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62-450: As of 2003, the town had an estimated population of 3,465 and a density of 18.65 persons per km. The total area was 185.79 km. On March 20, 2006, Ashio, along with the city of Imaichi , the town of Fujihara , and the village of Kuriyama (both from Shioya District ), was merged into the expanded city of Nikkō . The Ashio Copper Mine was located in Ashio. This copper mine caused

124-446: A {\displaystyle \scriptstyle x={\tfrac {-b}{2a}}} , and the y -coordinate of the vertex may be found by substituting this x -value into the function. The y -intercept is located at the point (0, c ) . The solutions of the quadratic equation ax + bx + c = 0 correspond to the roots of the function f ( x ) = ax + bx + c , since they are the values of x for which f ( x ) = 0 . If

186-443: A ( x − − b + b 2 − 4 a c 2 a ) ( x − − b − b 2 − 4 a c 2 a ) . {\displaystyle ax^{2}+bx+c=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right).} In

248-454: A t 2 {\textstyle x=x_{0}+v_{0}t+{\frac {1}{2}}at^{2}} . In chemistry , the pH of a solution of weak acid can be calculated from the negative base-10 logarithm of the positive root of a quadratic equation in terms of the acidity constant and the analytical concentration of the acid. Babylonian mathematicians , as early as 2000 BC (displayed on Old Babylonian clay tablets ) could solve problems relating

310-411: A x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} may be deduced from the graph of the quadratic function f ( x ) = a x 2 + b x + c , {\displaystyle f(x)=ax^{2}+bx+c,} which is a parabola . If the parabola intersects the x -axis in two points, there are two real roots , which are

372-449: A ) / R {\displaystyle (c/a)/R} where R is the root that is bigger in magnitude. This is equivalent to using the formula x = − 2 c b ± b 2 − 4 a c {\displaystyle x={\frac {-2c}{b\pm {\sqrt {b^{2}-4ac}}}}} using the plus sign if b > 0 {\displaystyle b>0} and

434-406: A c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} expresses the solutions in terms of a , b , and c . Completing the square is one of several ways for deriving the formula. 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

496-492: A c 4 a 2 . {\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.} Taking the square root of both sides, and isolating x , gives: x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} Some sources, particularly older ones, use alternative parameterizations of

558-458: A and q = c / a . 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 x = − p 2 ± ( p 2 ) 2 − q . {\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}\,.} In

620-458: A , b , and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x - coordinates of the points where the graph touches the x -axis. If the discriminant is positive, the graph touches the x -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x -axis. The term x − r {\displaystyle x-r}

682-562: A census to quantify the size of a resident population within a given jurisdiction. The term is also applied to non-human animals , microorganisms , and plants , and has specific uses within such fields as ecology and genetics . The word population is derived from the Late Latin populatio (a people, a multitude), which itself is derived from the Latin word populus (a people). In sociology and population geography , population refers to

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744-437: A population is often defined as a set of organisms in which any pair of members can breed together. They can thus routinely exchange gametes in order to have usually fertile progeny, and such a breeding group is also known therefore as a gamodeme. This also implies that all members belong to the same species. If the gamodeme is very large (theoretically, approaching infinity), and all gene alleles are uniformly distributed by

806-435: A ≠ 0 . (If a = 0 and b ≠ 0 then the equation is linear , not quadratic.) The numbers a , b , and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient , the linear coefficient and the constant coefficient or free term . The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of

868-466: A double root is counted for two. A quadratic equation can be factored into an equivalent equation a x 2 + b x + c = a ( x − r ) ( x − s ) = 0 {\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0} where r and s are the solutions for x . The quadratic formula x = − b ± b 2 − 4

930-471: A few programs, most notably the Chinese government's one-child per family policy, have resorted to coercive measures. In the 1970s, tension grew between population control advocates and women's health activists who advanced women's reproductive rights as part of a human rights -based approach. Growing opposition to the narrow population control focus led to a significant change in population control policies in

992-408: A group of human beings with some predefined feature in common, such as location, race , ethnicity , nationality , or religion . In ecology , a population is a group of organisms of the same species which inhabit the same geographical area and are capable of interbreeding . The area of a sexual population is the area where interbreeding is possible between any opposite-sex pair within

