty, start of Fourteenth Dynasty 1766... The term Old Babylonian is a period in Mesopotamian history that refers, roughly, to the period between the end of the Third Dynasty of Ur (c. ... Small tablets made out of clay were used from late 4th millennium BC onwards as a writing medium in Sumerian, Mesopotamian, Hittite, and Minoan/Mycenaean civilizations. ...

which are equivalent to the equation:^[1]

The original pair of equations were solved as follows:

1. Form
2. Form
3. Form
4. Form
5. Find by inspection of the values in (1) and (4).^[2]

In the Sulba Sutras in ancient India circa 8th century BCE quadratic equations of the form ax² = c and ax² + bx = c were explored using geometric methods. Babylonian mathematicians from circa 400 BCE and Chinese mathematicians from circa 200 BCE used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BCE. The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Map of South Asia (see note) This article deals with the geophysical region in Asia. ... (9th century BC - 8th century BC - 7th century BC - other centuries) (800s BC - 790s BC - 780s BC - 770s BC - 760s BC - 750s BC - 740s BC - 730s BC - 720s BC - 710s BC - 700s BC - other decades) (2nd millennium BC - 1st millennium BC - 1st millennium AD) Events Golden age in Armenia Assyria... Babylonian clay tablet YBC 7289 with annotations. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 450s BC 440s BC 430s BC 420s BC 410s BC - 400s BC - 390s BC 380s BC 370s BC 360s BC 350s BC Years: 405 BC 404 BC 403 BC 402 BC 401 BC - 400 BC - 399 BC 398 BC... Knowledge of Chinese mathematics before 100 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. ... (Redirected from 200 BCE) Centuries: 3rd century BC - 2nd century BC - 1st century BC Decades: 250s BC 240s BC 230s BC 220s BC 210s BC - 200s BC - 190s BC 180s BC 170s BC 160s BC 150s BC Years: 205 BC 204 BC 203 BC 202 BC 201 BC - 200 BC... Completing the square is an algebra technique, also used in many types of calculus. ... For other uses, see Euclid (disambiguation). ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC...

In 628 CE, Brahmagupta gave the first explicit (although still not completely general) solution of the quadratic equation: Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ...

This is equivalent to:

The Bakhshali Manuscript dated to have been written in India in the 7th century CE contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Mohammad bin Musa Al-kwarismi (Persia, 9th century) developed a set of formulae that worked for positive solutions. Abraham bar Hiyya Ha-Nasi (also known by the Latin name Savasorda) introduced the complete solution to Europe in his book Liber embadorum in the 12th century. Bhāskara II (1114–1185), an Indian mathematician–astronomer, gave the first general solution to the quadratic equation with two roots.^[3] The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in what is now Pakistan in 1881. ... In mathematics, more precisely in algebra, an indeterminate is a quantity that is not known, and cannot be solved for. ... A stamp issued September 6, 1983 in the Soviet Union, commemorating al-KhwārizmÄ«s (approximate) 1200th anniversary. ... For other uses of this term see: Persia (disambiguation) The Persian Empire is the name used to refer to a number of historic dynasties that have ruled the country of Persia (Iran). ... As a means of recording the passage of time the 9th century was the century that lasted from 801 to 900. ... Abraham bar Hiyya Ha-Nasi (Abraham son of [Rabbi] Hiyya the Prince/President) (1070–1136) was a Spanish Jewish mathematician and astronomer, also known as Savasorda (from the Arabic Sâhib as-Shurta). ... For other uses, see Latin (disambiguation). ... Abraham bar Hiyya Ha-Nasi (1070 - 1136) was a Spanish Jewish mathematician and astronomer, also known as Savasorda (from the Arabic Sâhib as-Shurta). ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician and astronomer. ... Events January 7 - Matilda, daughter of Henry I of England, marries Henry IV, Holy Roman Emperor Births Deaths Categories: 1114 ... Events April 25 - Genpei War - Naval battle of Dan-no-ura leads to Minamoto victory in Japan Templars settle in London and begin the building of New Temple Church End of the Heian Period and beginning of the Kamakura period in Japan. ... This article is under construction. ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ...

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. Yang Hui (楊輝, c. ... Events In the Iberian peninsula, James I of Aragon captures the city of Valencia September 28 from the Moors; the Moors retreat to Granada. ... Events July 2 - The Battle of Göllheim is fought between Albert I of Habsburg and Adolf of Nassau-Weilburg. ...

Derivation

Complete the square

The quadratic formula can be derived by the method of completing the square, so as to make use of the algebraic identity: In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Completing the square is an algebra technique, also used in many types of calculus. ...

Dividing the quadratic equation

by a (which is allowed because a is non-zero), gives:

or

The quadratic equation is now in a form in which the method of completing the square can be applied. To "complete the square" is to find some constant k such that

for another constant y. In order for these equations to be true,

and

thus

Adding this constant to equation (1) produces

The left side is now a perfect square because The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ...

The right side can be written as a single fraction, with common denominator 4a². This gives

Taking the square root of both sides yields In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

Isolating x, gives

Alternative derivation

Start with the general form of a quadratic:

Multiply both sides by 4a:

Subtract 4ac from both sides:

Add b² to both sides:

Factorise the left-hand side:

Take the square root of both sides:

Subtract b from both sides:

Divide both sides by 2a:

Alternative formula

In some situations it is preferable to express the roots in an alternate 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. (When c is zero we have division of zero by zero, which is indeterminate.)

The roots are the same regardless of which expression we use; the alternate form is merely an algebraic variation of the common form:

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. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

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 r₂ uses the fact that the product of the roots is c/a. Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... Loss of significance is an undesirable effect in calculations using floating-point arithmetic. ...

Viète's formulas

Viète'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: In mathematics, more specifically in algebra, Viètes formulas, named after François Viète, are formulas which relate the roots of a polynomial to its coefficients. ...

and

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: f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ...

The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving

Generalizations

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.) In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

The symbol

in the formula should be understood as "either of the two elements whose square is

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. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

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 In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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 mathematics, a number q is called a quadratic residue modulo p if there exists an integer x such that: Otherwise, q is called a quadratic non-residue. ...

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 In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

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 x + ax + 1 is irreducible over F₄, but splits over F₁₆, where it has the two roots ab and ab + a, where b is a root of x² + x + a in F₁₆. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

This is a special case of Artin-Schreier theory. See Artin-Schreier theorem for theory about real-closed fields In mathematics, Artin-Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for extensions of degree equal to the characteristic p. ...

References

1. ^ Stillwell, John. 2004. Mathematics and its History. Berlin and New York: Springer-Verlag. 542 pages. p. 86
2. ^ ^{a b} Stillwell, John. 2004. Mathematics and its History. Berlin and New York: Springer-Verlag. 542 pages. p. 87
3. ^ http://www.bbc.co.uk/dna/h2g2/A2982567

Book

Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, by Swami Sankaracarya (1884-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965; reprinted in Delhi, India, 1975, 1978. 367 pages.

See also

Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... Graph of a polynomial of degree 5, with 4 critical points. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... This article on periodic points of complex quadratic mappings describes periodic points of some quadratic polynomial mappings on the complex numbers. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ...

External links

• Eric W. Weisstein, Quadratic equations at MathWorld.
• L. Euler's Elements of Algebra
• Quadratic equation solver, plus solvers for cubic and quartic equations
• 101 uses of a quadratic equation part I Part II
• Quadratic graphical explorer Interactive applet. Sliders for a,b,c show effects on a graph.
• Trigonometric solutions: You Could Learn a Lot from a Quadratic
• Factoring, completing the square, solving quadratic equations, worked-out solutions.