Quadratic equation history of origin. d) Quadratic equations in Europe XIII-XVII centuries

From the history of occurrence quadratic equations

Algebra arose in connection with the solution of various problems using equations. Usually in problems it is required to find one or several unknowns, while knowing the results of some actions performed on the desired and given quantities. Such problems are reduced to solving one or a system of several equations, to finding the desired ones with the help of algebraic operations on given quantities. Algebra studies the general properties of actions on quantities.

Some algebraic techniques for solving linear and quadratic equations were known as early as 4000 years ago in Ancient Babylon.

Quadratic Equations in Ancient Babylon

The need to solve equations not only of the first, but also of the second degree in ancient times was caused by the need to solve problems related to finding the areas of land and earthworks of a military nature, as well as the development of astronomy and mathematics itself. The Babylonians knew how to solve quadratic equations around 2000 BC. Applying modern algebraic notation, we can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations:

https://pandia.ru/text/78/002/images/image002_15.gif" width="93" height="41 src=">

The rule for solving these equations, stated in the Babylonian texts, coincides essentially with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions stated in the form of recipes, with no indication of how they were found. In spite of high level development of algebra in Babylon, in cuneiform texts there is no concept of a negative number and general methods for solving quadratic equations.

Diophantus' Arithmetic does not contain a systematic exposition of algebra, but it contains a systematic series of problems, accompanied by explanations and solved by formulating equations. different degrees.

When compiling equations, Diophantus skillfully chooses unknowns to simplify the solution.

Here, for example, is one of his tasks.

Task 2. "Find two numbers, knowing that their sum is 20 and their product is 96."

Diophantus argues as follows: it follows from the condition of the problem that the desired numbers are not equal, since if they were equal, then their product would be equal not to 96, but to 100. Thus, one of them will be more than half of their sum, i.e. .10 + x. The other is smaller, i.e. 10 - x. The difference between them is 2x. Hence the equation:

(10+x)(10-x)=96,

Hence x = 2. One of the desired numbers is 12, the other is 8. The solution x = - 2 for Diophantus does not exist, since Greek mathematics knew only positive numbers.

If we solve this problem, choosing one of the unknown numbers as the unknown, then we can come to the solution of the equation:

It is clear that Diophantus simplifies the solution by choosing the half-difference of the desired numbers as the unknown; he manages to reduce the problem to solving an incomplete quadratic equation.

Quadratic equations in India

Problems for quadratic equations are already found in the astronomical treatise Aryabhattam, compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scholar, Brahmagupta (7th century), expounded general rule solutions of quadratic equations reduced to a single canonical form:

ax2 + bx = c, a>

In equation (1) coefficients can be negative. Brahmagupta's rule essentially coincides with ours.

In India, public competitions in solving difficult problems were common. In one of the old Indian books, the following is said about such competitions: “As the sun outshines the stars with its brilliance, so scientist man eclipse the glory popular assemblies, suggesting and solving algebraic problems". Tasks were often dressed in poetic form.

Here is one of the problems of the famous Indian mathematician of the XII century. Bhaskara.

Bhaskara's solution indicates that the author was aware of the two-valuedness of the roots of quadratic equations.

The equation corresponding to problem 3 is:

https://pandia.ru/text/78/002/images/image004_11.gif" width="12" height="26 src=">x2 - 64x = - 768

and, to complete the left side of this equation to the square, he adds 322 to both sides, getting then:

x2 - b4x + 322 = -768 + 1024,

(x - 32) 2 = 256,

x1 = 16, x2 = 48.

Al-Khwarizmi's Quadratic Equations

Al-Khwarizmi's algebraic treatise gives a classification of linear and quadratic equations. The author lists 6 types of equations, expressing them as follows:

1) “Squares are equal to roots”, i.e. ax2 = bx.

2) “Squares are equal to number”, i.e. ax2 = c.

3) "The roots are equal to the number", i.e. ax \u003d c.

4) “Squares and numbers are equal to roots”, i.e. ax2 + c = bx.

5) “Squares and roots are equal to number”, i.e. ax2 + bx = c.

6) “Roots and numbers are equal to squares”, i.e. bx + c == ax2.

For Al-Khwarizmi, who avoided the use negative numbers, the terms of each of these equations are terms, not subtracts. In this case, equations that do not have positive solutions are obviously not taken into account. The author outlines ways to solve the indicated equations, using the techniques of al-jabr and al-muqabala. His decision, of course, does not completely coincide with ours. Not to mention the fact that it is purely rhetorical, it should be noted, for example, that when solving an incomplete quadratic equation of the first type, Al-Khwarizmi, like all mathematicians before the 17th century, does not take into account the zero solution, probably because in specific practical tasks, it does not matter. When solving the complete quadratic equations of Al-Khwarizmi on partial numerical examples sets out the decision rules, and then their geometric proofs.

Let's take an example.

Problem 4. “The square and the number 21 are equal to 10 roots. Find the root ”(the root of the equation x2 + 21 \u003d 10x is implied).

Solution: divide the number of roots in half, you get 5, multiply 5 by itself, subtract 21 from the product, 4 remains. Take the root of 4, you get 2. Subtract 2 from 5, you get 3, this will be the desired root. Or add 2 to 5, which will give 7, this is also a root.

