Representatives of various civilizations: Ancient Egypt, Ancient Babylon, Ancient Greece, Ancient India, Ancient China, Medieval East, Europe mastered the methods of solving quadratic equations.
For the first time, the mathematicians of Ancient Egypt were able to solve a quadratic equation. One of the mathematical papyri contains the following problem:
“Find the sides of a field shaped like a rectangle if its area is 12 and its lengths are equal to its width.” “The length of the field is 4,” the papyrus states.
Millennia passed, and negative numbers entered algebra. Solving the equation x²= 16, we get two numbers: 4, –4.
Of course, in the Egyptian problem we would take X = 4, since the length of the field can only be a positive quantity.
Sources that have reached us indicate that ancient scientists had some general techniques for solving problems with unknown quantities. The rule for solving quadratic equations set forth in the Babylonian texts is essentially the same as the modern one, but it is not known how the Babylonians “got this far.” But in almost all papyri and cuneiform texts found, only problems with solutions are given. The authors only occasionally supplied their numerical calculations with skimpy comments such as: “Look!”, “Do this!”, “You found the right one!”
The Greek mathematician Diophantus composed and solved quadratic equations. His Arithmetic does not contain a systematic presentation of algebra, but it contains a systematic series of problems, accompanied by explanations and solved by constructing equations of various degrees.
Problems on composing quadratic equations are found already in the astronomical treatise “Aria-bhatiam”, compiled in 499 by the Indian mathematician and astronomer Aryabhatta.
Another Indian scientist Brahmagupta (7th century) outlined general rule solving quadratic equations of the form ax² + bx = c.
In ancient India, public competitions in solving difficult problems were common. One of the old Indian books says the following about such competitions: “As the sun eclipses the stars with its brilliance, so learned man eclipse the glory of another in popular assemblies by proposing and solving algebraic problems.” Problems were often presented in poetic form.
This is one of the problems of the famous Indian mathematician of the 12th century. Bhaskars:
A flock of frisky monkeys
Having eaten to my heart's content, I had fun.
Part eight of them were playing in the clearing in the square.
And twelve on the vines... began to jump, hanging...
How many monkeys were there?
Tell me, in this pack?
Bhaskara's solution shows that he knew that the roots of quadratic equations are two-valued.
The most ancient Chinese mathematical texts that have come down to us date back to the end of the 1st century. BC. In the II century. BC. Mathematics in Nine Books was written. Later, in the 7th century, it was included in the collection “Ten Classical Treatises,” which was studied for many centuries. The treatise "Mathematics in Nine Books" explains how to extract Square root using the formula for the square of the sum of two numbers.
The method was called “tian-yuan” (literally “heavenly element”) - this is how the Chinese designated an unknown quantity.
The first manual for solving problems that became widely known was the work of the Baghdad scientist of the 9th century. Muhammad bin Musa al-Khwarizmi. The word “al-jabr” over time turned into the well-known word “algebra”, and al-Khorezmi’s work itself became the starting point in the development of the science of solving equations. Al-Khwarizmi's algebraic treatise gives a classification of linear and quadratic equations. The author counts six types of equations, expressing them as follows:
-squares equal roots, that is, ah ² = bх;
-squares equal number, that is, ah ² = s;
-the roots are equal to the number, that is, ax = c;
-squares and numbers are equal to roots, that is, ah ²+ с = bх;
-squares and roots are equal to the number, that is, ah ² + bх = с;
-roots and numbers are equal to squares, that is, bx + c = ax ²;
Al-Khwarizmi's treatise is the first book that has come down to us, which systematically sets out the classification of quadratic equations and gives formulas for their solution.
Formulas for solving quadratic equations modeled after al-Khwarizmi in Europe were first set forth in the Book of Abacus, written in 1202 by the Italian mathematician Leonardo Fibonacci. The author independently developed some new algebraic examples solving problems and was the first in Europe to introduce negative numbers. His book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many problems from the “Book of Abacus” were included in almost all European textbooks of the 16th-17th centuries. and partly of the 18th century.
General rule for solving quadratic equations reduced to a single equation canonical form X ² + bх = с, for all possible combinations of signs of the coefficients b and с was formulated in Europe only in 1544 by M. Stiefel.
Derivation of the formula for solving a quadratic equation in general view Viet has it, but he also recognized only positive roots. Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century. In addition to positive and negative roots, they are taken into account. Only in the 17th century, thanks to the works of Girard, Descartes, Newton and other scientists, the method of solving quadratic equations took on its modern form.
Ministry of Education of the Russian Federation
Municipal educational institution
"Secondary school No. 22"
Quadratic and higher order equations
Completed:
Pupils of 8 "B" class
Kuznetsov Evgeniy and Rudi Alexey
Supervisor:
Zenina Alevtina Dmitrievna
mathematics teacher
Introduction
1.1 Equations in Ancient Babylon
1.2 Arab equations
1.3 Equations in India
Chapter 2. Theory of quadratic equations and higher order equations
2.1 Basic concepts
2.2 Formulas for even coefficient at x
2.3 Vieta's theorem
2.4 Quadratic equations of a particular nature
2.5 Vieta's theorem for polynomials (equations) higher degrees
2.6 Equations reducible to quadratic (biquadratic)
2.7 Study of biquadratic equations
2.8 Cordano formulas
2.9 Symmetric equations of the third degree
2.10 Reciprocal equations
2.11 Horner scheme
Conclusion
Bibliography
Annex 1
Appendix 2
Appendix 3
Introduction
Equations in school course algebras occupy leading place. More time is devoted to their study than to any other topic. Indeed, equations not only have important theoretical significance, but also serve purely practical purposes. The overwhelming number of problems about spatial forms and quantitative relationships real world comes down to a decision various types equations. By mastering ways to solve them, we find answers to various questions from science and technology (transport, Agriculture, industry, communications, etc.).