1054-497: A major pollution problem in Japan at the beginning of the 20th century. Subsequent environmental problems related to the mine are still evident along the river, in Tochigi , Gunma and Ibaraki Prefectures . In 1907 the Ashio miners rioted. During World War II a POW camp was based here to supply slave labour to the copper mines. The following communities agreed to seek the permission of

1116-415: A point on the parabola with the same y -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 h , and their imaginary part are ± d . That is, the roots are h + i d and h − i d , {\displaystyle h+id\quad {\text{and}}\quad h-id,} or in

1178-400: A quadratic equation can be found by several alternative methods. It may be possible to express a quadratic equation ax + bx + c = 0 as a product ( px + q )( rx + s ) = 0 . 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

1240-438: A quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral . Critical points of a cubic function and inflection points of a quartic function are found by solving

1302-1250: A quadratic equation in standard form, ax + bx + c = 0 We illustrate use of this algorithm by solving 2 x + 4 x − 4 = 0 2 x 2 + 4 x − 4 = 0 {\displaystyle 2x^{2}+4x-4=0}   x 2 + 2 x − 2 = 0 {\displaystyle \ x^{2}+2x-2=0}   x 2 + 2 x = 2 {\displaystyle \ x^{2}+2x=2}   x 2 + 2 x + 1 = 2 + 1 {\displaystyle \ x^{2}+2x+1=2+1} ( x + 1 ) 2 = 3 {\displaystyle \left(x+1\right)^{2}=3}   x + 1 = ± 3 {\displaystyle \ x+1=\pm {\sqrt {3}}}   x = − 1 ± 3 {\displaystyle \ x=-1\pm {\sqrt {3}}} The plus–minus symbol "±" indicates that both x = − 1 + 3 {\textstyle x=-1+{\sqrt {3}}} and x = − 1 − 3 {\textstyle x=-1-{\sqrt {3}}} are solutions of

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1364-579: A quadratic equation. In physics , for motion with constant acceleration a {\displaystyle a} , the displacement or position x {\displaystyle x} of a moving body can be expressed as a quadratic function of time t {\displaystyle t} given the initial position x 0 {\displaystyle x_{0}} and initial velocity v 0 {\displaystyle v_{0}} : x = x 0 + v 0 t + 1 2

1426-401: Is a factor of the polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} if and only if r is a root of the quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.} It follows from the quadratic formula that a x 2 + b x + c =

1488-411: Is called " univariate ". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation . In particular, it is a second-degree polynomial equation, since the greatest power is two. A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots . When there

1550-600: Is not even known to the nearest million, so there is a considerable margin of error in such estimates. Researcher Carl Haub calculated that a total of over 100 billion people have probably been born in the last 2000 years. Population growth increased significantly as the Industrial Revolution gathered pace from 1700 onwards. The last 50 years have seen a yet more rapid increase in the rate of population growth due to medical advances and substantial increases in agricultural productivity, particularly beginning in

1612-611: Is often referred to as the demographic transition . Human population planning is the practice of altering the rate of growth of a human population. Historically, human population control has been implemented with the goal of limiting the rate of population growth. In the period from the 1950s to the 1980s, concerns about global population growth and its effects on poverty, environmental degradation , and political stability led to efforts to reduce population growth rates. While population control can involve measures that improve people's lives by giving them greater control of their reproduction,

1674-509: Is only one distinct root, it can be interpreted as two roots with the same value, called a double root . When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part , and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct. The solutions of

1736-440: Is that it yields one valid root when a = 0 , while the other root contains division by zero, because when a = 0 , 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 0/0 for the other root. On the other hand, when c = 0 , the more common formula yields two correct roots whereas this form yields

1798-703: Is very likely that the world's population will stop growing before the end of the 21st century. Further, there is some likelihood that population will actually decline before 2100. Population has already declined in the last decade or two in Eastern Europe, the Baltics and in the former Commonwealth of Independent States. The population pattern of less-developed regions of the world in recent years has been marked by gradually declining birth rates. These followed an earlier sharp reduction in death rates. This transition from high birth and death rates to low birth and death rates

