what is another word for the solution of a quadratic equation
roots
339122
equation. In particular, it is a second-degree polynomial equation, since the greatest power is two. Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as "factorising", by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called "roots". These two solutions may or may not be distinct, and they may or may not be real. It may be possible to express
Quadratic equation
339136
lead to loss of up to half of correct significant figures in the roots. The golden ratio is found as the positive solution of the quadratic equation formula_26 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.
Quadratic equation
9310039
converges. This general solution of monic quadratic equations with complex coefficients is usually not very useful for obtaining rational approximations to the roots, because the criteria are circular (that is, the relative magnitudes of the two roots must be known before we can conclude that the fraction converges, in most cases). But this solution does find useful applications in the further analysis of the convergence problem for continued fractions with complex elements. Solving quadratic equations with continued fractions In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is where "a" ≠ 0. The
Solving quadratic equations with continued fractions
1825143
. The combination of these two equations leads to a quadratic equation (since they are the sum and the product of and ). Thus and are the two roots of the quadratic equation . Cardano assumed that . He suggested that his readers consult another of his books, "De Regula Aliza", which was published only in 1570, for the case in which . Solving this quadratic equation and using the fact that and may be exchanged, we find Therefore, is equal to: In his book "L'Algebra", published in 1572, Rafael Bombelli explained that what was done above still works, with
Cubic function
4729748
Algebraic solution An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots). The most well-known example is the solution introduced in secondary school, of the quadratic equation (where "a" ≠ 0). There exist more complicated algebraic solutions for the general cubic equation and quartic equation. The Abel–Ruffini theorem states that the general quintic equation
Algebraic solution
339123
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 ,
Quadratic equation
1347231
is no "general solution in radicals" that applies to "all" equations of a given degree greater than . The solution of any second-degree polynomial equation can be expressed in terms of its coefficients, using only addition, subtraction, multiplication, division, and square roots, in the familiar quadratic formula: the roots of the equation (with ) are Analogous formulas for third-degree equations and fourth-degree equations (using square roots and cube roots) have been known since the 16th century. What the Abel–Ruffini theorem says is that there is no similar formula for general equations of fifth degree or higher. In principle, it could
Abel–Ruffini theorem
4729750
logarithmic function, and the trigonometric functions and their inverses. Algebraic solution An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots). The most well-known example is the solution introduced in secondary school, of the quadratic equation (where "a" ≠ 0). There exist more complicated algebraic solutions for the general cubic equation and quartic equation.
Algebraic solution
339140
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 [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the
Quadratic equation
339133
have become common in schools and graphical methods have started to appear in textbooks, but they are generally not highly emphasized. Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a graph of , the ability to scale the graph appropriately to the dimensions of the graphing surface, and the recognition that when , is a solution to the equation. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function. Since an arbitrary function may cross the -axis
Quadratic equation
339155
extension field. In a field of characteristic , the quadratic formula, which relies on being a unit, 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,
Quadratic equation
339139
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", 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
Quadratic equation
2609074
equation, , with , and coefficients in the finite field formula_18 has either 0, 1 or 2 roots in GF("q") (and two roots, counted with multiplicity, in the quadratic extension GF("q")). If the characteristic of GF("q") is odd, the discriminant, indicates the number of roots in GF("q") and the classical quadratic formula gives the roots. However, when GF("q") has even characteristic (i.e., for some positive integer "h"), these formulas are no longer applicable. Consider the quadratic equation with coefficients in the finite field GF(2). If "b" = 0 then this equation has the unique solution formula_19 in GF("q"). If then
Field trace
339156
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 and 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
Quadratic equation
339150
[4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] 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. If the quadratic equation formula_52 with real coefficients has two complex roots—the case where formula_53 requiring "a" and "c" to have the same sign as each other—then the solutions for the roots can be expressed in
Quadratic equation
696579
BC to 1650 BC), contains the solution to a two-term quadratic equation. The Greek mathematician Euclid (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his "Elements", an influential mathematical treatise. Rules for quadratic equations appear in the Chinese "The Nine Chapters on the Mathematical Art" circa 200 BC. In his work "Arithmetica", the Greek mathematician Diophantus (circa 250 BC) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. His solution gives only one root, even when both roots are positive. The Indian mathematician Brahmagupta (597–668 AD) explicitly described
