TheInfoList

In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, a quadratic equation (from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

for "
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
") is any
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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'' (19 ...

, and , , and represent known numbers, where . If , then the equation is
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

, not quadratic, as there is no $ax^2$ term. The numbers , , and are the ''
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s'' of the equation and may be distinguished by calling them, respectively, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant'' or ''free term''. The values of that satisfy the equation are called ''
solutions Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent. In chemistry, a solution is a special type of Homogeneous and ...
'' of the equation, and ''
roots 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 most often ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, there are either two real solutions, or a single real double root, or two
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

solutions. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

into an equivalent equation :$ax^2+bx+c=a\left(x-r\right)\left(x-s\right)=0$ where and are the solutions for .
Completing the square : In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

on a quadratic equation in standard form results in the
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

, which expresses the solutions in terms of , , and . 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 In mathematics, a univariate object is an expression, equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
". The quadratic equation contains only
powers Powers (stylized as POWERS) is a musical duo composed of Mike Del Rio and Crista Ru. Their music has been described as alternative pop, electropop, and Progressive pop, progressive pop. ''Time'' has called their music "groovy and futuristic". ...
of that are non-negative integers, and therefore it is a
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. In particular, it is a second-degree polynomial equation, since the greatest power is two.

real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

coefficients In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). 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 and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

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 :$1\right) \ x^2+2x-2=0$ :$2\right) \ x^2+2x=2$ :$3\right) \ x^2+2x+1=2+1$ :$4\right) \ \left\left(x+1 \right\right)^2=3$ :$5\right) \ x+1=\pm\sqrt$ :$6\right) \ x=-1\pm\sqrt$ The plus–minus symbol "±" indicates that both and are solutions of the quadratic equation.

Completing the square : In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

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 In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
will now be briefly summarized. It can easily be seen, by
polynomial expansion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, that the following equation is equivalent to the quadratic equation: :$\left\left(x+\frac\right\right)^2=\frac.$ Taking the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 Muller's method is a root-finding algorithm, a numerical analysis, numerical method for solving equations of the form ''f''(''x'') = 0. It was first presented by David E. Muller in 1956. Muller's method is based on the secant method, which construc ...
provides the same roots via the equation :$x = \frac.$ This can be deduced from the standard quadratic formula by
Vieta's formulas In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
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 .

It is sometimes convenient to reduce a quadratic equation so that its
leading coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' of the quadratic equation, and is often represented using an upper case or an upper case Greek
delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, a landform at the mouth of a river * D (NATO phonetic alphabet: "Delta"), the fourth letter of the modern English alphabet * Delta Air Lines, an Ame ...
: :$\Delta = b^2 - 4ac.$ 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 or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
coefficients, if the discriminant is a
square number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, then the roots are rational—in other cases they may be quadratic irrationals. *If the discriminant is zero, then there is exactly one
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
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 The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

roots ::$-\frac + i \frac \quad\text\quad -\frac - i \frac,$ :which are
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
. 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 algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

. The graph of any quadratic function has the same general shape, which is called a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. 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 Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
. 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 A root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a lar ...
of the function , since they are the values of for which . As shown in Figure 2, if , , and are
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

and the
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of is the set of real numbers, then the roots of are exactly the -
coordinates In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

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 plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

of the
quadratic function In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

:$y=ax^2+bx+c,$ which is a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. If the parabola intersects the -axis in two points, there are two real
roots 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 most often ...
, which are the -coordinates of these two points (also called -intercept). If the parabola is
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
roots. Although these roots cannot be visualized on the graph, their
real and imaginary parts In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

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 :$h+id \quad \text \quad x-id,$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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). Numerical analysis ...
, where real numbers are approximated by
floating point number In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often used in system ...
s (called "reals" in many
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s). In this context, the quadratic formula is not completely
stable A stable is a building in which livestock Livestock are the domesticated Domestication is a sustained multi-generational relationship in which one group of organisms assumes a significant degree of influence over the reproduction and c ...
. This occurs when the roots have different
order of magnitude An order of magnitude is an approximation of the logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contain ...
, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause
loss of significance Loss of significance is an undesirable effect in calculations using finite-precision arithmetic such as floating-point arithmetic. It occurs when an operation on two numbers increases relative error substantially more than it increases absol ...
or
catastrophic cancellation Catastrophe or catastrophic comes from the Greek κατά (''kata'') = down; στροφή (''strophē'') = turning ( el, καταστροφή). It may refer to: A general or specific event * Disaster, a devastating event * The Asia Minor A ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

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 A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

and the other
conic sections In mathematics, a conic section (or simply 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, the par ...

ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

s,
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s, and
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) 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 ...

s—are quadratic equations in two variables. Given the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

or
sine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mut ...
states that for every four kissing (mutually tangent) circles, their
radii In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

satisfy a particular quadratic equation. The equation given by Fuss' theorem, giving the relation among the radius of a
bicentric quadrilateral In Euclidean geometry, a bicentric quadrilateral is a convex polygon, convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''ci ...

's
inscribed circle (I), excircles, excenters (J_A, J_B, J_C), internal angle bisector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldes ...
circumscribed circle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
, 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
of an
ex-tangential quadrilateral In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; ...

. Critical points of a
cubic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

and
inflection point In differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

s of a
quartic function In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A quarti ...

are found by solving a quadratic equation.

History

Babylonian mathematicians, as early as 2000 BC (displayed on
Old Babylonian Old Babylonian may refer to: *the period of the First Babylonian dynasty (20th to 16th centuries BC) *the historical stage of the Akkadian language of that time See also

*Old Assyrian (disambiguation) {{disambig ...
clay tablet In the Ancient Near East The ancient Near East was the home of early civilization A civilization (or civilisation) is any complex society that is characterized by urban development, social stratification, a form of government, and sym ...
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 BC ( middle chronology) Sumerian ruling dynasty based in the city of Ur and a short-lived territorial-political state which some historians consider to h ...
. 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 Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

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 surv ...
'', a Chinese treatise on mathematics. These early geometric methods do not appear to have had a general formula.
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach
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and Euclid created a general procedure to find solutions of the quadratic equation. In his work ''
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'', 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 autho ...
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, doc ...

, 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 ''
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'' written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic
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s (originally of type ).
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(
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, 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. He also described the method of completing the square and recognized that the
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
must be positive, which was proven by his contemporary
'Abd al-Hamīd ibn Turk ( fl. 830), known also as ( ar, ابومحمد عبدالحمید بن واسع بن ترک الجیلی) was a ninth-century Muslim mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim Abū al-Faraj Muḥamm ...
(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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
that succeeded him accepted negative solutions, as well as
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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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

,
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or fourth root) as solutions to quadratic equations or as
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s in an equation. The 9th century Indian mathematician
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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
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compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by
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in 1594. In 1637
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published ''
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'' 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) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
) are the relations :$x_1 + x_2 = -\frac, \quad x_1 x_2 = \frac$ between the roots of a quadratic polynomial and its coefficients. These formulas result immediately from the relation: :$\left\left( x - x_1 \right\right) \left\left( x-x_2 \right \right) = x^2 - \left\left( x_1+x_2 \right\right)x +x_1 x_2 = 0,$ which can be compared term by term with :$x^2 + \left(b/a\right)x +c/a = 0.$ 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 Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...

, 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 Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

) :$ax^2+bx+c=a\left\left(\left\left(x-\frac b\right\right)^2-\frac\right\right).$ 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 In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems ...
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
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).

Trigonometric solution

In the days before calculators, people would use
mathematical table Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Table (information), Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy and celestial navigatio ...
s—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 of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorit ...
, 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 that deals with the motion (physics), motions of celestial object, objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical obje ...
calculations. It is within this context that we may understand the development of means of solving quadratic equations by the aid of
trigonometric substitution Trigonometry (from Ancient Greek, Greek ''wikt:τρίγωνον, trigōnon'', "triangle" and ''wikt:μέτρον, metron'', "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The fie ...
. 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. 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, named after Thomas Carlyle, 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 polygons.

The formula and its derivation remain correct if the coefficients , and are complex numbers, or more generally members of any field (mathematics), field whose characteristic (algebra), 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 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 (ring theory), unit, does not hold. Consider the monic polynomial, 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 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 polynomial, 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 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 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.