In this abstract I would like to display formulas and methods of solving different equations. For this purpose, equations are given that are not studied in school curriculum. These are mainly equations of a particular nature and equations of higher degrees. To expand on this topic, proofs of these formulas are given.
Objectives of our essay:
Improve equation solving skills
Develop new ways to solve equations
Learn some new ways and formulas to solve these equations.
The object of study is elementary algebra. The object of study is equations. The choice of this topic was based on the fact that equations are included both in the elementary curriculum and in each subsequent grade secondary schools, lyceums, colleges. Many geometric problems, problems in physics, chemistry and biology are solved using equations. The equations were solved twenty-five centuries ago. They are still being created today - both for use in the educational process, and for competitive exams at universities, for olympiads of the highest level.
Chapter 1. History of quadratic and higher order equations
1.1 Equations in Ancient Babylon
Algebra arose in connection with solving various problems using equations. Typically, problems require finding one or more unknowns, while knowing the results of some actions performed on the desired and given quantities. Such problems come down to solving one or a system of several equations, to finding the required ones using algebraic operations on given quantities. Algebra studies the general properties of operations on quantities.
Some algebraic techniques for solving linear and quadratic equations were known 4000 years ago in Ancient Babylon. The need to solve equations not only of the first, but also of the second degree, even in ancient times, was caused by the need to solve problems related to finding the areas of land plots and land works of a military nature, as well as with the development of astronomy and mathematics itself. As mentioned earlier, quadratic equations were able to be solved around 2000 BC by the Babylonians. Using modern algebraic notation, we can say that both incomplete and complete quadratic equations occur in their cuneiform texts.
The rule for solving these equations, set out in Babylonian texts, essentially coincides with modern ones, but it is not known how the Babylonians arrived at this rule. Almost all cuneiform texts found so far provide only problems with solutions laid out in the form of recipes, with no indication as to how they were found.
Despite high level development of algebra in Babylon, the cuneiform texts lack the concept of a negative number and general methods for solving a quadratic equation.
1.2 Arab equations
Some methods for solving both quadratic and higher-order equations were developed by the Arabs. Thus, the famous Arab mathematician Al-Khorezmi in his book “Al-jabar” described many ways to solve various equations. Their peculiarity was that Al-Khorezmi used complex radicals to find roots (solutions) of equations. The need to solve such equations was needed in questions about the division of inheritance.
1.3 Equations in India
Quadratic equations were also solved in India. Problems on quadratic equations are found already in the astronomical treatise “Aryabhattiam”, compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta (7th century), set out a general rule for solving quadratic equations reduced to a single conic form:
aх² + bx= c, where a > 0
In this equation, the coefficients, except a, can be negative. Brahmagupta's rule is essentially the same as ours.
In ancient India, public competitions in solving difficult problems were common. One of the old Indian books says the following about such competitions: “As the sun outshines the stars with its brilliance, so a learned man will outshine the glory of another in public assemblies, proposing and solving algebraic problems.” Problems were often presented in poetic form.
Various equations, both quadratic and equations of higher degrees, were solved by our distant ancestors. These equations were solved in very different and distant countries. The need for equations was great. The equations were used in construction, in military affairs, and in everyday situations.
Chapter 2. Quadratic equations and higher order equations
2.1 Basic concepts
A quadratic equation is an equation of the form
where coefficients a, b, c are any real numbers, and a ≠ 0.
A quadratic equation is called reduced if its leading coefficient is 1.
Example :
x 2 + 2x + 6 = 0.
A quadratic equation is called unreduced if the leading coefficient is different from 1.
Example :
2x 2 + 8x + 3 = 0.
A complete quadratic equation is a quadratic equation in which all three terms are present, in other words, it is an equation in which the coefficients b and c are non-zero.
Example :
3x 2 + 4x + 2 = 0.
An incomplete quadratic equation is a quadratic equation in which at least one coefficient b, c is equal to zero.
Thus, there are three types of incomplete quadratic equations:
1) ax² = 0 (has two coinciding roots x = 0).
2) ax² + bx = 0 (has two roots x 1 = 0 and x 2 = -)
Example :
x 1 = 0, x 2 = -5.
Answer: x 1 =0, x 2 = -5.
If -<0 - уравнение не имеет корней.
Example :
Answer: The equation has no roots.
If –> 0, then x 1,2 = ±
Example :
Answer: x 1.2 =±
Any quadratic equation can be solved using the discriminant (b² - 4ac). Usually the expression b² - 4ac is denoted by the letter D and is called the discriminant of the quadratic equation ax² + bx + c = 0 (or the discriminant of the quadratic three term ax² + bx + c)
Example :
x 2 +14x – 23 = 0
D = b 2 – 4ac = 144 + 92 = 256
x 2 = 
Answer: x 1 = 1, x 2 = - 15.
Depending on the discriminant, the equation may or may not have a solution.
1) If D< 0, то не имеет решения.
2) If D = 0, then the equation has two coinciding solutions x 1,2 =
3) If D > 0, then it has two solutions found according to the formula:
x 1.2 = 
2.2 Formulas for even coefficient at x
We are accustomed to the fact that the roots of a quadratic equation
ax² + bx + c = 0 are found by the formula
x 1.2 =
But mathematicians will never pass up the opportunity to make their calculations easier. They found that this formula can be simplified in the case where the coefficient b is b = 2k, in particular if b is an even number.