Al-Khwarizmi's treatise is the first book that has come down to us, in which the classification of quadratic equations is systematically presented and formulas for their solution are given.

Quadratic equations in EuropeXII- XVIIin.

Forms for solving quadratic equations on the model of Al-Khwarizmi in Europe were first described in the "Book of the Abacus", written in 1202. Italian mathematician Leonard Fibonacci. The author independently developed some new algebraic examples problem solving and was the first in Europe to approach the introduction of negative numbers.

This book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many tasks from this book were transferred to almost all European textbooks of the 14th-17th centuries. The general rule for solving quadratic equations reduced to a single canonical form x2 + bx = c with all possible combinations of signs and coefficients b, c, was formulated in Europe in 1544 by M. Stiefel.

Derivation of the formula for solving a quadratic equation in general view Viet has, but Viet recognized only positive roots. The Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century. take into account, in addition to positive, and negative roots. Only in the XVII century. thanks to the works of Girard, Descartes, Newton and others scientists way solving quadratic equations takes on a modern form.

origins algebraic methods solutions to practical problems are related to science ancient world. As is known from the history of mathematics, a significant part of the problems of a mathematical nature, solved by Egyptian, Sumerian, Babylonian scribes-computers (XX-VI centuries BC), had a calculated nature. However, even then, from time to time, problems arose in which the desired value of a quantity was specified by some indirect conditions requiring, with our modern point vision, drawing up an equation or system of equations. Initially, arithmetic methods were used to solve such problems. Later, the beginnings of algebraic representations began to form. For example, Babylonian calculators were able to solve problems that can be reduced in terms of modern classification to equations of the second degree. Solution method was created word problems, which later served as the basis for the selection of the algebraic component and its independent study.

This study was already carried out in another era, first by Arab mathematicians (VI-X centuries AD), who singled out the characteristic actions by which the equations were reduced to standard form reduction of similar terms, transfer of terms from one part of the equation to another with a sign change. And then by the European mathematicians of the Renaissance, as a result of a long search, they created the language of modern algebra, the use of letters, the introduction of symbols for arithmetic operations, brackets, etc. At the turn of the 16th-17th centuries. Algebra as a specific part of mathematics, which has its own subject, method, areas of application, has already been formed. Its further development, up to our time, consisted in improving the methods, expanding the scope of applications, clarifying the concepts and their connections with the concepts of other branches of mathematics.

So, in view of the importance and vastness of the material associated with the concept of an equation, its study in modern methodology mathematics is associated with three main areas of its origin and functioning.

In order to solve any quadratic equation, you need to know:

the formula for finding the discriminant;

the formula for finding the roots of a quadratic equation;

· Algorithms for solving equations of this type.

solve incomplete quadratic equations;

solve complete quadratic equations;

solve the given quadratic equations;

find errors in the solved equations and correct them;

Do a check.

The solution to each equation consists of two main parts:

transformation of this equation to the simplest ones;

solving equations according to known rules, formulas or algorithms.

The generalization of the methods of students' activity in solving quadratic equations occurs gradually. The following stages can be distinguished when studying the topic "Quadratic Equations":

Stage I - "Solving incomplete quadratic equations."

Stage II - "Solution of complete quadratic equations."

Stage III - "Solution of the reduced quadratic equations."

At the first stage, incomplete quadratic equations are considered. Since at first mathematicians learned to solve incomplete quadratic equations, since for this they did not have to, as they say, invent anything. These are equations of the form: ax2 = 0, ax2 + c = 0, where c≠ 0, ax2 + bx = 0, where b ≠ 0. Consider the solution of several of these equations:

1. If ax2 = 0. Equations of this type are solved according to the algorithm:

1) find x2;

2) find x.

For example, 5x2 = 0 . Dividing both sides of the equation by 5, it turns out: x2 = 0, hence x = 0.

2. If ax2 + c = 0, c≠ 0 Equations of this type are solved according to the algorithm:

1) move the terms to the right side;

2) find all the numbers whose squares are equal to the number c.

For example, x2 - 5 = 0, This equation is equivalent to the equation x2 = 5. Therefore, you need to find all the numbers whose squares are equal to the number 5..gif" width="16" height="19">..gif" width=" 16" height="19 src="> and has no other roots.

3. If ах2 + bх = 0, b ≠ 0. Equations of this kind are solved according to the algorithm:

1) move the common factor out of brackets;

2) find x1, x2.

For example, x2 - 3x \u003d 0. Let us rewrite the equation x2 - 3x \u003d 0 in the form x (x - 3) \u003d 0. This equation obviously has roots x1 \u003d 0, x2 \u003d 3. It has no other roots, because if in substitute any number other than zero and 3 instead of x, then on the left side of the equation x (x - 3) \u003d 0 you get a number that is not equal to zero.

So, these examples show how incomplete quadratic equations are solved:

1) if the equation has the form ax2 = 0, then it has one root x = 0;

2) if the equation has the form ax2 + bx = 0, then the factorization method is used: x (ax + b) = 0; so either x = 0 or ax + b = 0..gif" width="16" height="41"> In case -< 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

https://pandia.ru/text/78/002/images/image010_9.gif" width="29" height="24 src=">.gif" width="29" height="24 src=">, (in this case, a shorter notation = is allowed.