Quadratic formula
9310030
alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions. Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation and manipulate it directly. Subtracting one from both sides we obtain This is easily factored into from which we obtain and finally Now comes the crucial step. We substitute this expression for "x" back into itself, recursively, to obtain But now we
Solving quadratic equations with continued fractions
7011201
sexagesimal representation of : As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on pre-calculated tables. To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form where here "b" and "c" were not necessarily integers, but "c" was always positive. They knew that a solution to this form of equation is and they found square roots efficiently using division and averaging. They always used the positive root because this made sense when solving "real" problems. Problems of this type included
Babylonian mathematics
17312751
Carlyle circle for the solution of the quadratic equation "x" + "x" − 2 = 0. According to Howard Eves (1911–2004) the mathematician John Leslie (1766-1832) described the geometric construction of roots of a quadratic equation with a circle in his book "Elements of Geometry" and noted that this idea was provided by his former student Thomas Carlyle (1795–1881). However while the description in Leslie's book contains an analogous circle construction, it was presented solely in elementary geometric terms without the notion of a Cartesian coordinate system or a quadratic function and its roots: In 1867 the Austrian engineer Eduard
Carlyle circle
696583
write the quadratic formula is: which may be simplified to: This version of the formula is convenient when complex roots are acceptable. Then the expression outside the square root will be the real part and the square root expression will be the imaginary part. The expression inside the square root is a discriminant. 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. These result in slightly different forms for the solution, but are otherwise equivalent. A lesser known quadratic formula, as used
Quadratic formula
7295309
its own terminology to describe parts of an expression: <br> 1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, formula_5 - variables The roots of a polynomial expression of degree "n", or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if "n" < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if "n" formula_6 5.
Algebraic expression
339157
two roots and , where is a root of in . This is a special case of Artin–Schreier theory. Quadratic equation In algebra, a quadratic equation (from the Latin for "square") is any equation having the form where represents an unknown, and , , and represent known numbers, with . If , then the equation is linear, not quadratic. The numbers , , and are the "coefficients" of the equation and may be distinguished by calling them, respectively, the "quadratic coefficient", the "linear coefficient" and the "constant" or "free term". Because the quadratic equation involves only one unknown, it is
Quadratic equation
1825149
from the depressed cubic (), , we make the substitution , known as Vieta's substitution. This results in the equation Multiplying by , it becomes a sextic equation in , which is in fact a quadratic equation in : The quadratic formula allows equation () to be solved for . If , and are the three cube roots of one of the solutions in , then the roots of the original depressed cubic are , , and . Another way of expressing the roots is to take ; then the roots of the original depressed cubic are , , and,
Cubic function
339128
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 is one. This is done by dividing both sides by "a", which is always possible since "a" is non-zero. This produces the "reduced quadratic equation": where "p" = "b"/"a" and "q" = "c"/"a". This monic 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: In the quadratic formula,
Quadratic equation
5848105
which nested radicals can be denested. Earlier algorithms worked in some cases but not others. In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example, and Nested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simplified form without a quadratic term, as whose general solution for one of the roots is In the case in which the cubic has only one real root, the real root is given by this expression with the radicands of the cube roots being real and with
Nested radical
5005092
and σ = 1, equivalent to the following substitutitons: The density of this restricted one-parameter family is a standard Student's "t": If the quadratic function (2) has a non-negative discriminant (formula_35), it has real roots "a" and "a" (not necessarily distinct): In the presence of real roots the quadratic function (2) can be written as and the solution to the differential equation is therefore Pearson (1895, p. 362) called this the "logarithmic case", because the integral involves only the logarithm function, and not the arctan function as in the previous case. Using the substitution we obtain the following solution to
Pearson distribution
339144
By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published "La Géométrie" containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton. Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form: and These results follow immediately from the relation: which can
Quadratic equation
5005087
main cases, determined by the sign of the discriminant (and hence the number of real roots) of the quadratic function If the discriminant of the quadratic function (2) is negative (formula_9), it has no real roots. Then define Observe that α is a well-defined real number and α ≠ 0, because by assumption formula_12 and therefore "b" ≠ 0. Applying these substitutions, the quadratic function (2) is transformed into The absence of real roots is obvious from this formulation, because α is necessarily positive. We now express the solution to the differential equation (1) as a function of "y": Pearson