In fact, let the coefficient b of the quadratic equation ax² + bx + c = 0 be b = 2k. Substituting the number 2k instead of b into our formula, we get:
So, the roots of the quadratic equation ax² + 2kx + c = 0 can be calculated using the formula:
x 1.2 = 
Example :
5x 2 - 2x + 1 = 0
The advantage of this formula is that it is not the number b that is squared, but its half; it is not 4ac that is subtracted from this square, but simply ac, and, finally, that the denominator contains not 2a, but simply a.
If the quadratic equation is reduced, then our formula will look like this:
Example :
x 2 – 4x + 3 = 0
Answer: x 1 = 3, x 2 = 1.
2.3 Vieta's theorem
A very interesting property of the roots of a quadratic equation was discovered by the French mathematician Francois Viète. This property was called Vieta's theorem:
So that the numbers x 1 and x 2 are the roots of the equation:
ax² + bx + c = 0
it is necessary and sufficient to fulfill the equality
x 1 + x 2 = -b/a and x 1 x 2 = c/a
Vieta's theorem allows us to judge the signs and absolute value of a quadratic equation
x² + bx + c = 0
1. If b>0, c>0 then both roots are negative.
2. If b<0, c>0 then both roots are positive.
3. If b>0, c<0 то уравнение имеет корни разных знаков, причём отрицательный корень по абсолютной величине больше положительного.
4. If b<0, c<0 то уравнение имеет корни разных знаков, причём отрицательный корень по абсолютной величине меньше положительного.
2.4 Quadratic equations of a particular nature
1) If a + b + c = 0 in the equation ax² + bx + c = 0, then
x 1 = 1, and x 2 = .
Proof :
In the equation ax² + bx + c = 0, its roots
x 1.2 = (1).
Let us represent b from the equality a + b + c = 0
Let's substitute this expression into formula (1):
=
If we consider the two roots of the equation separately, we get:
1) x 1 = 
2) x 2 = 
It follows: x 1 = 1, and x 2 =.
1. Example :
2x² - 3x + 1 = 0
a = 2, b = -3, c = 1.
a + b + c = 0, therefore
2. Example :
418x² - 1254x + 836 = 0
This example is very difficult to solve using a discriminant, but knowing the above formula it can be easily solved.
a = 418, b = -1254, c = 836.
x 1 = 1 x 2 = 2
2) If a - b + c = 0, in the equation ax² + bx + c = 0, then:
x 1 =-1, and x 2 =-.
Proof :
Consider the equation ax² + bx + c = 0, it follows that:
x 1.2 = (2).
Let us represent b from the equality a - b + c = 0
b = a + c, substitute into formula (2):
=
We get two expressions:
1) x 1 = 
2) x 2 = 
This formula is similar to the previous one, but it is also important because... Examples of this type are common.
1) Example :
2x² + 3x + 1 = 0
a = 2, b = 3, c = 1.
a - b + c = 0, therefore
2)Example :
Answer: x 1 = -1; x 2 = -
3) Method “ transfers ”
The roots of the quadratic equations y² + by + ac = 0 and ax² + bx + c = 0 are related by the following relations:
x 1 = and x 2 =
Proof :
a) Consider the equation ax² + bx + c = 0
x 1.2 = = 
b) Consider the equation y² + by + ac = 0
y 1,2 =
Note that the discriminants of both solutions are equal; let us compare the roots of these two equations. They differ from each other by a leading factor, the roots of the first equation are less than the roots of the second by a. Using Vieta's theorem and the above rule, it is not difficult to solve various equations.
Example :
We have an arbitrary quadratic equation
10x² - 11x + 3 = 0
Let's transform this equation according to the given rule
y² - 11y + 30 = 0
We obtain the reduced quadratic equation, which can be solved quite easily using Vieta’s theorem.
Let y 1 and y 2 be the roots of the equation y² - 11y + 30 = 0
y 1 y 2 = 30 y 1 = 6
y 1 + y 2 = 11 y 2 = 5
Knowing that the roots of these equations differ from each other by a, then
x 1 = 6/10 = 0.6
x 2 = 5/10 = 0.5
In some cases it is convenient to decide not to first given equation ax² + bx + c = 0, and the reduced y² + by + ac = 0, which is obtained from the given “transfer” coefficient a, and then divide the found roots by a to find the original equation.
2.5 Vieta formula for polynomials (equations) of higher degrees
The formulas derived by Viète for quadratic equations are also true for polynomials of higher degrees.
Let the polynomial
P(x) = a 0 x n + a 1 x n -1 + … +a n
Has n different roots x 1, x 2..., x n.
In this case, it has a factorization of the form:
a 0 x n + a 1 x n-1 +…+ a n = a 0 (x – x 1)(x – x 2)…(x – x n)
Let's divide both sides of this equality by a 0 ≠ 0 and open the brackets in the first part. We get the equality:
x n + ()x n -1 + … + () = x n – (x 1 + x 2 + … + x n) x n -1 + (x 1 x 2 + x 2 x 3 + … + x n -1 x n)x n - 2 + … +(-1) n x 1 x 2 … x n
But two polynomials are identically equal if and only if the coefficients of the same powers are equal. It follows that the equality
x 1 + x 2 + … + x n = -
x 1 x 2 + x 2 x 3 + … + x n -1 x n =
x 1 x 2 … x n = (-1) n
For example, for polynomials of third degree
a 0 x³ + a 1 x² + a 2 x + a 3
We have identities
x 1 + x 2 + x 3 = -
x 1 x 2 + x 1 x 3 + x 2 x 3 =
x 1 x 2 x 3 = -
As with quadratic equations, this formula is called Vieta's formula. The left-hand sides of these formulas are symmetric polynomials from the roots x 1, x 2 ..., x n of this equation, and the right-hand sides are expressed through the coefficient of the polynomial.