So an incomplete quadratic equation can have two roots, one root, no roots.

At the second stage, the transition to the solution of the complete quadratic equation is carried out. These are equations of the form ax2 + bx + c = 0, where a, b, c are given numbers, a ≠ 0, x is the unknown.

Any complete quadratic equation can be converted to the form , in order to determine the number of roots of a quadratic equation and find these roots. Considered following cases solutions of complete quadratic equations: D< 0, D = 0, D > 0.

1. If D< 0, то квадратное уравнение ах2 + bx + c = 0 не имеет действительных корней.

For example, 2x2 + 4x + 7 = 0. Solution: here a = 2, b = 4, c = 7.

D \u003d b2 - 4ac \u003d 42 - 4 * 2 * 7 \u003d 16 - 56 \u003d - 40.

Since D< 0, то данное квадратное уравнение не имеет корней.

2. If D \u003d 0, then the quadratic equation ax2 + bx + c \u003d 0 has one root, which is found by the formula.

For example, 4x - 20x + 25 = 0. Solution: a = 4, b = - 20, c = 25.

D \u003d b2 - 4ac \u003d (-20) 2 - 4 * 4 * 25 \u003d 400 - 400 \u003d 0.

Since D = 0, then given equation has one root. This root is found using the formula ..gif" width="100" height="45">.gif" width="445" height="45 src=">.

An algorithm for solving an equation of the form ax2 + bx + c = 0 is compiled.

1. Calculate the discriminant D using the formula D = b2 - 4ac.

2. If D< 0, то квадратное уравнение ах2 + bx + c = 0 не имеет корней.

3. If D = 0, then the quadratic equation has one root, which is found by the formula

4..gif" width="101" height="45">.

This algorithm is universal, it is applicable to both incomplete and complete quadratic equations. However, incomplete quadratic equations are usually not solved by this algorithm.

Mathematicians are practical, economical people, so they use the formula: https://pandia.ru/text/78/002/images/image022_5.gif" width="155" height="53">. (4)

2..gif" width="96" height="49 src="> having the same sign as D..gif" width="89" height="49"> then equation (3) has two roots ;

2) if then the equation has two coinciding roots;

3) if then the equation has no roots.

An important point in the study of quadratic equations is the consideration of the Vieta theorem, which states the existence of a relationship between the roots and coefficients of the reduced quadratic equation.

Vieta's theorem. The sum of the roots of the given quadratic equation is equal to the second coefficient, taken from 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

These formulas are called Vieta formulas in honor of French mathematician F. Vieta (), who introduced a system of algebraic symbols, developed the foundations of elementary algebra. He was one of the first who began to designate numbers with letters, which significantly developed the theory of equations.

For example, the above equation x2 - 7x +10 \u003d 0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. It can be seen that the sum of the roots is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.

Also true is the theorem converse theorem Vieta.

Theorem inverse to Vieta's theorem. If formulas (5) are valid for the numbers x1, x2, p, q, then x1 and x2 are the roots of the equation x2 + px + q = 0.

Vieta's theorem and its inverse theorem are often used in solving various problems.

For example. Let's write the given quadratic equation, the roots of which are the numbers 1 and -3.

According to Vieta's formulas

– p = x1 + x2 = - 2,

Therefore, the desired equation has the form x2 + 2x - 3 = 0.

The complexity of mastering the Vieta theorem is associated with several circumstances. First of all, it is necessary to take into account the difference between direct and inverse theorems. In Vieta's direct theorem, a quadratic equation and its roots are given; in the inverse there are only two numbers, and the quadratic equation appears at the conclusion of the theorem. Students often make the mistake of substantiating their reasoning with an incorrect reference to the direct or inverse Vieta theorem.

For example, when finding the roots of a quadratic equation by selection, you need to refer to the inverse Vieta theorem, and not to the direct one, as students often do. In order to extend the Vieta theorems to the case of zero discriminant, we have to agree that in this case the quadratic equation has two equal root. The convenience of such an agreement is manifested in the decomposition square trinomial for multipliers.

History of the development of solutions to quadratic equations

Aristotle

D.I. Mendeleev

Find the sides of a field that has the shape of a rectangle if its area is 12 , a

Let's consider this problem.

Let x be the length of the field, then be its width,
is its area.
Let's make a quadratic equation:
The papyrus gives the rule for his decision: "Divide 12 by".
12: .
So, .
"The length of the field is 4", - stated in the papyrus.

Reduced quadratic equation
where are any real numbers.

In one of the Babylonian tasks, it was also required to determine the length of a rectangular field (let's denote it) and its width ().

Adding the length and two widths of a rectangular field, you get 14, and the area of \u200b\u200bthe field is 24. Find its sides.

Let's make a system of equations:

From here we get a quadratic equation.

To solve it, we add a certain number to the expression,

To obtain full square:

Consequently, .

