A quadratic equation is a second-order polynomial equation in a single variable x
with
. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.
The roots x can be found by completing the square,
Solving for x then gives
This equation is known as the quadratic formula.
The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt. This problem reduces to solving
(Smith 1953, p. 443). The Greeks were able to solve the quadratic equation by geometric methods, and Euclid's(ca. 325-270 BC) Data contains three problems involving quadratics. In his work Arithmetica, the Greek mathematician Diophantus(ca. 210-290) solved the quadratic equation, but giving only one root, even when both roots were positive (Smith 1951, p. 134).
A number of Indian mathematicians gave rules equivalent to the quadratic formula. It is possible that certain altar constructions dating from ca. 500 BC represent solutions of the equation, but even should this be the case, there is no record of the method of solution (Smith 1953, p. 444). The Hindu mathematician Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions (Smith 1951, p. 159; Smith 1953, p. 444), while Brahmagupta (ca. 628) appears to have considered only one of them (Smith 1951, p. 159; Smith 1953, pp. 444-445). Similarly, Mahavra (ca. 850) had substantially the modern rule for the positive root of a quadratic. Srdhara (ca. 1025) gave the positive root of the quadratic formula, as stated by Bhaskara (ca. 1150; Smith 1953, pp. 445-446). The Persian mathematicians al-Khwarizm(ca. 825) and Omar Khayyám(ca. 1100) also gave rules for finding the positive root.
Viète was among the first to replace geometric methods of solution with analytic ones, although he apparently did not grasp the idea of a general quadratic equation (Smith 1953, pp. 449-450).
An alternate form of the quadratic equation is given by dividing (1) through by
:
Therefore,
This form is helpful if
, in which case the usual form of the quadratic formula can give inaccurate numerical results for one of the roots. This can be avoided by defining
so that b and the term under the square root sign always have the same sign. Now, if b > 0, then
so
Similarly, if b < 0, then
so
Therefore, the roots are always given by
and 
Now consider the equation expressed in the form
with solutions
and 
. These solutions satisfy Vièta's formulas
The properties of the symmetric polynomials appearing in Vièta's theorem then give
Given a quadratic integer polynomial
, consider the number such polynomials that are factorable over the integers for a and b taken from some set of integers 
. For example, for 
, there are four such polynomials,
The following table summarizes the counts of such factorable polynomials for simpleand small n. Plots of the fractions of factorable polynomials for 
(red), 
(blue), and 
(green) are also illustrated above. Amazingly, the sequence for 
has the recurrence equation
where
is the number of divisors of n and 
is the characteristic function of the square numbers.
[Ctrl+A 全部选择 提示：你可先修改部分代码，再按运行]
(1)
with
. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex. The roots x can be found by completing the square,
(2)
(3)
(4)
Solving for x then gives
(5)
This equation is known as the quadratic formula.
The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt. This problem reduces to solving
(6)
(7)
(Smith 1953, p. 443). The Greeks were able to solve the quadratic equation by geometric methods, and Euclid's(ca. 325-270 BC) Data contains three problems involving quadratics. In his work Arithmetica, the Greek mathematician Diophantus(ca. 210-290) solved the quadratic equation, but giving only one root, even when both roots were positive (Smith 1951, p. 134).
A number of Indian mathematicians gave rules equivalent to the quadratic formula. It is possible that certain altar constructions dating from ca. 500 BC represent solutions of the equation, but even should this be the case, there is no record of the method of solution (Smith 1953, p. 444). The Hindu mathematician Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions (Smith 1951, p. 159; Smith 1953, p. 444), while Brahmagupta (ca. 628) appears to have considered only one of them (Smith 1951, p. 159; Smith 1953, pp. 444-445). Similarly, Mahavra (ca. 850) had substantially the modern rule for the positive root of a quadratic. Srdhara (ca. 1025) gave the positive root of the quadratic formula, as stated by Bhaskara (ca. 1150; Smith 1953, pp. 445-446). The Persian mathematicians al-Khwarizm(ca. 825) and Omar Khayyám(ca. 1100) also gave rules for finding the positive root.
Viète was among the first to replace geometric methods of solution with analytic ones, although he apparently did not grasp the idea of a general quadratic equation (Smith 1953, pp. 449-450).
An alternate form of the quadratic equation is given by dividing (1) through by
:(8)
(9)
(10)
Therefore,
(11)
(12)
(13)
This form is helpful if
, in which case the usual form of the quadratic formula can give inaccurate numerical results for one of the roots. This can be avoided by defining (14)
so that b and the term under the square root sign always have the same sign. Now, if b > 0, then
(15)
(16)
so
(17)
(18)
Similarly, if b < 0, then
(19)
(20)
so
(21)
(22)
Therefore, the roots are always given by
and Now consider the equation expressed in the form
(23)
with solutions
and . These solutions satisfy Vièta's formulas(24)
(25)
The properties of the symmetric polynomials appearing in Vièta's theorem then give
(26)
(27)
(28)
Given a quadratic integer polynomial
, consider the number such polynomials that are factorable over the integers for a and b taken from some set of integers . For example, for , there are four such polynomials, (29)
(30)
(31)
The following table summarizes the counts of such factorable polynomials for simple
and small n. Plots of the fractions of factorable polynomials for (red), (blue), and (green) are also illustrated above. Amazingly, the sequence for has the recurrence equation(32)
where
is the number of divisors of n and is the characteristic function of the square numbers.[Ctrl+A 全部选择 提示：你可先修改部分代码，再按运行]
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