2.6 Equations reducible to quadratic (biquadratic)
Equations of the fourth degree are reduced to quadratic equations:
ax 4 + bx 2 + c = 0,
called biquadratic, and a ≠ 0.
It is enough to put x 2 = y in this equation, therefore,
ay² + by + c = 0
let's find the roots of the resulting quadratic equation
y 1,2 =
To immediately find the roots x 1, x 2, x 3, x 4, replace y with x and get
x² =
x 1,2,3,4 = 
.
If a fourth degree equation has x 1, then it also has a root x 2 = -x 1,
If has x 3, then x 4 = - x 3. The sum of the roots of such an equation is zero.
Example :
2x 4 - 9x² + 4 = 0
Let's substitute the equation into the formula for the roots of biquadratic equations:
x 1,2,3,4 = 
,
knowing that x 1 = -x 2, and x 3 = -x 4, then:
x 3.4 = 
Answer: x 1.2 = ±2; x 1.2 =
2.7 Study of biquadratic equations
Let's take the biquadratic equation
ax 4 + bx 2 + c = 0,
where a, b, c are real numbers, and a > 0. By introducing the auxiliary unknown y = x², we examine the roots of this equation and enter the results into the table (see Appendix No. 1)
2.8 Cardano formula
If we use modern symbolism, the derivation of the Cardano formula can look like this:
x = 
This formula determines the roots of a general third-degree equation:
ax 3 + 3bx 2 + 3cx + d = 0.
This formula is very cumbersome and complex (it contains several complex radicals). It will not always apply, because... very difficult to fill out.
2.9 Symmetric equations of the third degree
Symmetric equations of the third degree are equations of the form
ax³ + bx² +bx + a = 0 ( 1 )
ax³ + bx² - bx – a = 0 ( 2 )
where a and b are given numbers, with a¹0.
Let us show how the equation ( 1 ).
ax³ + bx² + bx + a = a(x³ + 1) + bx(x + 1) = a(x + 1) (x² - x + 1) + bx(x + 1) = (x + 1) (ax² +(b – a)x + a).
We find that the equation ( 1 ) is equivalent to the equation
(x + 1) (ax² +(b – a)x + a) = 0.
This means that its roots will be the roots of the equation
ax² +(b – a)x + a = 0
and number x = -1
the equation ( 2 )
ax³ + bx² - bx - a = a(x³ - 1) + bx(x - 1) = a(x - 1) (x² + x + 1) + bx(x - 1) = (x - 1) (ax 2 + ax + a + bx) = (x - 1) (ax² +(b + a)x + a).
1) Example :
2x³ + 3x² - 3x – 2 = 0
It is clear that x 1 = 1, and
x 2 and x 3 roots of the equation 2x² + 5x + 2 = 0,
Let's find them through the discriminant:
x 1.2 = 
x 2 = -, x 3 = -2
2) Example :
5x³ + 21x² + 21x + 5 = 0
It is clear that x 1 = -1, and
x 2 and x 3 roots of the equation 5x² + 26x + 5 = 0,
Let's find them through the discriminant:
x 1.2 = 
x 2 = -5, x 3 = -0.2.
2.10 Reciprocal equations
Reciprocal equation – algebraic equation
a 0 x n + a 1 x n – 1 + … + a n – 1 x + a n =0,
in which a k = a n – k, where k = 0, 1, 2 …n, and a ≠ 0.
The problem of finding the roots of a reciprocal equation is reduced to the problem of finding solutions to an algebraic equation of a lower degree. The term reciprocal equations was introduced by L. Euler.
Fourth degree equation of the form:
ax 4 + bx 3 + cx 2 + bmx + am² = 0, (a ≠ 0).
Reducing this equation to the form
a (x² + m²/x²) + b(x + m/x) + c = 0, and y = x + m/x and y² - 2m = x² + m²/x²,
from where the equation is reduced to quadratic
ay² + by + (c-2am) = 0.
3x 4 + 5x 3 – 14x 2 – 10x + 12 = 0
Dividing it by x 2 gives the equivalent equation
3x 2 + 5x – 14 – 5 ×, or
Where and 
3(y 2 - 4) + 5y – 14 = 0, whence
y 1 = y 2 = -2, therefore
And where
Answer: x 1.2 = x 3.4 = .
A special case of reciprocal equations are symmetric equations. We talked about symmetric equations of the third degree earlier, but there are symmetric equations of the fourth degree.
Symmetric equations of the fourth degree.
1) If m = 1, then this is a symmetric equation of the first kind, having the form
ax 4 + bx 3 + cx 2 + bx + a = 0 and solved by a new substitution
2) If m = -1, then this is a symmetric equation of the second kind, having the form
ax 4 + bx 3 + cx 2 - bx + a = 0 and solved by a new substitution
2.11 Horner scheme
To divide polynomials, the “division by angle” rule, or Horner’s scheme, is used . For this purpose, polynomials are arranged in descending degrees X and find the leading term of the quotient Q(x) from the condition that when multiplied by the leading term of the divisor D(x), the leading term of the dividend P(x) is obtained. The found term of the quotient is multiplied, then by the divisor and subtracted from the dividend. The leading term of the quotient is determined from the condition that, when multiplied by the leading term of the divisor, it gives the leading term of the difference polynomial, etc. The process continues until the degree of the difference is less than the degree of the divisor (see Appendix No. 2).