In general, the quadratic equation

Has two roots:

DIOPHANT
An ancient Greek mathematician who probably lived in the 3rd century BC. e. Author of "Arithmetic" - a book dedicated to solving algebraic equations.
Nowadays, "Diophantine equations" are usually understood as equations with integer coefficients, the solutions of which must be found among integers. Diophantus was also one of the first to develop mathematical notation.

"Find two numbers knowing that their sum is 20 and their product is 96."

One of the numbers will be more than half of their sum, that is, 10+, the other less, that is, 10-.

Hence the equation ()()=96

Here is one of the problems of the famous

12th-century Indian mathematician Bhaskara:

Frisky flock of monkeys

Eating well, having fun.

Them squared part eight

Having fun in the meadow.

And twelve in vines ...

They began to jump, hanging ...

How many monkeys were

You tell me, in this flock?

Bhaskara's solution indicates that he was aware of the two-valuedness of the roots of quadratic equations.
The corresponding solution to the equation

Bhaskara writes in the form and, to complete the left side of this equation to a square, we add 32 2 to both sides, getting

"AL-JEBR" - RESTORATION - AL-KHOREZMI CALLED THE OPERATION OF EXCLUSION FROM BOTH PARTS OF THE EQUATION OF NEGATIVE MEMBERS BY ADDING EQUAL MEMBERS, BUT OPPOSITE IN SIGN.

"AL-MUKABALA" - OPPOSITION - REDUCTION IN THE PARTS OF THE EQUATION OF THE SAME MEMBERS.

THE RULE OF "AL-JABR"

WHEN SOLVING THE EQUATION

IF IN PART ONE,

IT DOES NOT MATTER WHAT

MEET NEGATIVE MEMBER,

WE ARE TO BOTH PARTS

WE GIVE AN EQUAL MEMBER,

ONLY WITH ANOTHER SIGN,

AND WE WILL FIND A POSITIVE RESULT.

1) the squares are equal to the roots, that is;

2) squares are equal to a number, that is;

3) the roots are equal to the number, that is;

4) squares and numbers are equal to the roots, i.e.;

5) squares and roots are equal to a number, i.e.;

6) the roots and numbers are equal to squares, i.e. .

A task . The square and the number 21 are equal to 10 roots. Find a root.

Decision. Divide the number of roots in half - you get 5, multiply 5 by itself,

Subtract 21 from the product, leaving 4.

Take the square root of 4 and you get 2.

Subtract 2 from 5 - you get 3, this will be the desired root. Or add to 5, which will give 7, this is also a root.

Fibonacci was born in Italian mall the city of Pisa, presumably in the 1170s. . In 1192 he was appointed to represent the Pisan trading colony in North Africa. At the request of his father, he moved to Algeria and studied mathematics there. In 1200, Leonardo returned to Pisa and began writing his first work, The Book of the Abacus. [ . According to the historian of mathematics A.P. Yushkevich The book of the abacus” rises sharply above the European arithmetic and algebraic literature of the XII-XIV centuries by the variety and strength of methods, the richness of problems, the evidence of presentation ... Subsequent mathematicians widely drew from it both problems and methods for solving them ».

Let's plot the function

The graph is a parabola whose branches are directed upwards, since

2) Parabola vertex coordinates

W. Sauer spoke :

“It is often more useful for a student of algebra to solve the same problem in three different ways than to solve three or four different problems. Solving one problem various methods, you can find out by comparison which one is shorter and more efficient. That's how experience is made."

"The city is a unity of the dissimilar"

Aristotle

“A number expressed in a decimal sign will be read by a German, a Russian, an Arab, and a Yankee in the same way”

