In
algebra, a quadratic equation () is any
equation that can be rearranged in standard form as
where represents an
unknown value, and , , and represent known numbers, where . (If and then the equation is
linear, not quadratic.) The numbers , , and are the ''
coefficients'' 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
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
'' of the equation, and ''
roots'' or ''
zeros'' of the
expression
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, o ...
on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a
double root. If all the coefficients are
real numbers, there are either two real solutions, or a single real double root, or two
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
solutions that are
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
where and are the solutions for .
The
quadratic formula
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 perfe ...
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
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate ...
". 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.
Solving the quadratic equation
A quadratic equation with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
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). 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
:
which represents a well-defined algorithm 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
:
:
:
:
:
:
:
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 perfe ...
can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
:
Taking the square root of both sides, and isolating , gives:
:
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
:
This can be deduced from the standard quadratic formula by Vieta's formulas, 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
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
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,
:
can be verified by cross multiplication, and similarly for the other choice of signs.
Reduced quadratic equation
It is sometimes convenient to reduce a quadratic equation so that its leading coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves v ...
is one. This is done by dividing both sides by , which is always possible since is non-zero. This produces the ''reduced quadratic equation'':
:
where and . This monic polynomial 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:
:
or equivalently:
:
Discriminant
In the quadratic formula, the expression underneath the square root sign is called the '' discriminant'' of the quadratic equation, and is often represented using an upper case or an upper case Greek delta:
:
A quadratic equation with ''real'' coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
*If the discriminant is positive, then there are two distinct roots
::
:both of which are real numbers. For quadratic equations with rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...
s.
*If the discriminant is zero, then there is exactly one real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
root 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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
roots
:which are complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s of each other. In these expressions is the imaginary unit.
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
Geometric interpretation
The function is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. 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 , 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 and the domain of is the set of real numbers, then the roots of are exactly the - coordinates 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.
Quadratic factorization
The term
:
is a factor of the polynomial
:
if and only if is a root of the quadratic equation
:
It follows from the quadratic formula that
:
In the special case where the quadratic has only one distinct root (''i.e.'' the discriminant is zero), the quadratic polynomial can be factored as
:
Graphical solution
The solutions of the quadratic equation
:
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 discre ...
of the quadratic function
:
which is a parabola.
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 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, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
roots. Although these roots cannot be visualized on the graph, their real and imaginary parts
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 ...
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
:
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
:
or in the case of the example of the figure
:
Avoiding loss of significance
Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point number
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
s (called "reals" in many programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s). In this context, the quadratic formula is not completely stable.
This occurs when the roots have different order of magnitude, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, , can be computed as 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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
is found as the positive solution of the quadratic equation
The equations of the circle and the other conic sections— ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.
Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large 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 radii
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
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 quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
'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 vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, 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, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
of an ex-tangential quadrilateral
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the ''extensions'' of all four sides are tangent to a circle outside the quadrilateral.Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tan ...
.
Critical points of a cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
and inflection points of a quartic function
In algebra, a quartic function is a 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 four, called a quartic polynomial.
A '' quartic equation'', or equation of the fourth de ...
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
Akkadian (, Akkadian: )John Huehnergard & Christopher Woods, "Akkadian and Eblaite", ''The Camb ...
clay tablets) 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. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
:
which is equivalent to the statement that and are the roots of the equation:
:
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 , which is equivalent to the modern day quadratic formula for the larger real root (if any) 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
Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
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, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work '' Arithmetica'', the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.
In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation as follows: "To the absolute number multiplied by four times the 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
:
The '' Bakhshali Manuscript'' written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic ]indeterminate equation In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation ax + by =c is a simple indeterminate equation, as is x^2=1. Indeterminate equations cannot be solv ...
s (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 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) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...
that succeeded him accepted negative solutions,[ as well as ]irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
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, cube root or fourth root) as solutions to quadratic equations or as coefficients 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 (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 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 published '' La Géométrie'' containing the quadratic formula in the form we know today.
Advanced topics
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 whose work on new algebra was an important step towards modern algebra, due to i ...
) are the relations
:
between the roots of a quadratic polynomial and its coefficients. They result from comparing term by the relation
:
with the equation
:
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
:
The -coordinate can be obtained by substituting the above result into the given quadratic equation, giving
:
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 perfe ...
)
:
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:
:
The second Vieta's formula then provides:
:
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large ), which causes round-off error in a numerical evaluation. The figure shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient 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
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
).
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 using formulas from trigonometry. For the 25 years preceding 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 calculations.
It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities ...
. Consider the following alternate form of the quadratic equation,
''
where the sign of the ± symbol is chosen so that and may both be positive. By substituting
''
and then multiplying through by , we obtain
''
Introducing functions of and rearranging, we obtain
''
''
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:
:::
#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.
#
#
#
#
#
# (rounded to six significant figures)
::
Solution for complex roots in polar coordinates
If the quadratic equation with real coefficients has two complex roots—the case where 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
:
where and
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 construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
s of regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s.
Generalization of quadratic equation
The formula and its derivation remain correct if the coefficients , and are complex numbers, 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
:
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 fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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'' (a ...
, does not hold. Consider the monic quadratic polynomial
:
over a field of characteristic . If , then the solution reduces to extracting a square root, so the solution is
:
and there is only one root since
:
In summary,
:
See quadratic residue for more information about extracting square roots in finite fields.
In the case that , 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 is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
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
:
and
:
For example, let denote a multiplicative generator of the group of units of , the Galois field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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
In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for ex ...
.
See also
* Solving quadratic equations with continued fractions
* Linear equation
* Cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
* 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, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
* 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 single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
References
External links
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*
101 uses of a quadratic equation
{{DEFAULTSORT:Quadratic Equation
Elementary algebra
Equations