Pearson distribution
5090257
pairs satisfying this constraint. Another ancient problem involving quadratic forms asks us to solve Pell's equation. For instance, we may seek integers "x" and "y" so that formula_72. Changing signs of "x" and "y" in a solution gives another solution, so it is enough to seek just solutions in positive integers. One solution is formula_73, that is, there is an equality formula_74. If formula_60 is any solution to formula_76, then formula_77 is another such pair. For instance, from the pair formula_78, we compute and we can check that this satisfies formula_80. Iterating this process, we find further pairs formula_60 with
Binary quadratic form
5090250
nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to quadratic and more general number fields, but advances specific to binary quadratic forms still occur on occasion. Pierre Fermat stated that if p is an odd prime then the equation formula_2 has a solution iff formula_3, and he made similar statement about the equations formula_4, formula_5, formula_6 and formula_7 formula_8 and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems Another instance of quadratic forms is Pell's equation formula_9 Binary quadratic forms
Binary quadratic form
6755061
a variety of solution algorithms based on spherical trigonometry and the Gauss-Newton NLLS iterative method. For Cartesian coordinates, when four TOAs are available, Bancroft's or another closed-form algorithm are one option, even if the stations are moving. When the four stations are stationary, extension of Fang's algorithm (based on DTOAs) to three dimensions is a second option. A third option, and likely the most utilized in practice, is the Gauss-Newton NLLS method. Most closed-form algorithms reduce finding the user vehicle location from measured TOAs to the solution of a quadratic equation. One solution of the quadratic equation yields the user's
Multilateration
12287254
equation formula_8 can be any numbers whatsoever (except that formula_9 cannot be formula_10), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity formula_2 which satisfy the equation. That is to say, to find all the solutions of the equation. Historically, and in current teaching, the study of algebra starts with the solving of equations such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered.
Algebra
339143
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.
Quadratic equation
696577
Quadratic formula In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way. The general quadratic equation is Here represents an unknown, while , , and are constants with not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation by inserting the former into the latter. With the above parameterization, the quadratic formula is: Each of the solutions given by
Quadratic formula
696600
physical quantities has been set up correctly, prior to solving it. Quadratic formula In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way. The general quadratic equation is Here represents an unknown, while , , and are constants with not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation by inserting the former into the latter. With the above
Quadratic formula
2608238
to infinity. Solving the quadratic equation 1/"g" = 0 yields the solution: Another singularity occurs where the purely temporal component "g" of the metric changes sign from positive to negative. Again solving a quadratic equation "g"=0 yields the solution: Due to the cosθ term in the square root, this outer surface resembles a flattened sphere that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; the space between these two surfaces is called the ergosphere. Within this volume, the purely temporal component "g" is negative, i.e., acts like a purely
Kerr metric
122446
each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation has solutions Since formula_120 is not any real number, both of these solutions for "x" are complex numbers. An exponential equation is one which has the form formula_121 for formula_122, which has solution when formula_124. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if then, by subtracting 1 from both sides
Elementary algebra
2060461
Quadratic irrational number 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 irreducible over the set of rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be
Quadratic irrational number
10103378
by analogy with the Nielsen transformation for groups. For example, starting with an equation "xα" = "yβ" where "x" and "y" are the unknowns, we can transform it (assuming "|x| ≥ |y|", so there exists "t" such that "x"="yt") to "ytα" = "yβ", thus to "tα" = "β". This approach results in a graph of substitutions generated by repeatedly applying Levi's lemma. If each unknown appears at most twice, then word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has
Levi's lemma
1859995
even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as formula_19 where "f" is the multi-valued implicit function. While explicit solutions can be found for equations that are quadratic, cubic, and quartic in "y", the same is not in general true for quintic and higher degree equations, such as Nevertheless, one can still refer to the implicit solution "y" = "f"("x") involving the multi-valued implicit function "f". Not every equation "R"("x", "y") = 0 implies a graph of a single-valued function, the circle equation being one prominent example. Another
Implicit function
18458938
integrate and fire neuron is defined by the autonomous differential equation, where formula_2 is a real positive constant. Note that a solution to this differential equation is the tangent function, which blows up in finite time. Thus a "spike" is said to have occurred when the solution reaches positive infinity, and the solution is reset to negative infinity. When implementing this model in computers, a threshold crossing value (formula_3) and a reset value (formula_4) is assigned, so that when the solution rises above the threshold, formula_5, the solution is immediately reset to formula_4 Quadratic integrate and fire The quadratic integrate