From the history of quadratic equations Author: student of class 9 "A" Radchenko Svetlana Supervisor: Alabugina I.A. teacher of mathematics MBOU “Secondary school No. 5 of Guryevsk” of the Kemerovo region Presentation subject area: 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 Al-Khwarizmi’s Quadratic Equations……………………………………………6 How Diophantus Compiled and Solved Quadratic Equations……………………..... 7 Quadratic equations in Europe Xll - XVll centuries…………………………………………………………………………………………………………………………………………………………………………………………………. .10 Methodology for studying quadratic equations………………………………………………………………………………………………………………………………………………………………………..10 10 ways to solve quadratic equations………………………………….12 Algorithm for solving incomplete quadratic equations………… ………………13 Algorithm for solving the complete quadratic equation…………………………..14 solving applied problems……………………………………………………………………………………….16 5. Conclusion. …………………………………………………………………………… 18 1. 2. 6. List of used literature………………………………… …………….19 2 Introduction To consider unfortunate 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 in the school algebra course take leading place. A lot of school time in mathematics is devoted to studying them. Basically, quadratic equations serve specific practical purposes. Most problems about spatial forms and quantitative relations the real world comes down to solving various kinds equations, including quadratic ones. By mastering the ways of solving 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 for solving both complete and incomplete quadratic equations were known. For example, in Ancient Babylon, the following quadratic equations were solved: 4 India Problems solved with the help of quadratic equations are found in the treatise on astronomy "Aryabhattiam", written by the Indian astronomer and mathematician Aryabhata in 499 AD. Another Indian scientist, Brahmagupta, outlined a universal rule for solving a quadratic equation reduced to canonical form: ax2+bx=c; moreover, it was assumed that all coefficients in it, except for "a", can be negative. The rule formulated by the scientist essentially coincides with the modern one. 5 Al-Khwarizmi's quadratic equations: Al-Khwarizmi's algebraic treatise gives a classification of linear and quadratic equations. The author lists 6 types of equations, expressing them as follows: “Squares are equal to roots”, i.e. ax2 = bx.; "Squares are equal to number", i.e. ax2 = c; "The roots are equal to the number", i.e. ax \u003d c; "Squares and numbers are equal to roots", i.e. ax2 + c = bx; "Squares and roots are equal to number", i.e. ax2 + bx = c; "Roots and numbers are equal to squares", i.e. bx + c = ax2. 6 How Diophantus compiled and solved quadratic equations: One of the most peculiar ancient Greek mathematicians was Diophantus of Alexandria. Until now, neither the year of birth nor the date of death of Diophantus have been clarified; He is believed to have 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 exposition of algebra, but it contains a number of problems accompanied by explanations and solved by compiling equations of various degrees. When compiling equations, Diophantus skillfully chooses unknowns to simplify the solution. 7 Quadratic equations in Europe XII-XVII centuries: The Italian mathematician Leonard Fibonacci independently developed some new algebraic examples of problem solving and was the first in Europe to approach the introduction of negative numbers. The general rule for solving quadratic equations reduced to a single canonical form x2 + bx = c with all possible combinations of signs and coefficients b, c, was formulated in Europe in 1544 by Michael Stiefel. 8 François Viet French mathematician F. Viet (1540-1603), introduced a system of algebraic symbols, developed the foundations of elementary algebra. He was one of the first who began to designate numbers with letters, which significantly developed the theory of equations. Vieta has a general derivation of the formula for solving a quadratic equation, but Vieta 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 causes the gradual accumulation of both their types and the “fund” of identical and equivalent transformations, which can be used to reduce an arbitrary equation to the simplest ones. In this direction, one should also build the process of forming generalized methods for solving equations in school course algebra. In a high school mathematics course, students encounter new classes of equations, systems, or in-depth study already known equations 10 Methods of studying quadratic equations Since the beginning of the study of a systematic course in algebra, the main attention is paid to methods for solving quadratic equations, which become a special object of study. This topic is characterized by a great depth of presentation and the richness of the connections established with its help in learning, 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 the Vieta theorem, which states the existence of a relationship between the roots and coefficients of the reduced quadratic equation. The complexity of mastering the Vieta theorem is associated with several circumstances. First of all, it is necessary to take into account the difference between direct and inverse theorems. 11 10 ways to solve quadratic equations: Factoring the left side of the equation. Full square selection method. Solution of quadratic equations by formula. Solution of equations using Vieta's theorem. Solving equations by the method of "transfer" Properties of the coefficients of a quadratic equation. Graphical solution of a quadratic equation. Solving quadratic equations with a compass and straightedge. 12 Solving quadratic equations using a nomogram. Geometric way of solving quadratic equations. Algorithm for solving incomplete quadratic equations 1) if the equation has the form ax2 = 0, then it has one root x = 0; 2) if the equation has the form ax2 + bx = 0, then the factorization method is used: x (ax + b) = 0; so either x = 0 or ax + b = 0. As a result, two roots are obtained: x1 = 0; x2 \u003d 3) if the equation has the form ax2 + c \u003d 0, then it is converted to the form ax2 \u003d - c and then x2. = In the case when -< 0, уравнение х2 =- не имеет корней (значит, не имеет корней и исходное уравнение ах2 + с = 0). В случае, когда - >0, i.e. - \u003d m, where m>0, the equation x2 \u003d m has two roots. Thus, an incomplete quadratic equation can have two roots, one root, no root. 13 Algorithm for solving the complete quadratic equation. These are equations of the form ax2 + bx + c = 0, where a, b, c are given numbers, and ≠ 0, x is the unknown. Any complete quadratic equation can be converted to the 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 the 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. (*) Converse 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 Bhaskar problems ( 1114-1185) - the largest Indian mathematician and astronomer of the XII century. He headed the astronomical observatory in Ujjain. Bhaskara wrote the treatise "Siddhanta-shiromani" ("The Crown of Teaching"), consisting of four parts: "Lilavati" is devoted to arithmetic, "Bizhdaganita" - to algebra, "Goladhaya" - to the sphere, "Granhaganita" - to the theory of planetary movements. Bhaskara received negative roots of equations, although he doubted their significance. He owns one of the earliest perpetual motion projects. 16 One of the problems of the famous Indian mathematician of the XII century. Bhaskara: Bhaskara's solution indicates that the author was aware of the two-valuedness of the roots of quadratic equations. 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, Newton did the science of solving quadratic equations take on a modern form. The value of quadratic equations lies not only in the elegance and brevity of solving problems, although this is very important. No less important is the fact that as a result of the use of quadratic equations in solving problems, new details are often discovered, interesting generalizations can be made and refinements made, which are prompted by an analysis of the obtained formulas and relationships. Studying the 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 do 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 at all times thinking, inquisitive, reasonable people. People have strived to understand themselves, their place in the world at all times. Look into yourself too, maybe your natural curiosity suffers, because you have succumbed to everyday life, laziness? The fate of many scientists - 18 examples to follow. Not all names are well known and popular. Think: what am I to the people around me? But the most important thing is how do I feel about myself, do I deserve respect? Think about it... References 1. Zvavich L.I. “Algebra Grade 8”, M., 2002. 2. Savin Yu.P. “ encyclopedic Dictionary young mathematician”, M., 1985. 3. Yu.N. Makarychev “Algebra Grade 8”, M, 2012. /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

