From the history of quadratic equations. History of the development of quadratic equations

1.1. 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 military nature, as well as with 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:

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 common methods solutions of 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:

ax 2 + bx \u003d c, a> 0. (1)

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 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.

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.

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. ax 2 = bx.

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

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

4) “Squares and numbers are equal to roots”, i.e. ax 2 + c = bx.

5) "Squares and roots are equal to the number", i.e. ax 2 + bx \u003d c.

6) “Roots and numbers are equal to squares”, i.e. bx + c == ax 2.

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 the 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).

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 Europe XII-XVII centuries.

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 x 2 + bx \u003d 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 other scientists, the method for solving quadratic equations takes modern look..

The origins of algebraic methods for solving 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 character. However, even then, from time to time, problems arose in which the desired value of a quantity was specified 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 can be reduced in terms of modern classification 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 the characteristic actions by which the equations were reduced to standard view 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 the modern methodology of mathematics is associated with three main areas of its occurrence and functioning.

INTRODUCTION

Equations in the school course of algebra occupy a leading place. More time is devoted to their study than to any other topic of the school mathematics course. The strength of the theory of equations is that it not only has theoretical significance for the knowledge of natural laws, but also serves specific practical purposes. Most problems about spatial forms and quantitative relations of the real world come down to solving various types of equations. By mastering the ways of solving them, people find answers to various questions from science and technology (transport, agriculture, industry, communications, etc.). Also, for the formation of the ability to solve equations, the student's independent work in learning to solve equations is of great importance. When studying any topic, equations can be used as an effective means of consolidating, deepening, repeating and expanding theoretical knowledge, for the development of creative mathematical activity of students.

In the modern world, equations are widely used in various branches of mathematics, in solving important applied problems. 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. Students begin to study the topic "Square trinomials" having already accumulated some experience, owning a fairly large stock of algebraic and general mathematical concepts, concepts, and skills. To a large extent, it is on the material of this topic that it is necessary to synthesize the material related to equations, to implement the principles of historicism and accessibility.

Relevance The topic is the need to implement the principles of historicism and the lack of material for the implementation of this on the topic "Solution of quadratic equations".

Research problem: identifying historical material for learning to solve quadratic equations.

Objective: the formation of ideas about working on quadratic equations in mathematics lessons, the selection of a set of lessons with elements of historicism on the topic "Quadric equations".

Object of study: solving quadratic equations in grade 8 using elements of historicism.

Subject of study: quadratic equations and development of lessons on learning to solve quadratic equations using historical materials.

Tasks:

perform an analysis of scientific and methodological literature on the research problem;

analyze school textbooks and highlight in them the place of learning to solve quadratic equations;

pick up a set of lessons on solving quadratic equations using historical materials.

Research methods:

analysis of literature on the topic "Solution of quadratic equations";

observation of students during a lesson on the topic "Solving quadratic equations";

selection of material: lessons on the topic "Solving quadratic equations" using historical reference.

§ 1. From the history of the emergence of quadratic equations

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:

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. 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.

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. .
. The other is smaller, i.e.
. The difference between them
. Hence the equation:

From here
. One of the desired numbers is 12, the other is 8. Solution
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:

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 scientist, Brahmagupta (7th century), outlined the general rule for solving quadratic equations reduced to a single canonical form:

(1)

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

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:

Bhaskara writes under the guise of:

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

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:

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 outlines the 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
).

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 of 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. General rule for solving quadratic equations reduced to a single canonical form
with all possible combinations of signs and coefficients b, c, was formulated in Europe in 1544 by M. Stiefel.

Vieta has a general derivation of the formula for solving a quadratic equation, but 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 specified 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, 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.

Kovalchuk Kirill

The Quadratic Equations Through Centuries and Countries project introduces students to mathematicians whose discoveries are the basis scientific and technological progress, develops interest in mathematics as a subject based on acquaintance with historical material, broadens the horizons of students, stimulates their cognitive activity and creativity.

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Design work of a student of the 8th grade of the MOU secondary school No. 17 of the village of Borisovka Kovalchuk Kirill Head Mulyukova G.V.

Quadratic equations through centuries and countries

The purpose of the project: To acquaint students with scientists of mathematics, whose discoveries are the basis of scientific and technological progress. Show the significance of the work of scientists for the development of geometry and physics.??????????? Demonstrate the application of scientific discoveries in life. Develop interest in mathematics as a subject based on familiarity with historical material. To broaden the horizons of students, stimulate their cognitive activity and creativity

The need to solve equations not only of the first degree, but also of the second, even in ancient times, was caused by the need to solve problems related to finding the areas of land, with the development of astronomy and mathematics itself. Quadratic equations were able to solve about 2000 BC. e. Babylonians. The rules for solving these equations set forth in the Babylonian texts essentially coincide with modern ones, but these texts lack the concept of a negative number and general methods for solving quadratic equations.

. (c. 365 - 300 BC) - ancient Greek mathematician, author of the first theoretical treatises on mathematics that have come down to us. Euclid, or Euclid

Euclid Beginning Where the Nile merges with the sea, In the ancient hot land of the Pyramids, the Greek mathematician lived - the Knowledgeable, Wise Euclid. Geometry he studied, Geometry he taught. He wrote a great work. This book is called "Beginnings".