Quadratic integrate and fire
14886019
is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case. A prototypical question, posed at a global level, is whether some polynomial equation has a solution in "F". If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of "F", which is often easier, since analytic methods
Algebraic number field
1857682
considered what is now called Pell's equation, , and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange. In 1801 Gauss published "Disquisitiones Arithmeticae," a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated. Any "n"×"n" real symmetric matrix "A" determines a quadratic form "q" in "n" variables by the formula Conversely, given a quadratic form in "n"
Quadratic form
339141
[coefficient of the] middle term, being divided by twice the [coefficient of the] square 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 equations (originally of type ). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while
Quadratic equation
15533800
the quadratic equation for , and inserting , into the last equation, one gets This is the equation of the hyperbola above. Its branches bear the two isoptics of the parabola for the two angles and . In the case of an ellipse one can adopt the idea for the orthoptic for the quadratic equation Now, as in the case of a parabola, the quadratic equation has to be solved and the two solutions , must be inserted into the equation Rearranging shows that the isoptics are parts of the degree-4 curve: The solution for the case of a hyperbola
Orthoptic (geometry)
406580
and later versions of TeX, notably pdfTeX, XeTeX and LuaTeX, all support output directly to PDF. TeX provides a different text syntax specifically for mathematical formulas. For example, the quadratic formula (which is the solution of the quadratic equation) appears as: The formula is printed in a way a person would write by hand, or typeset the equation. In a document, entering "mathematics mode" is done by starting with a $ symbol, then entering a formula in TeX syntax, and closing again with another of the same symbol. Knuth explained in jest that he chose the dollar sign to indicate
TeX
2300783
the equation "x" = 9, one may give the solution as "x" = ±3. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations "x" = +3 or "x" = −3. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula describing the two solutions to the quadratic equation "ax" + "bx" + "c" = 0. Similarly, the trigonometric identity can be interpreted as a shorthand for two equations: one with
Plus-minus sign
9310036
the continued fraction solution to the general monic quadratic equation with real coefficients given by converges or not depending on both the coefficient "b" and the value of the discriminant, "b" − 4"c". If "b" = 0 the general continued fraction solution is totally divergent; the convergents alternate between 0 and formula_17. If "b" ≠ 0 we distinguish three cases. When the monic quadratic equation with real coefficients is of the form "x" = "c", the general solution described above is useless because division by zero is not well defined. As long as "c" is positive, though, it is always
Solving quadratic equations with continued fractions
339137
Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation. The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, 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
Quadratic equation
331348
the concentration of the acid, so [A] = [H]. After some further algebraic manipulation an equation in the hydrogen ion concentration may be obtained. Solution of this quadratic equation gives the hydrogen ion concentration and hence p[H] or, more loosely, pH. This procedure is illustrated in an ICE table which can also be used to calculate the pH when some additional (strong) acid or alkaline has been added to the system, that is, when C ≠ C. For example, what is the pH of a 0.01M solution of benzoic acid, pK = 4.19? For alkaline solutions an additional term is
PH
696581
by Simon Stevin in 1594. In 1637 René Descartes published "La Géométrie" containing special cases of the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton. The quadratic formula can be derived with a simple application of technique of completing the square. The two derivations are as follows: Divide the quadratic equation by , which is allowed because is non-zero: Subtract from both sides of the equation, yielding: The quadratic equation is now in a form to which the method of
Quadratic formula
3279340
is given by and the cost accumulates at rate formula_36, the HJB equation is given by with optimal action given by Assuming a quadratic form for the value function, we obtain the usual Riccati equation for the Hessian of the value function as is usual for Linear-quadratic-Gaussian control. Hamilton–Jacobi–Bellman equation The Hamilton–Jacobi–Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the value function which gives the minimum cost for a given dynamical system with an associated cost function. When solved locally, the HJB is a necessary condition,
Hamilton–Jacobi–Bellman equation
3853199
In both cases, the solution formula_29 may be found by setting formula_30. Hence, in the first case, and in the second case, Given we substitute the simple solution "x": For "x" to be a solution, either "x" = 0, which gives the trivial solution, or the coefficient of "x" is zero. Solving the quadratic equation, we get "m" = 1, 3. The general solution is therefore There is a difference equation analogue to the Cauchy–Euler equation. For a fixed "m" > 0, define the sequence "ƒ"("n") as Applying the difference operator to formula_37, we find that If we do this
Cauchy–Euler equation
339126
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
Quadratic equation
5090272
definition is a special case of the genus of a quadratic form in n variables. This states that forms are in the same genus if they are locally equivalent at all rational primes (including the Archimedean place). There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian
Binary quadratic form
122445
precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example, has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as: For this equation, −1 is a root of multiplicity 2. This means −1 appears two times, since the equation can be rewritten in factored form as All quadratic equations have exactly two solutions in complex numbers (but they may be equal to
Elementary algebra
1296053
language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all formula_17, i.e., that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring formula_8 is a UFD. Ernst Kummer had shown three years before that this was not the case already for formula_19 (the full, finite list of values for which formula_8 is a UFD is now
Dedekind domain
113646
to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 'absurd' because it would lead to a negative value for . One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. Diophantus made important advances in mathematical notation, becoming the first person known to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an
Diophantus
11869718
their own evolution) and/or the control response matrix of the state equation, but still with a linear state equation and quadratic objective function, a Riccati equation can still be obtained for iterating backward to each period's solution even though certainty equivalence does not apply. The discrete-time case of a non-quadratic loss function but only additive disturbances can also be handled, albeit with more complications. A typical specification of the discrete-time stochastic linear quadratic control problem is to minimize where E is the expected value operator conditional on "y", superscript "T" indicates a matrix transpose, and "S" is the time horizon,
Stochastic control
6864444
With the backward Euler method one finds the implicit equation for formula_18 (compare this with formula (3) where formula_18 was given explicitly rather than as an unknown in an equation). This is a quadratic equation, having one negative and one positive root. The positive root is picked because in the original equation the initial condition is positive, and then formula_23 at the next time step is given by In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no analytical solution exists. Then one uses
Explicit and implicit methods
11771861
controlled variable right now, knowing that one will also behave optimally at all times in the future. The optimal current values of the problem's control variables at any time can be found using the solution of the Riccati equation and the current observations on evolving state variables. With multiple state variables and multiple control variables, the Riccati equation will be a matrix equation. The algebraic Riccati equation determines the solution of the infinite-horizon time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These are two of the most fundamental problems in
Algebraic Riccati equation
696595
equation The value of in the extreme point is then added to both sides of the equation This gives the quadratic formula. This way one avoids the technique of completing the square and much more complicated math is not needed. Note this solution is very similar to solving deriving the formula by substitution. Consider the equation where formula_56 is a complex number and where formula_27, formula_58, and formula_59 are real numbers. Then This splits into two equations, the real part: and the imaginary part: Assuming that formula_63 then divide the second equation by formula_64: and solve for formula_66: Substitute this
Quadratic formula
113619
produce all other solutions in the following way. Let be a homogeneous Diophantine equation, where formula_7 is a quadratic form (that is, a homogeneous polynomial of degree 2), with integer coefficients. The "trivial solution" is the solution where all formula_8 are zero. If formula_9 is a non-trivial integer solution of this equation, then formula_10 are the homogeneous coordinates of a rational point of the hypersurface defined by . Conversely, if formula_11 are homogeneous coordinates of a rational point of this hypersurface, where formula_12 are integers, then formula_13 is an integer solution of the Diophantine equation. Moreover, the integer solutions that
Diophantine equation
14723905
This has slope (in output space) formula_21 and analogously for firm Y which (we assume) has the same conjecture. To see what consistency means, consider the simple Cournot conjecture formula_8 with constant marginal cost "a=0". In this case the slope of the reaction functions is −1/2 which is "inconsistent" with the conjecture. The Bresnehan consistency condition is that the conjectured slope formula_17 equals the actual slope formula_24 which means that formula_25 This is a quadratic equation which gives us the unique consistent conjecture formula_26 This is the positive root of the quadratic: the negative solution would be a conjecture more
Conjectural variation
2060471
which this is true, hence the smallest number for which "q" is also an integer. Then: is also an integer. But 0 < ( − "n") < 1 so ( − "n")"q" < "q". Hence ( − "n")"q" is an integer smaller than "q". This is a contradiction since "q" was defined to be the smallest number with this property; hence cannot be rational. Quadratic irrational number 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
Quadratic irrational number
7924404
the Kalman gain equals The matrix formula_33 is called the feedback gain matrix. This matrix is determined by the matrices formula_34 and formula_35 through the following associated matrix Riccati differential equation: Given the solution formula_38 the feedback gain equals Observe the similarity of the two matrix Riccati differential equations, the first one running forward in time, the second one running backward in time. This similarity is called duality. The first matrix Riccati differential equation solves the linear–quadratic estimation problem (LQE). The second matrix Riccati differential equation solves the linear–quadratic regulator problem (LQR). These problems are dual and together they solve
Linear–quadratic–Gaussian control
335422