From the history of quadratic equations.

a) Quadratic equations in ancient Babylon

The need to solve equations not only of the first, but also of the second degree in ancient times was caused by the need to solve problems related to finding areas land plots and with earthworks of a military nature, as well as with the development of astronomy and mathematics itself. Quadratic equations were able to solve about 2000 BC. Babylonians. Applying 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 \u003d, x 2 - x \u003d 14

The rule for solving these equations, set forth in the Babylonian texts, essentially coincides with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions stated in the form of recipes, with no indication of 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.

Diophantus' Arithmetic does not contain a systematic presentation of algebra, but it contains a systematic series of problems, accompanied by explanations and solved by compiling equations of various degrees.

When compiling equations, Diophantus skillfully chooses unknowns to simplify the solution.

Here, for example, is one of his tasks.

Task 2. "Find two numbers, knowing that their sum is 20 and their product is 96."

Diophantus argues as follows: it follows from the condition of the problem that the desired numbers are not equal, since if they were equal, then their product would not be 96, but 100. Thus, one of them will be more than half of their sum, i.e. .10 + x. The other is smaller, i.e. 10 - x. The difference between them is 2x. Hence the equation:

(10+x)(10-x)=96,

100 -x 2 = 96.

Hence x = 2. One of the desired numbers is 12, the other is 8. The solution x = - 2 for Diophantus does not exist, since Greek mathematics knew only positive numbers.

If we solve this problem, choosing one of the unknown numbers as an unknown, then we can come to the solution of the equation:

Problems for quadratic equations are already found in the astronomical tract "Aryabhattayam", compiled in 499 by the Indian mathematician and astronomer Aryabahatta. Another Indian scientist, Brahmagupta (7th century), outlined the general rule for solving quadratic equations reduced to a single canonical form

Oh 2 + bx = c, a > 0

In the equation, the coefficients , except a, can be negative. Brahmagupta's rule essentially coincides with ours.

In India, public competitions in solving difficult problems were common. In one of the old Indian books, the following is said about such competitions: “As the sun outshines the stars with its brilliance, so a learned person will outshine the glory in public meetings, proposing and solving algebraic problems.” Tasks were often dressed in poetic form.

Here is one of the problems of the famous Indian mathematician of the XII century. Bhaskara.

Task 3.

Bhaskara's solution indicates that the author was aware of the two-valuedness of the roots of quadratic equations.

The equation corresponding to problem 3 is:

Bhaskara writes under the guise of:

x 2 - 64x = - 768

and, to complete the left side of this equation to the square, adds 32 2 to both sides, then getting:

x 2 - b4x + 32 2 = -768 + 1024,

(x - 32) 2 = 256,

x 1 = 16, x 2 = 48.

c) Al-Khwarizmi's quadratic equations

Al-Khwarizmi's algebraic treatise gives a classification of linear and quadratic equations. The author lists 6 types of equations, expressing them as follows:

“The squares are equal to the roots”, i.e. ax 2 = bx.
"Squares are equal to number", i.e. ax 2 = c.
"The roots are equal to the number", i.e. ax = c.
"Squares and numbers are equal to roots", i.e. ax 2 + c \u003d bx.
"Squares and roots are equal to the number", i.e. ax 2 + bx \u003d c.
“Roots and numbers are equal to squares”, i.e. bx + c == ax 2.

For Al-Khwarizmi, who avoided the use of negative numbers, the terms of each of these equations are addends, not subtractions. 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 methods of al-jabr and al-muqabala. His decision, of course, does not completely coincide with ours. Not to mention the fact that it is purely rhetorical, it should be noted, for example, that when solving an incomplete quadratic equation of the first type, Al-Khwarizmi, like all mathematicians before the 17th century, does not take into account the zero solution, probably because in specific practical tasks, it does not matter. When solving complete quadratic equations, Al-Khwarizmi sets out the rules for solving them using particular numerical examples, and then their geometric proofs.

Let's take an example.

Problem 4. “The square and the number 21 are equal to 10 roots. Find the root "(meaning the root of the equation x 2 + 21 \u003d 10x).

Al-Khwarizmi's treatise is the first book that has come down to us, in which the classification of quadratic equations is systematically presented and formulas for their solution are given.

d) Quadratic equations in Europe XIII-XVII centuries.

Formulas for solving quadratic equations on the model of al-Khwarizmi in Europe were first set forth in the "Book of the Abacus", written in 1202 by the Italian mathematician Leonardo Fibonacci. This voluminous work, which reflects the influence of mathematics from both the countries of Islam and Ancient Greece, differs in both completeness and clarity of presentation. The author independently developed some new algebraic examples of problem solving 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 tasks from the Book of the Abacus passed into almost all European textbooks of the 16th-17th centuries. and partly XVIII.