In the case of equations R = 0, this algorithm is replaced by Horner's scheme.
Example :
x 3 + 4x 2 + x – 6 = 0
Find the divisors of the free term ±1; ± 2; ± 3; ± 6.
Let's denote the left side of the equation by f(x). Obviously, f(1) = 0, x1 = 1. Divide f(x) by x – 1. (see Appendix No. 3)
x 3 + 4x 2 + x – 6 = (x – 1) (x 2 + 5x + 6)
We denote the last factor by Q(x). We solve the equation Q(x) = 0.
x 2.3 = 
Answer : 1; -2; -3.
In this chapter, we have given some formulas for solving various equations. Most of these formulas for solving partial equations. These properties are very convenient because it is much easier to solve equations using a separate formula for this equation, rather than using the general principle. We have provided a proof and several examples for each method.
Conclusion
The first chapter examined the history of the emergence of quadratic equations and equations of higher orders. Various equations were solved more than 25 centuries ago. Many methods for solving such equations were created in Babylon, India. There has been and will continue to be a need for equations.
The second chapter provides various ways to solve (find roots) quadratic equations and higher order equations. Basically, these are methods for solving equations of a particular nature, that is, for each group of equations united by some common properties or type, a special rule is given that applies only to this group of equations. This method (selecting your own formula for each equation) is much easier than finding roots through a discriminant.
In this abstract, all goals have been achieved and the main tasks have been completed, new, previously unknown formulas have been proven and learned. We worked through many examples of examples before including them in the abstract, so we already have an idea of how to solve some equations. Each solution will be useful to us in further studies. This essay helped to classify old knowledge and learn new ones.
Bibliography
1. Vilenkin N.Ya. “Algebra for 8th grade”, M., 1995.
2. Galitsky M.L. “Collection of problems in algebra”, M. 2002.
3. Daan-Dalmedico D. “Paths and labyrinths”, M., 1986.
4. Zvavich L.I. “Algebra 8th grade”, M., 2002.
5. Kushnir I.A. “Equations”, Kyiv 1996.
6. Savin Yu.P. “ encyclopedic Dictionary young mathematician”, M., 1985.
7. Mordkovich A.G. “Algebra 8th grade”, M., 2003.
8. Khudobin A.I. “Collection of problems in algebra”, M., 1973.
9. Sharygin I.F. “Optional course in algebra”, M., 1989.
Annex 1
Study of biquadratic equations
| C | b | conclusions | ||
| On the roots of the auxiliary equation ay² +by+c=0 | About the roots of this equation a(x²)² +bx² +c=0 | |||
C< 0 |
b- any real number | y< 0 ; y > 01 2 |
x = ±Öy |
|
| C > 0 | b<0 | D > 0 | x = ±Öy |
|
| D=0 | y > 0 | x = ±Öy |
||
| D< 0 | No roots | No roots | ||
| b ≥ 0 | No roots | |||
| No roots | No roots | |||
y > 0 ; y< 01 2 |
x = ±Öy |
|||
| C=0 | b > 0 | y = 0 | x = 0 | |
| b = 0 | y = 0 | x = 0 | ||
| b< 0 | y = 0 | x = 0 | ||
Appendix 2
Dividing a polynomial into a polynomial using a corner
| A 0 | a 1 | a 2 | ... | a n | c |
| + | |||||
| b 0 c | b 1 c | … | b n-1 c | ||
| B 0 | b 1 | b 2 | … | b n | = R (remainder) |
Appendix 3
Horner scheme
| Root | |||||
| 1 | 4 | 1 | -6 | 1 | |
| x 1 = 1 | |||||
| demolishing | 5 | 6 | 0 | ||
| 1 | 1×1 +4 = 5 | 5×1 + 1 = 6 | 6×1 – 6 = 0 | ||
| root | |||||
| x 1 = 1 |
From the history of quadratic equations Author: 9th “A” class student Svetlana Radchenko Supervisor: Alabugina I.A. mathematics teacher MBOU “Secondary school No. 5 of Guryevsk” Kemerovo region Subject area of presentation: mathematics Made to help the teacher Total 20 slides Contents Introduction…………………………………………………………… ……………3 From the history of the emergence of quadratic equations Quadratic equations in Ancient Babylon………………………………….4 Quadratic equations in India…………………………………… …………5 Quadratic equations in Al-Khwarizmi…………………………………6 How Diophantus composed and solved quadratic equations………………………..... 7 Quadratic equations in Europe XII – XVIII centuries……………………………………...8 3. Quadratic equations today……………………………………………………………… .10 Methodology for studying quadratic equations……………………………………11 10 ways to solve quadratic equations…………………………….12 Algorithm for solving incomplete quadratic equations………… ………………13 Algorithm for solving a complete quadratic equation…………………………..14 Solving the given quadratic equations…………………………………15 4. Practical applications of quadratic equations for solving applied problems………………………………………………………………………………….16 5. Conclusion. ……………………………………………………………………………………18 1. 2. 6. List of references used…………………………………… …………….19 2 Introduction Consider as unhappy that day or that hour in which you did not learn anything new, did not add anything to your education. Jan Amos Comenius 3 Quadratic equations are the foundation on which the majestic edifice of algebra rests. They are widely used in solving trigonometric, exponential, logarithmic, irrational and transcendental equations and inequalities. Quadratic equations occupy a leading place in the school algebra course. A lot of time in the school mathematics course is devoted to their study. Basically, quadratic equations serve specific practical purposes. Most problems about spatial forms and quantitative relationships in the real world come down to solving various types of equations, including quadratic ones. By mastering ways to solve them, people find answers to various questions from science and technology. From the history of the emergence of quadratic equations Ancient Babylon: already about 2000 years BC, the Babylonians knew how to solve quadratic equations. Methods were known for solving both complete and incomplete quadratic equations. For example, in Ancient Babylon, the following quadratic equations were solved: 4 India Problems solved using quadratic equations are found in the treatise on astronomy "Aryabhattiam", written