1860-437: 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

1922-423: The quadratic function 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 solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and

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1984-400: The x -coordinate and the y -coordinate of the vertex of the parabola (that is the point with maximal or minimal y -coordinate. The quadratic function may be rewritten y = a ( x − h ) 2 + k . {\displaystyle y=a(x-h)^{2}+k.} Let d be the distance between the point of y -coordinate 2 k on the axis of the parabola, and

2046-445: The x -coordinates of these two points (also called x -intercept). If the parabola is tangent to the x -axis, there is a double root, which is the x -coordinate of the contact point between the graph and parabola. If the parabola does not intersect the x -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be. Let h and k be respectively

2108-415: The "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0 . 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 x + bx + c = 0 , the sought factorization has

2170-547: The 1960s, made by the Green Revolution . In 2017 the United Nations Population Division projected that the world's population would reach about 9.8 billion in 2050 and 11.2 billion in 2100. In the future, the world's population is expected to peak at some point, after which it will decline due to economic reasons, health concerns, land exhaustion and environmental hazards. According to one report, it

2232-485: The United States Census Bureau, the world population hit 6.5 billion on 24 February 2006. The United Nations Population Fund designated 12 October 1999 as the approximate day on which world population reached 6 billion. This was about 12 years after the world population reached 5 billion in 1987, and six years after the world population reached 5.5 billion in 1993. The population of countries such as Nigeria

2294-477: The area and more probable than cross-breeding with individuals from other areas. In humans , interbreeding is unrestricted by racial differences, as all humans belong to the same species of Homo sapiens. In ecology, the population of a certain species in a certain area can be estimated using the Lincoln index to calculate the total population of an area based on the number of individuals observed. In genetics,

2356-465: 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: x + y = p ,     x y = q , {\displaystyle x+y=p,\ \ xy=q,} which is equivalent to the statement that x and y are

2418-480: The breaking up of a large sexual population (panmictic) into smaller overlapping sexual populations. This failure of panmixia leads to two important changes in overall population structure: (1) the component gamodemes vary (through gamete sampling) in their allele frequencies when compared with each other and with the theoretical panmictic original (this is known as dispersion, and its details can be estimated using expansion of an appropriate binomial equation ); and (2)

2480-486: The case of the example of the figure 5 + 3 i and 5 − 3 i . {\displaystyle 5+3i\quad {\text{and}}\quad 5-3i.} 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 numbers (called "reals" in many programming languages ). In this context,

2542-427: The discriminant determines the number and nature of the roots. There are three cases: Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. The function f ( x ) = ax + bx + c 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

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2604-443: The early 1980s. Quadratic equation In mathematics , a quadratic equation (from Latin quadratus  ' square ') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0\,,} where the variable x represents an unknown number, and a , b , and c represent known numbers, where

2666-621: The form ( x + q )( x + s ) , and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule" and is related to Vieta's formulas ). As an example, x + 5 x + 6 factors as ( x + 3)( x + 2) . The more general case where a does not equal 1 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 b = 0 or c = 0 , factoring by inspection only works for quadratic equations that have rational roots. This means that

2728-691: The gametes within it, the gamodeme is said to be panmictic. Under this state, allele ( gamete ) frequencies can be converted to genotype ( zygote ) frequencies by expanding an appropriate quadratic equation , as shown by Sir Ronald Fisher in his establishment of quantitative genetics. This seldom occurs in nature: localization of gamete exchange – through dispersal limitations, preferential mating, cataclysm, or other cause – may lead to small actual gamodemes which exchange gametes reasonably uniformly within themselves but are virtually separated from their neighboring gamodemes. However, there may be low frequencies of exchange with these neighbors. This may be viewed as

2790-400: The governor of the prefecture to merge on March 1, 2006: 36°38′N 139°27′E  /  36.633°N 139.450°E  / 36.633; 139.450 This Tochigi Prefecture location article is a stub . You can help Misplaced Pages by expanding it . Population Population is the term typically used to refer to the number of people in a single area. Governments conduct

2852-484: The great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the algebraic identity x 2 + 2 h x + h 2 = ( x + h ) 2 , {\displaystyle x^{2}+2hx+h^{2}=(x+h)^{2},} which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with