General rule for solving quadratic equations reduced to a single canonical form

x 2 + bx \u003d c,

for all possible combinations of signs of the coefficients b, With was formulated in Europe only in 1544 by M. Stiefel.

Vieta has a general derivation of the formula for solving a quadratic equation, however, Vieta recognized only positive roots. The Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century. Take into account, in addition to positive, and negative roots. Only in the XVII century. thanks to the works of Girard, Descartes, Newton and other scientists, the method of solving quadratic equations takes on a modern form.

The origins of algebraic methods for solving practical problems are connected with the science of the ancient world. As is known from the history of mathematics, a significant part of the problems of a mathematical nature, solved by Egyptian, Sumerian, Babylonian scribes-computers (XX-VI centuries BC), had a calculated character. However, even then, from time to time, problems arose in which the desired value of a quantity was set by some indirect conditions, requiring, from our modern point of view, the formulation of an equation or a system of equations. Initially, arithmetic methods were used to solve such problems. Later, the beginnings of algebraic representations began to form. For example, Babylonian calculators were able to solve problems that, from the point of view of modern classification, are reduced to equations of the second degree. A method for solving text problems was created, which later served as the basis for highlighting the algebraic component and its independent study.

This study was already carried out in another era, first by Arab mathematicians (VI-X centuries AD), who singled out characteristic actions by which equations were reduced to a standard form, reduction of similar terms, transfer of terms from one part of the equation to another with a sign change. And then by the European mathematicians of the Renaissance, as a result of a long search, they created the language of modern algebra, the use of letters, the introduction of symbols for arithmetic operations, brackets, etc. At the turn of the 16th-17th centuries. algebra as a specific part of mathematics, having its own subject, method, application areas, has already been formed. Its further development, up to our time, consisted in improving the methods, expanding the scope of applications, clarifying the concepts and their connections with the concepts of other branches of mathematics.

So, in view of the importance and vastness of the material associated with the concept of equation, its study in the modern methodology of mathematics is associated with three main areas of its occurrence and functioning.

Home > Report

MOU secondary school named after Heroes Soviet Union
Sotnikova A.T. and Shepeleva N. G. s. Uritskoe

Report on the topic:

"The history of the emergence

quadratic equations"

Prepared by:Izotova Julia,
Ampleeva Elena,
Shepelev Nikolay,

Dyachenko Yuri.

Oh mathematics. For centuries you are covered with glory,

Luminary of all earthly luminaries.

You majestic queen

No wonder Gauss christened.

Strict, logical, majestic,

Slender in flight, like an arrow,

Your everlasting glory

Through the ages, she gained immortality.

We praise the human mind

The works of his magical hands,

The hope of this age

Queen of all earthly sciences.

We want to tell you today

History of occurrence

What every student should know

History of quadratic equations.

Euclid, in the III century BC. e. took geometric algebra in his "Principles" throughout the second book, which contains all the necessary material for solving quadratic equations.

Euclid (Eνκλειδηζ), ancient Greek mathematician, author of the first theoretical treatise on mathematics that has come down to us

Information about Euclid is extremely scarce. The only thing that can be considered reliable is that scientific activity flowed in Alexandria in the III century BC. e. Euclid - the first mathematician Alexandrian school. His main job"Beginnings" (in the Latinized form - "Elements") contains a presentation of planimetry, stereometry and a number of issues in number theory; in it he summed up the previous development of Greek mathematics and created the foundation further development mathematics. Heron - Greek mathematician and engineer for the first time in Greece in the 1st century AD. gives a purely algebraic way of solving a quadratic equation.

Heron of Alexandria; Heron, I c. n. e., Greek mechanic and mathematician. The time of his life is uncertain, it is only known that he quoted Archimedes (who died in 212 BC), he himself was quoted by Pappus (c. 300 AD). At present, the prevailing opinion is that he lived in the 1st century. n. e. He studied geometry, mechanics, hydrostatics, optics; invented the prototype steam engine and precision leveling instruments. The most popular automatons were automatic theaters, fountains, and others. G. described the theodolite, relying on the laws of statics and kinetics, and gave a description of the lever, block, propeller, and military vehicles. In optics, he formulated the laws of light reflection, in mathematics - methods for measuring the most important geometric shapes. Major works G. is Ietrik, Pneumatics, Autopoietics, Mechanics (French; the work has been preserved entirely in Arabic), Catoptics (the science of mirrors; preserved only in Latin translation) and others. G. used the achievements of his predecessors: Euclid, Archimedes, Strato from Lampsak. His style is simple and clear, although sometimes too laconic or unstructured. Interest in the writings of G. arose in the III century. n. e. Greek and then Byzantine and Arabic students commented on and translated his works.

Diophantus- a Greek scientist in the 3rd century AD, without resorting to geometry, solved some quadratic equations in a purely algebraic way, and the equation itself and its solution were written in symbolic form

“I will tell you how the Greek mathematician Diophantus composed and solved quadratic equations. Here, for example, is one of his tasks:"Find two numbers knowing that their sum is 20 and their product is 96."

