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In
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, a quadratic equation () is any
equation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
that can be rearranged in standard form as $ax^2 + bx + c = 0\,,$ where represents an
unknown Unknown or The Unknown may refer to: Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), a silent boxing film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film) * The Unknown (1927 film), ''The Unknown'' (1 ...
value, and , , and represent known numbers, where . (If and then the equation is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, ...
, not quadratic.) The numbers , , and are the ''
coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
s'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant'' or ''free term''. The values of 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 number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, there are either two real solutions, or a single real double root, or two complex solutions that are
complex conjugate In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent equation $ax^2+bx+c=a(x-r)(x-s)=0$ where and are the solutions for . The
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as Factorization, factoring (direct ...
$x=\frac$ expresses the solutions in terms of , , and .
Completing the square : In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a Square ...
is one of several ways for getting it. Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. Because the quadratic equation involves only one unknown, it is called " univariate". The quadratic equation contains only powers of that are non-negative integers, and therefore it is a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
. In particular, it is a second-degree polynomial equation, since the greatest power is two.

A quadratic equation with real or complex
coefficients In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
has two solutions, called ''roots''. These two solutions may or may not be distinct, and they may or may not be real.

## Factoring by inspection

It may be possible to express a quadratic equation as a product . In some cases, it is possible, by simple inspection, to determine values of ''p'', ''q'', ''r,'' and ''s'' that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if or . Solving these two linear equations provides the roots of the quadratic. For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form , the sought factorization has the form , and one has to find two numbers and that add up to and whose product is (this is sometimes called "Vieta's rule" and is related to
Vieta's formulas In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its Root of a function, roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta" ...
). As an example, factors as . The more general case where does not equal can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where or , factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.

## Completing the square

The process of completing the square makes use of the algebraic identity :$x^2+2hx+h^2 = \left(x+h\right)^2,$ which represents a well-defined
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, #Divide each side by , the coefficient of the squared term. #Subtract the constant term from both sides. #Add the square of one-half of , the coefficient of , to both sides. This "completes the square", converting the left side into a perfect square. #Write the left side as a square and simplify the right side if necessary. #Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side. #Solve each of the two linear equations. We illustrate use of this algorithm by solving :$2x^2+4x-4=0$ :$\ x^2+2x-2=0$ :$\ x^2+2x=2$ :$\ x^2+2x+1=2+1$ :$\left\left(x+1 \right\right)^2=3$ :$\ x+1=\pm\sqrt$ :$\ x=-1\pm\sqrt$ The plus–minus symbol "±" indicates that both and are solutions of the quadratic equation.

## Quadratic formula and its derivation

Completing the square : In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a Square ...
can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every p ...
will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: :$\left\left(x+\frac\right\right)^2=\frac.$ Taking the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
of both sides, and isolating , gives: :$x=\frac.$ Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as or  , where has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent. A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics. A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation :$x = \frac.$ This can be deduced from the standard quadratic formula by
Vieta's formulas In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its Root of a function, roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta" ...
, which assert that the product of the roots is . One property of this form is that it yields one valid root when , while the other root contains division by zero, because when , the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form for the other root. On the other hand, when , the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form . When neither nor is zero, the equality between the standard quadratic formula and Muller's method, :$\frac = \frac\,,$ can be verified by cross multiplication, and similarly for the other choice of signs.

It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by , which is always possible since is non-zero. This produces the ''reduced quadratic equation'': :$x^2+px+q=0,$ where and . This
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cd ...
equation has the same solutions as the original. The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is: :$x = \frac \left\left( - p \pm \sqrt \right\right),$ or equivalently: :$x = - \frac \pm \sqrt.$

## Discriminant

In the quadratic formula, the expression underneath the square root sign is called the ''
discriminant In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
'' of the quadratic equation, and is often represented using an upper case or an upper case Greek delta: :$\Delta = b^2 - 4ac.$ 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 ::$\frac \quad\text\quad \frac,$ :both of which are real numbers. For quadratic equations with
rational Rationality is the Quality (philosophy), quality of being guided by or based on reasons. In this regard, a person Action (philosophy), acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong e ...
coefficients, if the discriminant is a
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
, then the roots are rational—in other cases they may be quadratic irrationals. *If the discriminant is zero, then there is exactly one real root $-\frac,$ sometimes called a repeated or double root. *If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots$-\frac + i \frac \quad\text\quad -\frac - i \frac,$ :which are
complex conjugate In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s of each other. In these expressions is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimen ...
. 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.