by the Indian astronomer and mathematician Aryabhatta in 499 AD. Another Indian scientist, Brahmagupta, outlined a universal rule for solving a quadratic equation reduced to its canonical form: ax2+bx=c; Moreover, it was assumed that all coefficients in it, except for “a,” could be negative. The rule formulated by the scientist essentially coincides with the modern one. 5 Quadratic equations in Al-Khorezmi: In the algebraic treatise of Al-Khorezmi, a classification of linear and quadratic equations is given. The author counts 6 types of equations, expressing them as follows: “Squares are equal to roots,” i.e. ax2 = bx.; “Squares are equal to numbers,” i.e. ax2 = c; “The roots are equal to the number,” i.e. ax = c; “Squares and numbers are equal to roots,” i.e. ax2 + c = bx; “Squares and roots are equal to the number,” i.e. ax2 + bx = c; “Roots and numbers are equal to squares,” i.e. bx + c = ax2. 6 How Diophantus composed and solved quadratic equations: One of the most original ancient Greek mathematicians was Diophantus of Alexandria. Neither the year of birth nor the date of death of Diophantus has been clarified; It is believed that he lived in the 3rd century. AD Of the works of Diophantus, the most important is Arithmetic, of which 13 books only 6 have survived to this day. Diophantus' Arithmetic does not contain a systematic presentation of algebra, but it contains a number of problems, accompanied by explanations and solved by constructing equations of various degrees. When composing equations, Diophantus skillfully selects unknowns to simplify the solution. 7 Quadratic equations in Europe in the 12th-17th centuries: The Italian mathematician Leonard Fibonacci independently developed some new algebraic examples of solving problems and was the first in Europe to introduce negative numbers. The general rule for solving quadratic equations reduced to a single canonical form x2 + bх = с for all possible combinations of signs and coefficients b, c was formulated in Europe in 1544 by Michael Stiefel. 8 François Viète French mathematician F. Viète (1540-1603), introduced a system of algebraic symbols, and developed the foundations of elementary algebra. He was one of the first to denote numbers by letters, which significantly developed the theory of equations. The derivation of the formula for solving a quadratic equation in general form is available from Vieth, but Vieth recognized only positive roots. 9 Quadratic equations today The ability to solve quadratic equations serves as the basis for solving other equations and their systems. Learning to solve equations begins with their simplest types, and the program determines the gradual accumulation of both their types and the “fund” of identical and equivalent transformations, with the help of which you can reduce an arbitrary equation to the simplest. The process of developing generalized techniques for solving equations in a school algebra course should also be built in this direction. In a high school mathematics course, students are faced with new classes of equations, systems, or with an in-depth study of already known equations. 10 Methods for studying quadratic equations With the beginning of studying a systematic algebra course, the main attention is paid to methods of solving quadratic equations, which become a special object of study. This topic is characterized by great depth of presentation and the richness of the connections established with its help in teaching, and the logical validity of the presentation. Therefore, it occupies an exceptional position in the line of equations and inequalities. An important point in the study of quadratic equations is the consideration of Vieta's theorem, which states the existence of a relationship between the roots and coefficients of the reduced quadratic equation. The difficulty of mastering Vieta's theorem is due to several circumstances. First of all, it is necessary to take into account the difference between the direct and inverse theorems. 11 10 ways to solve quadratic equations: Factoring the left side of the equation. Method for selecting a complete square. Solving quadratic equations using the formula. Solving equations using Vieta's theorem. Solving equations using the “throwing” method. Properties of the coefficients of a quadratic equation. Graphical solution of a quadratic equation.< 0, уравнение х2 =- не имеет корней (значит, не имеет корней и исходное уравнение ах2 + с = 0). В случае, когда - >0, i.e. - = m, where m>0, the equation x2 = m has two roots. Thus, an incomplete quadratic equation can have two roots, one root, or no roots. 13 Algorithm for solving a complete quadratic equation. These are equations of the form ax2 + bx + c = 0, where a, b, c are given numbers, and ≠ 0, x is an unknown. Any complete quadratic equation can be converted to form in order to determine the number of roots of the quadratic equation and find these roots. The following cases of solving complete quadratic equations are considered: D< 0, D = 0, D >0. 1. If D< 0, то квадратное уравнение ах2 + bx + c = 0 не имеет действительных корней. Так как D = 0, то данное уравнение имеет один корень. Этот корень находится по формуле. 3. Если D > 0, then the quadratic equation ax2 + bx + c = 0 has two roots, which are found by the formulas: ; 14 Solution of reduced quadratic equations F. Vieta's theorem: The sum of the roots of the reduced quadratic equation is equal to the second coefficient taken with the opposite sign, and the product of the roots is equal to the free term. In other words, if x1 and x2 are the roots of the equation x2 +px + q = 0, then x1 + x2 = - p, x1 x2 = q. (*) The inverse theorem to Vieta’s theorem: If the formulas (*) are valid for the numbers x1, x2, p, q, then x1 and x2 are the roots of the equation x2 + px + q = 0. 