1. From the condition of the problem it follows that the desired numbers are not equal, because if they were equal, then their product would not be 96, but 100.

2. Thus. one of them will be more than half of their sum, i.e. 10 + x, the other is less, i.e. 10 - x.

3. The difference between them is 2x.

4. Hence the equation (10 + x) * (10 - x) = 96

100 - x 2 = 96 x 2 - 4 = 0

5. Answer x = 2. One of the desired numbers is 12,
other - 8. Solution x = - 2 for Diophantus does not exist, because Greek mathematics knew only positive numbers.” Diophantus knew how to solve very complex equations, used letter designations for unknowns, introduced a special symbol for calculation, used abbreviations of words. Bhaskare - Akaria- Indian mathematician in the XII century AD. opened general method solutions of quadratic equations.

Let's analyze one of the problems of Indian mathematicians, for example, the problem of Bhaskara:

“A flock of monkeys is having fun: an eighth of the total number of them in a square frolics in the forest, the remaining twelve scream at the top of the mound. Tell me, how many monkeys are there?"

Commenting on the problem, I would like to say that the equation (x/8) 2 + 12 = x corresponds to the problem. Bhaskara writes as x 2 - 64x \u003d - 768. Adding square 32 to both parts, the equation will take the form:

x 2 - 64 x + 32 2 = - 768 + 1024

(x - 32) 2 = 256

After extraction square root we get: x - 32 \u003d 16.

"AT this case, says Bhaskara, - the negative units of the first part are such that the units of the second part are less than them, and therefore the latter can be considered both positive and negative, and we get double meaning unknown: 48 and 16".

It must be concluded that Bhaskara's solution indicates that he knew about the two-valuedness of the roots of quadratic equations.

It is proposed to solve the old Indian Bhaskara problem:

“The square of a fifth of the monkeys, reduced by three, hid in the grotto, one monkey climbed a tree, was visible. How many monkeys were there? It should be noted that given task solved elementarily, reducing to a quadratic equation.

Al - Khorezmi - an Arab scholar who in 825 wrote the book "The Book of Restoration and Opposition." It was the world's first algebra textbook. He also gave six kinds of quadratic equations and for each of the six equations in verbal form formulated special rule his decisions. In the treatise, Khorezmi lists 6 types of equations, expressing them as follows:

1. "Squares are equal to roots", i.e. ax 2 = in.

2. "Squares are equal to number", i.e. ax 2 = s.

3. "The roots are equal to the number", i.e. ah = s.

4. "Squares and numbers are equal to the roots", i.e. ax 2 + c \u003d in.

5. "Squares and roots are equal to the number", i.e. ax 2 + in = s.

6. "Roots and numbers are equal to squares", i.e. in + c \u003d ah 2.

Let's analyze the problem of al-Khwarizmi, which is reduced to solving a quadratic equation. "A square and a number are equal to the roots." For example, one square and the number 21 are equal to 10 roots of the same square, i.e. the question is, from what is a square formed, which, after adding 21 to it, becomes equal to 10 roots of the same square?

And using the 4th formula of al-Khwarizmi, students must write down: x 2 + 21 = 10x

François Viet - French mathematician, formulated and proved the theorem on the sum and product of the roots of the given quadratic equation.

The art I present is new, or at least has been so corrupted by the influence of the barbarians that I have seen fit to give it a completely new look.

François Viet

Yette François (1540-13.12. 1603) was born in the town of Fontenay-le-Comte in the province of Poitou, not far from famous fortress La Rochelle. Having received legal education, from the age of nineteen he successfully practiced as a lawyer in hometown. As a lawyer, Viet enjoyed prestige and respect among the population. He was wide an educated person. Knew astronomy and mathematics and all free time gave to these sciences.

Vieta's main passion was mathematics. He deeply studied the works of the classics Archimedes and Diophantus, the immediate predecessors of Cardano, Bombelli, Stevin and others. Vieta not only admired them, he saw a big flaw in them, which was the difficulty of understanding due to verbal symbolism: Almost all actions and signs were recorded in words, there was no hint of those convenient, almost automatic rules that we now use. It was impossible to write down and, therefore, to begin in a general form, algebraic comparisons or any other algebraic expressions. Each type of equation with numerical coefficients was solved according to a special rule. Therefore, it was necessary to prove that there are general actions over all numbers that do not depend on these numbers themselves. Viet and his followers established that it does not matter whether the number in question is the number of objects or the length of the segment. The main thing is that it is possible to perform algebraic operations with these numbers and, as a result, again obtain numbers of the same kind. Hence, they can be denoted by some abstract signs. Viet did just that. He not only introduced his literal calculus, but made a fundamentally new discovery, setting himself the goal of studying not numbers, but actions on them. This recording method allowed Viet to make important discoveries when studying common properties algebraic equations. It is no coincidence that Vieta is called the "father" of algebra, the founder of letter symbols.

Informational resources:

http :// som. fio. en/ resources/ Karpuhina/2003/12/ Completed%20 work/ Concert/ index1. htm

http :// pages. marsu. en/ iac/ school/ s4/ page74. html