## Geometric interpretation

The function is a
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...

. The graph of any quadratic function has the same general shape, which is called a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

. The location and size of the parabola, and how it opens, depend on the values of , , and . As shown in Figure 1, if , the parabola has a minimum point and opens upward. If , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The ''-coordinate'' of the vertex will be located at $\scriptstyle x=\tfrac$, and the ''-coordinate'' of the vertex may be found by substituting this ''-value'' into the function. The ''-intercept'' is located at the point . The solutions of the quadratic equation correspond to the roots of the function , since they are the values of for which . As shown in Figure 2, if , , and are
real numbers In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

and the domain of is the set of real numbers, then the roots of are exactly the -
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...

of the points where the graph touches the -axis. As shown in Figure 3, if the discriminant is positive, the graph touches the -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the -axis.

The term :$x - r$ is a factor of the polynomial : $ax^2+bx+c$ if and only if is a
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often b ...
of the quadratic equation : $ax^2+bx+c=0.$ It follows from the quadratic formula that : $ax^2+bx+c = a \left\left( x - \frac \right\right) \left\left( x - \frac \right\right).$ 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 :$ax^2+bx+c = a \left\left( x + \frac \right\right)^2.$

## Graphical solution

The solutions of the quadratic equation :$ax^2+bx+c=0$ may be deduced from the of the
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...

:$f\left(x\right)=ax^2+bx+c,$ which is a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

. If the parabola intersects the -axis in two points, there are two real roots, which are the -coordinates of these two points (also called -intercept). If the parabola is
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

to the -axis, there is a double root, which is the -coordinate of the contact point between the graph and parabola. If the parabola does not intersect the -axis, there are two
complex conjugate In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
roots. Although these roots cannot be visualized on the graph, their can be. Let and be respectively the -coordinate and the -coordinate of the vertex of the parabola (that is the point with maximal or minimal -coordinate. The quadratic function may be rewritten : $y = a\left(x - h\right)^2 + k.$ Let be the distance between the point of -coordinate on the axis of the parabola, and a point on the parabola with the same -coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is , and their imaginary part are . That is, the roots are :$h+id \quad \text \quad h-id,$ or in the case of the example of the figure :$5+3i \quad \text \quad 5-3i.$

## Avoiding loss of significance

Although the quadratic formula provides an exact solution, the result is not exact if
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s are approximated during the computation, as usual in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, where real numbers are approximated by floating point numbers (called "reals" in many
programming language A programming language is a system of notation for writing computer program, computer programs. Most programming languages are text-based formal languages, but they may also be visual programming language, graphical. They are a kind of computer ...

s). In this context, the quadratic formula is not completely stable. This occurs when the roots have different
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic d ...
, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or in the smaller root. To avoid this, the root that is smaller in magnitude, , can be computed as $\left(c/a\right)/R$ where is the root that is bigger in magnitude. A second form of cancellation can occur between the terms and of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.

# Examples and applications

The
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek let ...

is found as the positive solution of the quadratic equation $x^2-x-1=0.$ The equations of the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...

and the other
conic sections In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, ...

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

s,
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

s, and
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A h ...

s—are quadratic equations in two variables. Given the or
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

of an angle, finding the cosine or sine of involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. Descartes' theorem states that for every four kissing (mutually tangent) circles, their satisfy a particular quadratic equation. The equation given by Fuss' theorem, giving the relation among the radius of a 's inscribed circle, the radius of its
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertex (geometry), vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius ...
, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an . Critical points of a and
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a plane curve#Smooth plane curve, smooth plane curve at which the signed curvature, curv ...
s of a
quartic function In algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of alm ...
are found by solving a quadratic equation.