these records, see ("x"), and ("y")). The following is a list of the smallest solution to formula_46 with "n" ≤ 128. For square "n", there is no solution except (1, 0). (sequence ("x") and ("y") in OEIS, or ("x") and ("y") (for nonsquare "n")) Pell's equation has connections to several other important subjects in mathematics. Pell's equation is closely related to the theory of algebraic numbers, as the formula is the norm for the ring formula_48 and for the closely related quadratic field formula_49. Thus, a pair of integers formula_50 solves Pell's equation if and only if formula_51 is a
Pell's equation
335420
equation is more efficient than the continued fraction method, though it still takes more than polynomial time. Under the assumption of the generalized Riemann hypothesis, it can be shown to take time where "N" = log "n" is the input size, similarly to the quadratic sieve . showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time. Hallgren's algorithm, which can be interpreted as an algorithm for finding the group of units of a real quadratic number field, was extended to more general fields by . As an
Pell's equation
5402163
Meyer's theorem In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form "Q" in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation has a non-zero real solution, then it has a non-zero rational solution (the converse is obvious). By clearing the denominators, an integral solution "x" may also be found. Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement: Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms
Meyer's theorem
340414
Quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: .</math> This law allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form formula_1 for "p" an odd prime; that is, to determine the "perfect squares" mod "p". However, this is a non-constructive result: it gives no help at all for "finding"
Quadratic reciprocity
7924026
since formula_32 is driven to its final state formula_33 by formula_34. For a discrete-time linear system described by with a performance index defined as the optimal control sequence minimizing the performance index is given by where and formula_7 is the unique positive definite solution to the discrete time algebraic Riccati equation (DARE) This can be also written as with Note that one way to solve the algebraic Riccati equation is by iterating the dynamic Riccati equation of the finite-horizon case until it converges. Linear–quadratic regulator The theory of optimal control is concerned with operating a dynamic system at minimum cost.
Linear–quadratic regulator
20820313
the "method undetermined coefficients". The algorithm returns the general polynomial solution of a recurrence equation. Applying the formula for the degree bound on the recurrence equationformula_37over formula_38 yields formula_39. Hence one can use an ansatz with a quadratic polynomial formula_40 with formula_41. Plugging this ansatz into the original recurrence equation leads toformula_42This is equivalent to the following system of linear equationsformula_43with the solution formula_44. Therefore the only polynomial solution is formula_45. Polynomial solutions of P-recursive equations In mathematics a P-recursive equation can be solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm
Polynomial solutions of P-recursive equations
331343
strong acids and bases no calculations are necessary except in extreme situations. The pH of a solution containing a weak acid requires the solution of a quadratic equation. The pH of a solution containing a weak base may require the solution of a cubic equation. The general case requires the solution of a set of non-linear simultaneous equations. A complicating factor is that water itself is a weak acid and a weak base (see amphoterism). It dissociates according to the equilibrium with a dissociation constant, defined as where [H] stands for the concentration of the aqueous hydronium ion and [OH]
PH
7208823
data, for example where the data is obtained from a point source traversing a curve, typically a helix. More generally an "ultrahyperbolic" partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form where formula_10, such that the quadratic form can be reduced by a linear change of variables to the form It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of "u" can be extended to a solution. John's equation John's
John's equation
696586
literature. These derivations may be simpler than the standard completing the square method and represent interesting applications of other algebraic techniques or may offer insight into other areas of mathematics. Another technique is solution by substitution. In this technique, we substitute into the quadratic to get: Expanding the result and then collecting the powers of produces: We have not yet imposed a second condition on and , so we now choose so that the middle term vanishes. That is, or . Subtracting the constant term from both sides of the equation (to move it to the right hand side) and
Quadratic formula
20093748
"X" is a continuous function from the interval ["a", "b"] to ℝ with finite "p"-variation with "p" less than 2, then Young integration is enough to establish the solution of the equation formula_22 driven by the path "X". The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of "p"-variation. "p"-variation should be contrasted with the quadratic variation which is used in stochastic analysis, where it takes one stochastic process to another. Quadratic variation is defined as a limit as the partition gets finer, whereas "p"-variation is a supremum over
P-variation
339151
polar form as where formula_55 and formula_56 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
Quadratic equation
591348
The field of Gaussian rationals is the field of fractions of the ring of Gaussian integers. It consists of the complex numbers whose real and imaginary part are both rational. The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals. This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation with and integers. In fact is solution of the equation and this equation has integer coefficients if and only if and are both integers. As for