2914-463: The level of homozygosity rises in the entire collection of gamodemes. The overall rise in homozygosity is quantified by the inbreeding coefficient (f or φ). All homozygotes are increased in frequency – both the deleterious and the desirable. The mean phenotype of the gamodemes collection is lower than that of the panmictic original – which is known as inbreeding depression. It is most important to note, however, that some dispersion lines will be superior to

2976-526: The minus sign if b < 0. {\displaystyle b<0.} A second form of cancellation can occur between the terms b and 4 ac 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. The golden ratio is found as the positive solution of the quadratic equation x 2 − x − 1 = 0. {\displaystyle x^{2}-x-1=0.} The equations of

3038-407: The other choice of signs. It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a , which is always possible since a is non-zero. This produces the reduced quadratic equation : x 2 + p x + q = 0 , {\displaystyle x^{2}+px+q=0,} where p = b /

3100-418: The panmictic original, while some will be about the same, and some will be inferior. The probabilities of each can be estimated from those binomial equations. In plant and animal breeding , procedures have been developed which deliberately utilize the effects of dispersion (such as line breeding, pure-line breeding, backcrossing). Dispersion-assisted selection leads to the greatest genetic advance (ΔG=change in

3162-419: The parabola, and how it opens, depend on the values of a , b , and c . If a > 0 , the parabola has a minimum point and opens upward. If a < 0 , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex . The x -coordinate of the vertex will be located at x = − b 2

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3224-756: The phenotypic mean), and is much more powerful than selection acting without attendant dispersion. This is so for both allogamous (random fertilization) and autogamous (self-fertilization) gamodemes. According to the UN, the world's population surpassed 8 billion on 15 November 2022, an increase of 1 billion since 12 March 2012. According to a separate estimate by the United Nations, Earth's population exceeded seven billion in October 2011. According to UNFPA , growth to such an extent offers unprecedented challenges and opportunities to all of humanity. According to papers published by

3286-405: The quadratic equation such as ax + 2 bx + c = 0 or ax − 2 bx + c = 0  , where b 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

3348-456: The quadratic equation. Completing the square 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: ( x + b 2 a ) 2 = b 2 − 4

3410-438: The quadratic formula is not completely stable . This occurs when the roots have different order of magnitude , or, equivalently, when b and b − 4 ac 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, r , can be computed as ( c /

3472-475: 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 D or an upper case Greek delta : Δ = b 2 − 4 a c . {\displaystyle \Delta =b^{2}-4ac.} A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case

3534-504: The roots of the equation: z 2 + q = p z . {\displaystyle z^{2}+q=pz.} The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y , were as follows: In modern notation this means calculating x = p 2 + ( p 2 ) 2 − q {\displaystyle x={\frac {p}{2}}+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}} , which

3596-419: The special case b = 4 ac where the quadratic has only one distinct root ( i.e. the discriminant is zero), the quadratic polynomial can be factored as a x 2 + b x + c = a ( x + b 2 a ) 2 . {\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}.} The solutions of the quadratic equation

3658-539: The square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics. A lesser known quadratic formula, as used in Muller's method , provides the same roots via the equation x = 2 c − b ± b 2 − 4 a c . {\displaystyle x={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac}}}}.} This can be deduced from

3720-546: The standard quadratic formula by Vieta's formulas , which assert that the product of the roots is c / a . It also follows from dividing the quadratic equation by x 2 {\displaystyle x^{2}} giving c x − 2 + b x − 1 + a = 0 , {\displaystyle cx^{-2}+bx^{-1}+a=0,} solving this for x − 1 , {\displaystyle x^{-1},} and then inverting. One property of this form

3782-431: The two solutions of a quadratic equation. 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

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3844-557: The zero root and an indeterminate form 0/0 . When neither a nor c is zero, the equality between the standard quadratic formula and Muller's method, 2 c − b − b 2 − 4 a c = − b + b 2 − 4 a c 2 a , {\displaystyle {\frac {2c}{-b-{\sqrt {b^{2}-4ac}}}}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\,,} can be verified by cross multiplication , and similarly for

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