15 Practical applications of quadratic equations for solving applied problems Bhaskar ( 1114-1185) - the largest Indian mathematician and astronomer of the 12th century. He headed the astronomical observatory in Ujjain. Bhaskara wrote the treatise “Siddhanta-shiromani” (“Crown of Teaching”), consisting of four parts: “Lilavati” is devoted to arithmetic, “Bizhaganita” to algebra, “Goladhaya” to spherics, and “Granhaganita” to the theory of planetary motions. Bhaskara obtained negative roots of the equations, although he doubted their significance. He owns one of the earliest designs of a perpetual motion machine. 16 One of the problems of the famous Indian mathematician of the 12th century. Bhaskara: Bhaskara's solution shows that the author knew that the roots of quadratic equations are two-valued. 17 Conclusion The development of the science of solving quadratic equations has come a long and thorny path. Only after the works of Stiefel, Vieta, Tartaglia, Cardano, Bombelli, Girard, Descartes, and Newton did the science of solving quadratic equations take on its modern form. The significance of quadratic equations lies not only in the elegance and brevity of solving problems, although this is also very important. It is equally important that as a result of using quadratic equations when solving problems, new details are often discovered, interesting generalizations can be made and clarifications can be made, which are suggested by the analysis of the resulting formulas and relationships. Studying literature and Internet resources related to the history of the development of quadratic equations, I asked myself: “What motivated scientists who lived in such a difficult time to engage in science, even under the threat of death?” Probably, first of all, it is the inquisitiveness of the human mind, which is the key to the development of science. Questions about the essence of the World, about the place of man in this world haunt thinking, inquisitive, intelligent people at all times. People have always strived to understand themselves and their place in the world. Look inside yourself, maybe your natural curiosity is suffering because you have given in to everyday life and laziness? The fates of many scientists are 18 examples to follow. Not all names are well known and popular. Think about it: what am I like to the people close to me? But the most important thing is how I feel about myself, am I worthy of respect? Think about it... References 1. Zvavich L.I. “Algebra 8th grade”, M., 2002. 2. Savin Yu.P. “Encyclopedic Dictionary of a Young Mathematician”, M., 1985. 3. Yu.N. Makarychev “Algebra 8th grade”, M, 2012. 4. https://ru.wikipedia.org 5. http://www.ido. rudn.ru/nfpk/matemat/05/main_1.htm 6. http://rudocs.exdat.com/docs/index-14235.html 7. http://podelise.ru/docs/40825/index-2427. html 19 Thank you for your attention 20
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Kop'evsk rural secondary school
10 Ways to Solve Quadratic Equations
Head: Patrikeeva Galina Anatolyevna,
mathematic teacher
village Kopevo, 2007
1. History of the development of quadratic equations
1.1 Quadratic equations in Ancient Babylon
1.2 How Diophantus composed and solved quadratic equations
1.3 Quadratic equations in India
1.4 Quadratic equations by al-Khorezmi
1.5 Quadratic equations in Europe XIII - XVII centuries
1.6 About Vieta's theorem
2. Methods for solving quadratic equations
Conclusion
Literature
1. History of the development of quadratic equations
1.1 Quadratic equations in Ancient Babylon
The need to solve equations not only of the first, but also of the second degree, even in ancient times, was caused by the need to solve problems related to finding the areas of land plots and with earthworks of a military nature, as well as with the development of astronomy and mathematics itself. Quadratic equations could be solved around 2000 BC. e. Babylonians.
Using modern algebraic notation, we can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations:
X 2 + X = ¾; X 2 - X = 14,5
The rule for solving these equations, set out in the Babylonian texts, essentially coincides with the modern one, but it is not known how the Babylonians arrived at this rule. Almost all cuneiform texts found so far provide only problems with solutions laid out in the form of recipes, with no indication as to how they were found.
Despite the high level of development of algebra in Babylon, the cuneiform texts lack the concept of a negative number and general methods for solving quadratic equations.
1.2 How Diophantus composed and solved quadratic equations.
Diophantus' Arithmetic does not contain a systematic presentation of algebra, but it contains a systematic series of problems, accompanied by explanations and solved by constructing equations of various degrees.
When composing equations, Diophantus skillfully selects unknowns to simplify the solution.
Here, for example, is one of his tasks.
Problem 11.“Find two numbers knowing that their sum is 20 and their product is 96”
Diophantus reasons as follows: from the conditions of the problem it follows that the required numbers are not equal, since if they were equal, then their product would not be equal to 96, but to 100. Thus, one of them will be more than half of their sum, i.e. . 10 + x, the other is less, i.e. 10's. The difference between them 2x .
Hence the equation:
(10 + x)(10 - x) = 96
100 - x 2 = 96
x 2 - 4 = 0 (1)
From here x = 2. One of the required numbers is equal to 12 , other 8 . Solution x = -2 for Diophantus does not exist, since Greek mathematics knew only positive numbers.
If we solve this problem by choosing one of the required numbers as the unknown, then we will come to a solution to the equation
y(20 - y) = 96,
y 2 - 20y + 96 = 0. (2)
It is clear that by choosing the half-difference of the required numbers as the unknown, Diophantus simplifies the solution; he manages to reduce the problem to solving an incomplete quadratic equation (1).