Gaussian integer
5030076
Hilbert's eleventh problem Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900. A furthering of the theory of quadratic forms, he stated the problem as follows: As stated by Kaplansky, "The 11th Problem is simply this: classify quadratic forms over algebraic number fields." This is exactly what Minkowski did for quadratic form with fractional coefficients. A quadratic form (not quadratic equation) is any polynomial in which each term has variables appearing exactly twice. The general form of such an equation is ax^(2)+bxy+cy^(2). (All coefficients
Hilbert's eleventh problem
339121
Quadratic equation In algebra, a quadratic equation (from the Latin for "square") is any equation having the form where represents an unknown, and , , and represent known numbers, with . If , then the equation is linear, not quadratic. The numbers , , and are the "coefficients" of the equation and may be distinguished by calling them, respectively, the "quadratic coefficient", the "linear coefficient" and the "constant" or "free term". Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of that are non-negative integers, and therefore it is a polynomial
Quadratic equation
5394070
(k, k) are the basic eigenvectors. The constants "c" and "c" account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system. The above determinant leads to the characteristic polynomial: which is just a quadratic equation of the form: where; ("tr" denotes trace) and The explicit solution of the eigenvalues are then given by the quadratic formula: where The eigenvectors and nodes determine the profile of the phase paths, providing a pictorial interpretation of the solution to the dynamical system, as shown next. The phase plane is then first set-up by drawing
Phase plane
6432328
Feigenbaum function In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum: The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function "g" and a parameter "λ" by the relation with the initial conditions For a particular form of solution with a quadratic dependence of the solution near x=0, the inverse "1/λ=2.5029..." is
Feigenbaum function
2057239
into the Riccati equation yields the linear equation A set of solutions to the Riccati equation is then given by where z is the general solution to the aforementioned linear equation. Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form where formula_2 and formula_3. If formula_4 the equation reduces to a Bernoulli equation, while if formula_5 the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754). More
Riccati equation
17789978
of the two given equations one gets the line equation: The intersection of two disks (the interiors of the two circles) forms a shape called a lens. The problem of intersection of an ellipse/hyperbola/parabola with another conic section leads to a system of quadratic equations, which can be solved in special cases easily by elimination of one coordinate. Special properties of conic sections may be used to obtain a solution. In general the intersection points can be determined by solving the equation by a Newton iteration. If a) both conics are given implicitly (by an equation) a 2-dimensional Newton iteration
Intersection (Euclidean geometry)
627619
real axis). Being a quadratic polynomial with no multiple root, the defining equation has "two" distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. More precisely, once a solution of the equation has been fixed, the value , which is distinct from , is also a solution. Since the equation is the only definition of , it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity results as long as one or other of the solutions is chosen and labelled as "", with the other one then
Imaginary unit
5005086
density function of the normal distribution (which gives a linear function) and, secondly, from a recurrence relation for values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure). In Equation (1), the parameter "a" determines a stationary point, and hence under some conditions a mode of the distribution, since follows directly from the differential equation. Since we are confronted with a first order linear differential equation with variable coefficients, its solution is straightforward: The integral in this solution simplifies considerably when certain special cases of the integrand are considered. Pearson (1895, p. 367) distinguished two
Pearson distribution
6335662
formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears that Gauss already knew this formula in 1801. This exposition follows Davenport. Let "d" be a fundamental discriminant, and write "h(d)" for the number of equivalence classes of quadratic forms with discriminant "d". Let formula_2 be the Kronecker symbol. Then formula_3 is a Dirichlet character. Write formula_4 for the Dirichlet L-series based on formula_3. For "d > 0", let "t > 0", "u > 0" be the solution to the Pell equation formula_6 for which "u" is
Class number formula
14291509
formula_78 such that formula_79 Since formula_80, the equation formula_81 which leads to solving the equation formula_82. This has solution formula_83. Indeed, formula_84. Pocklington's algorithm Pocklington's algorithm is a technique for solving a congruence of the form where "x" and "a" are integers and "a" is a quadratic residue. The algorithm is one of the first efficient methods to solve such a congruence. It was described by H.C. Pocklington in 1917. Inputs: Outputs: Pocklington separates 3 different cases for "p": The first case, if formula_7, with formula_8, the solution is formula_9. The second case, if formula_10, with formula_8 and The third
Pocklington's algorithm
12506800
very acute, i.e., if is small relative to and or is small compared to 1. It is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for "a" in the quadratic equation . This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if , only one positive solution if , and no solution if or . These different cases are also explained by the side-side-angle congruence ambiguity.