# History

Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian
clay tablet In the Ancient Near East, clay tablets (Akkadian language, Akkadian ) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age. Cuneiform characters were imprinted on a wet clay t ...
s) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the
Third Dynasty of Ur The Third Dynasty of Ur, also called the Neo-Sumerian Empire, refers to a 22nd to 21st century Common Era, BC (middle chronology) Sumerian ruling dynasty based in the city of Ur and a short-lived territorial-political state which some historians c ...
. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: :$x+y=p,\ \ xy=q,$ which is equivalent to the statement that and are the roots of the equation: :$z^2+q=pz.$ The steps given by Babylonian scribes for solving the above rectangle problem, in terms of and , were as follows: #Compute half of ''p''. #Square the result. #Subtract ''q''. #Find the (positive) square root using a table of squares. #Add together the results of steps (1) and (4) to give . In modern notation this means calculating $x = \left\left(\frac\right\right) + \sqrt$, which is equivalent to the modern day
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as Factorization, factoring (direct ...
for the larger real root (if any) $x = \frac$ with , , and . Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. Rules for quadratic equations were given in ''
The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'', a Chinese treatise on mathematics. These early geometric methods do not appear to have had a general formula.
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
and Euclid created a general procedure to find solutions of the quadratic equation. In his work ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
'', the Greek mathematician
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
solved the quadratic equation, but giving only one root, even when both roots were positive. In 628 AD,
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
, 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 oefficient of thesquare, add the square of the oefficient of themiddle term; the square root of the same, less the oefficient of themiddle term, being divided by twice the oefficient of thesquare is the value." (''Brahmasphutasiddhanta'', Colebrook translation, 1817, page 346) This is equivalent to :$x = \frac.$ 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 ).
Muhammad ibn Musa al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
(9th century), possibly 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. He also described the method of completing the square and recognized that the
discriminant In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
must be positive, 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. While al-Khwarizmi himself did not accept negative solutions, later
Islamic mathematicians Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius of Perga, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress wa ...
that succeeded him accepted negative solutions, as well as
irrational number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s as solutions. 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 In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
,
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
or fourth root) as solutions to quadratic equations or as
coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or an expression (mathematics), expression; it is usually a number, but may be any expression (including variables su ...
s in an equation. The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations. 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. His solution was largely based on Al-Khwarizmi's work. The writing of the Chinese mathematician
Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circle (mathematics), magic ci ...
(1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. By 1545
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
in 1594. In 1637
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...
published ''
La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 159 ...
'' containing the quadratic formula in the form we know today.

## Alternative methods of root calculation

### Vieta's formulas

''Vieta's formulas'' (named after
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics ...
) are the relations :$x_1 + x_2 = -\frac, \quad x_1 x_2 = \frac$ between the roots of a quadratic polynomial and its coefficients. They result from comparing term by the relation :$\left\left( x - x_1 \right\right) \left\left( x-x_2 \right \right) = x^2 - \left\left( x_1+x_2 \right\right)x +x_1 x_2 = 0$ with the equation :$x^2 + \frac ba x +\frac ca = 0.$ The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's -coordinate is located at the average of the roots (or intercepts). Thus the -coordinate of the vertex is :$x_V = \frac = -\frac.$ The -coordinate can be obtained by substituting the above result into the given quadratic equation, giving :$y_V = - \frac + c = - \frac .$ These formulas for the vertex can also deduced directly from the formula (see
Completing the square : In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a Square ...
) :$ax^2+bx+c=a\left\left(\left\left(x-\frac b\right\right)^2-\frac\right\right).$ For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If , then , and we have the estimate: :$x_1 \approx -\frac .$ The second Vieta's formula then provides: :$x_2 = \frac \approx -\frac .$ 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 ), which causes
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are ...
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 increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as 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 ensure a stable operation (see Step response).