1.3 Quadratic Equations in India
Problems on quadratic equations are found already in the astronomical treatise “Aryabhattiam”, compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta (7th century), outlined the general rule for solving quadratic equations reduced to a single canonical form:
ah 2 + b x = c, a > 0. (1)
In equation (1), the coefficients, except A, can also be negative. Brahmagupta's rule is essentially the same as ours.
In ancient India, public competitions in solving difficult problems were common. One of the old Indian books says the following about such competitions: “As the sun outshines the stars with its brilliance, so a learned man will outshine the glory of another in public assemblies, proposing and solving algebraic problems.” Problems were often presented in poetic form.
This is one of the problems of the famous Indian mathematician of the 12th century. Bhaskars.
Problem 13.
“A flock of frisky monkeys, and twelve along the vines...
The authorities, having eaten, had fun. They started jumping, hanging...
There are them in the square, part eight. How many monkeys were there?
I was having fun in the clearing. Tell me, in this pack?
Bhaskara's solution indicates that he knew that the roots of quadratic equations are two-valued (Fig. 3).
The equation corresponding to problem 13 is:
( x /8) 2 + 12 = x
Bhaskara writes under the guise:
x 2 - 64x = -768
and, to complete the left side of this equation to square, adds to both sides 32 2 , then getting:
x 2 - 64x + 32 2 = -768 + 1024,
(x - 32) 2 = 256,
x - 32 = ± 16,
x 1 = 16, x 2 = 48.
1.4 Quadratic equations in al - Khorezmi
In the algebraic treatise of al-Khorezmi, a classification of linear and quadratic equations is given. The author counts 6 types of equations, expressing them as follows:
1) “Squares are equal to roots,” i.e. ax 2 + c = b X.
2) “Squares are equal to numbers”, i.e. ax 2 = c.
3) “The roots are equal to the number,” i.e. ah = s.
4) “Squares and numbers are equal to roots,” i.e. ax 2 + c = b X.
5) “Squares and roots are equal to numbers”, i.e. ah 2 + bx = s.
6) “Roots and numbers are equal to squares,” i.e. bx + c = ax 2 .
For al-Khorezmi, who avoided the use of negative numbers, the terms of each of these equations are addends and not subtractables. In this case, equations that do not have positive solutions are obviously not taken into account. The author sets out methods for solving these equations using the techniques of al-jabr and al-muqabala. His decisions, of course, do not completely coincide with ours. Not to mention that it is purely rhetorical, it should be noted, for example, that when solving an incomplete quadratic equation of the first type
al-Khorezmi, like all mathematicians before the 17th century, does not take into account the zero solution, probably because in specific practical problems it does not matter. When solving complete quadratic equations, al-Khorezmi sets out the rules for solving them using particular numerical examples, and then geometric proofs.
Problem 14.“The square and the number 21 are equal to 10 roots. Find the root" (implies the root of the equation x 2 + 21 = 10x).
The author's solution goes something like this: divide the number of roots in half, you get 5, multiply 5 by itself, subtract 21 from the product, what remains is 4. Take the root from 4, you get 2. Subtract 2 from 5, you get 3, this will be the desired root. Or add 2 to 5, which gives 7, this is also a root.
The treatise of al-Khorezmi is the first book that has come down to us, which systematically sets out the classification of quadratic equations and gives formulas for their solution.
1.5 Quadratic equations in Europe XIII - XVII bb
Formulas for solving quadratic equations along the lines of al-Khorezmi in Europe were first set forth in the Book of Abacus, written in 1202 by the Italian mathematician Leonardo Fibonacci. This voluminous work, which reflects the influence of mathematics, both from the countries of Islam and from ancient Greece, is distinguished by its completeness and clarity of presentation. The author independently developed some new algebraic examples of solving problems and was the first in Europe to approach the introduction of negative numbers. His book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many problems from the Book of Abacus were used in almost all European textbooks of the 16th - 17th centuries. and partly XVIII.
The general rule for solving quadratic equations reduced to a single canonical form:
x 2 + bx = c,
for all possible combinations of coefficient signs b , With was formulated in Europe only in 1544 by M. Stiefel.
The derivation of the formula for solving a quadratic equation in general form is available from Vieth, but Vieth recognized only positive roots. Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century. In addition to positive ones, negative roots are also taken into account. Only in the 17th century. Thanks to the work of Girard, Descartes, Newton and other scientists, the method of solving quadratic equations takes on a modern form.
1.6 About Vieta's theorem
The theorem expressing the relationship between the coefficients of a quadratic equation and its roots, named after Vieta, was formulated by him for the first time in 1591 as follows: “If B + D, multiplied by A - A 2 , equals BD, That A equals IN and equal D ».
To understand Vieta, we should remember that A, like any vowel letter, meant the unknown (our X), vowels IN, D- coefficients for the unknown. In the language of modern algebra, the above Vieta formulation means: if there is
(a + b )x - x 2 = ab ,
x 2 - (a + b )x + a b = 0,
x 1 = a, x 2 = b .
Expressing the relationship between the roots and coefficients of equations with general formulas written using symbols, Viète established uniformity in the methods of solving equations. However, the symbolism of Viet is still far from modern look. He did not recognize negative numbers and therefore, when solving equations, he considered only cases where all the roots were positive.
2. Methods for solving quadratic equations
Quadratic equations are the foundation on which the majestic edifice of algebra rests. Quadratic equations are widely used in solving trigonometric, exponential, logarithmic, irrational and transcendental equations and inequalities. We all know how to solve quadratic equations from school (8th grade) until graduation.