Law of cosines
2609075
the substitution converts the quadratic equation to the form: This equation has two solutions in GF("q") if and only if the absolute trace formula_21 In this case, if is one of the solutions, then is the other. Let "k" be any element of GF("q") with formula_22 Then a solution to the equation is given by: When "h" = 2"m" + 1, a solution is given by the simpler expression: When "L"/"K" is separable, the trace provides a duality theory via the trace form: the map from to "K" sending to Tr("xy") is a nondegenerate, symmetric, bilinear form called the trace
Field trace
696580
the quadratic formula in his treatise "Brāhmasphuṭasiddhānta" published in 628 AD, but written in words instead of symbols. His solution of the quadratic equation was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." This is equivalent to: The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically. The quadratic formula covering all cases was first obtained
Quadratic formula
335419
in the form using much smaller integers "a", "b", and "c". For instance, Archimedes' cattle problem is equivalent to the Pell equation formula_37, the fundamental solution of which has 206545 digits if written out explicitly. However, the solution is also equal to where and formula_40 and formula_41 only have 45 and 41 decimal digits, respectively. Methods related to the quadratic sieve approach for integer factorization may be used to collect relations between prime numbers in the number field generated by , and to combine these relations to find a product representation of this type. The resulting algorithm for solving Pell's
Pell's equation
24170
are often reversed in physics. In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation "y" = "x" corresponds to the set of all the points on the plane whose "x"-coordinate and "y"-coordinate are equal. These points form a line, and "y" = "x" is said to be the equation for this line. In general, linear equations involving "x" and "y" specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures. Usually, a single equation corresponds to a
Analytic geometry
9732928
equation is solvable. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. Also, no general method may be abstracted from all Diophantus' solutions. In "Arithmetica", Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as "syncopated" algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So, for example, what we
History of algebra
1857697
the group of isometries of into itself. If a quadratic space has a product so that "A" is an algebra over a field, and satisfies Every quadratic form "q" in "n" variables over a field of characteristic not equal to 2 is equivalent to a diagonal form Such a diagonal form is often denoted by Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms. Using Cartesian coordinates in three dimensions, let formula_30, and let formula_31 be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation
Quadratic form
627628
The cosine of is a real number: And the sine of is purely imaginary: Imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is . Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , which in turn provides at least one root for
Imaginary unit
2104279
"q"; actually, "q" is "p" divided with its leading coefficient. In this manner, then, any non-trivial polynomial equation "p"("x") = 0 may be replaced by an equivalent monic equation "q"("x") = 0. E.g., the general real second degree equation may be replaced by by putting "p" = "b"/"a" and "q" = "c"/"a". Thus, the equation is equivalent to the monic equation The general quadratic solution formula is then the slightly more simplified form of: On the other hand, if the coefficient ring is not a field, there are more essential differences. E.g., a monic polynomial equation with integer coefficients cannot
Monic polynomial
627615
Imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is . Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , which in turn provides at least one root for every nonconstant polynomial . (See Algebraic closure and Fundamental theorem of algebra.) The term
Imaginary unit
14613011
matrix. Then denoting the 2"n"×1 stacked vector of current and once-lagged variables as formula_31 and the 2"n"×2"n" block matrix as "L", we have as before the solution Also as before, this stacked equation and thus the original second-order equation are stable if and only if all eigenvalues of the matrix "L" are smaller than unity in absolute value. In linear-quadratic-Gaussian control, there arises a nonlinear matrix equation for the reverse evolution of a current-and-future-cost "matrix", denoted below as "H". This equation is called a discrete dynamic Riccati equation, and it arises when a variable vector evolving according to a linear
Matrix difference equation
118915
are denoted by letters at the beginning, "a", "b", "c", "d", … . For example, the general quadratic equation is usually written "ax" + "bx" + "c" = 0. The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters is called solving the equation. Such expressions of the solutions in terms of the parameters are also called "solutions". A system of equations is a set of "simultaneous equations", usually in several unknowns, for which the common solutions are sought. Thus a "solution to the system" is a set of values for
Equation
339142
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 that succeeded him accepted negative solutions, as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form
Quadratic equation