### Trigonometric solution

In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called
prosthaphaeresis Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division (mathematics), division using formulas from trigonometry. For the ...
, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in
celestial mechanics Celestial mechanics is the branch of astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronol ...
calculations. It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation, ''   $ax^2 + bx \pm c = 0 ,$ where the sign of the ± symbol is chosen so that and may both be positive. By substituting ''   $x = \sqrt \tan\theta$ and then multiplying through by , we obtain ''   $\sin^2\theta + \frac \sin\theta \cos\theta \pm \cos^2\theta = 0 .$ Introducing functions of and rearranging, we obtain ''   $\tan 2 \theta_n = + 2 \frac ,$ ''   $\sin 2 \theta_p = - 2 \frac ,$ where the subscripts and correspond, respectively, to the use of a negative or positive sign in equation ''. Substituting the two values of or found from equations '' or '' into '' gives the required roots of ''. Complex roots occur in the solution based on equation '' if the absolute value of exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. Calculating complex roots would require using a different trigonometric form. :To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy: :::$4.16130x^2 + 9.15933x - 11.4207 = 0$ #A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries. #$\log a = 0.6192290, \log b = 0.9618637, \log c = 1.0576927$ #$2 \sqrt/b = 2 \times 10^ = 1.505314$ #$\theta = \left(\tan^1.505314\right) / 2 = 28.20169^ \text -61.79831^$ #$\log , \tan \theta , = -0.2706462 \text 0.2706462$ #$\log\sqrt = \left(1.0576927 - 0.6192290\right) / 2 = 0.2192318$ #$x_1 = 10^ = 0.888353$ (rounded to six significant figures) ::$x_2 = -10^ = -3.08943$

### Solution for complex roots in polar coordinates

If the quadratic equation $ax^2+bx+c=0$ with real coefficients has two complex roots—the case where $b^2-4ac<0,$ requiring ''a'' and ''c'' to have the same sign as each other—then the solutions for the roots can be expressed in polar form as :$x_1, \, x_2=r\left(\cos \theta \pm i\sin \theta\right),$ where $r=\sqrt$ and $\theta =\cos ^\left\left(\tfrac\right\right).$

### Geometric solution

The quadratic equation may be solved geometrically in a number of ways. One way is via
Lill's method In mathematics, Lill's method is a visual method of finding the real number, real zero of a function, roots of a univariate polynomial of any degree of a polynomial, degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper ...
. The three coefficients , , are drawn with right angles between them as in SA, AB, and BC in Figure 6. 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 or SA. If is the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA. The
Carlyle circle In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the equation solving, solutions of the quadratic equation are the horizont ...
, named after
Thomas Carlyle Thomas Carlyle (4 December 17955 February 1881) was a Scottish essayist, historian and philosopher. A leading writer of the Victorian era, he exerted a profound influence on 19th-century art, literature and philosophy. Born in Ecclefechan, Dum ...
, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s.

The formula and its derivation remain correct if the coefficients , and are
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s, or more generally members of any field whose characteristic is not . (In a field of characteristic 2, the element is zero and it is impossible to divide by it.) The symbol :$\pm \sqrt$ 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 . Even if a field does not contain a square root of some number, there is always a quadratic
extension field In mathematics, particularly in algebra, a field extension is a pair of Field (mathematics), fields E\subseteq F, such that the operations of ''E'' are those of ''F'' Restriction (mathematics), restricted to ''E''. In this case, ''F'' is an extens ...
which does, so the quadratic formula will always make sense as a formula in that extension field.

### Characteristic 2

In a field of characteristic , the quadratic formula, which relies on being a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
, does not hold. Consider the monic quadratic polynomial :$x^ + bx + c$ over a field of characteristic . If , then the solution reduces to extracting a square root, so the solution is :$x = \sqrt$ and there is only one root since :$-\sqrt = -\sqrt + 2\sqrt = \sqrt.$ In summary, :$\displaystyle x^ + c = \left(x + \sqrt\right)^.$ See
quadratic residue In number theory, an integer ''q'' is called a quadratic residue modulo operation, modulo ''n'' if it is Congruence relation, congruent to a Square number, perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv ...
for more information about extracting square roots in finite fields. In the case that , 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 of to be a root of the polynomial , an element of the
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field (mathematics), field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitt ...
of that polynomial. One verifies that is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic are :$\fracR\left\left(\frac\right\right)$ and :$\frac\left\left(R\left\left(\frac\right\right)+1\right\right).$ For example, let denote a multiplicative generator of the group of units of , the
Galois field In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of order four (thus and are roots of over . Because , is the unique solution of the quadratic equation . On the other hand, the polynomial is irreducible over , but it splits over , where it has the two roots and , where is a root of in . This is a special case of Artin–Schreier theory.

* Solving quadratic equations with continued fractions *
Linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b,a_1,\ldots,a_n are the coefficients, ...
* Cubic function *
Quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
*
Quintic equation In algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost ...
*
Fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one co ...