Euclid 3rd century BC Euclid solved quadratic equations using the geometric method. Here is one of the tasks from the ancient Greek treatise: “There is a city with a border in the form of a square with a side of unknown size, in the center of each side there is a gate. There is a pillar at a distance of 20bu (1bu=1.6m) from the northern gate. If you go straight from the southern gate 14bu, then turn west and go through another 1775bu, you can see a pillar. The question is: what side of the border of the city? »

To determine the unknown side of the square, we get the quadratic equation x ² +(k+l)x-2kd =0 . In this case, the equation looks like x ² +34x-71000=0 , whence x=250bu l x d k

Quadratic equations in India Problems on quadratic equations are also found in the astronomical treatise "Aryabhattayam", compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta, outlined the general rule for solving quadratic equations reduced to a single canonical form: ax ² +bx=c , a>0 In ancient India, public competitions in solving difficult problems were widespread. 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 of another in public meetings, proposing and solving algebraic problems.”

One of the tasks of the famous Indian mathematician of the 12th century Bhaskara Frisky flock of monkeys Having eaten to their heart's content, had fun. Part eight of them in a square I had fun in the clearing. And twelve along the vines ... They began to jump hanging ... How many monkeys were You tell me, in this flock ?.

Solution. () 2 +12 = x, x 2 - 64x +768 = 0, a = 1, b = -64, c = 768, then D = (-64) 2 -4 1 768 = 1024 > 0. X 1, 2 \u003d, x 1 \u003d 48, x 2 \u003d 16. Answer. There were 16 or 48 monkeys. Let's solve it.

The formula for the roots of a quadratic equation has been "rediscovered" repeatedly. One of the first derivations of this formula that has survived to this day belongs to the Indian mathematician Brahmagupta. The Central Asian scientist al-Khwarizmi in the treatise "Kitab al-dzherb wal-muqabala" obtained this formula by selecting a full square.

How did al-Khwarizmi solve this equation. He wrote: “The rule is this: double the number of roots, x = 2x 5 get five in this problem, 5 multiply by this equal to it, it will be twenty-five, 5 5 = 25 add this to thirty-nine, 25 + 39 will be sixty-four , 64 take the root from this, it will be eight, 8 and subtract from this half the number of roots, i.e. five, 8-5 will remain three - this and 3 will be the root of the square that you were looking for. What about the second root? The second root was not found, since negative numbers were not known. x 2 +10 x = 39

Quadratic equations in Europe 13th-17th 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 solutions to 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 the Abacus" passed into almost all European textbooks of the 16th-17th centuries. and partially 18.

François Viet - the greatest mathematician of the 16th century

Before F. Vieta, the solution of a quadratic equation was carried out according to its own rules in the form of very long verbal reasoning and descriptions, rather cumbersome actions. Even the equation itself could not be written down, this required a rather long and complex verbal description. He coined the term "coefficient". He suggested that the required values be denoted by vowels, and the data by consonants. Thanks to the symbolism of Vieta, you can write a quadratic equation in the form: ax 2 + bx + c \u003d 0. 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. Despite the fact that this theorem is called "Vieta's Theorem", it was known before him, and he only transformed it into a modern form. Vieta is called the "father of algebra"

Mankind has gone a long way from ignorance to knowledge, constantly replacing incomplete and imperfect knowledge with more and more complete and perfect knowledge on this path. Final word

Us living in early XXI century, antiquity attracts. In our ancestors, we notice first of all what they lack from a modern point of view, and usually do not notice what we ourselves lack in comparison with them.

Let's not forget about them...

Thank you for your attention!

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Representatives of various civilizations: ancient egypt, Ancient Babylon, Ancient Greece, Ancient India, Ancient China, Medieval East, Europe mastered the techniques 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 problem:

"Find the sides of a field that has the shape of a rectangle, if its area is 12, and - the lengths are equal to the width." “The length of the field is 4,” says the papyrus.

Millennia passed, 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 value.

Sources that have come down to us indicate that ancient scientists owned some common tricks solving problems with unknown quantities. The rule for solving quadratic equations stated in the Babylonian texts is essentially the same as the modern one, but it is not known how the Babylonians "came to this point." But in almost all found papyri and cuneiform texts, only problems with solutions are given. The authors only occasionally provided their numerical calculations with mean comments like: “Look!”, “Do it!”, “You found it right!”.

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 compiling equations of various degrees.

Tasks for the compilation of quadratic equations are already found in the astronomical treatise "Aria-bhatiam", compiled in 499 by the Indian mathematician and astronomer Ariabhatta.

Another Indian scientist Brahmagupta (7th century) outlined the general rule for solving quadratic equations of the form ax² + bx = c.

In ancient India, public competitions in solving difficult problems were common. In one of the old Indian books about such competitions, the following is said: “As the sun outshines the stars with its brilliance, so a learned person will outshine the glory of another 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:

Frisky flock of monkeys

Eating well, having fun.

The eighth part of them in the square was amused in the clearing.

And twelve along the vines ... 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 most ancient Chinese mathematical texts that have come down to us date back to the end of the 1st century BC. 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") - as the Chinese denoted an unknown quantity.

The first manual for solving problems, which 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 the work of al-Khwarizmi 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 lists six types of equations, expressing them as follows:

-squares equal roots, that is ah ² = bx;

-squares equal number, that is ah ² = c;

-the roots are equal to the number, that is, ax = c;

-squares and numbers are equal to roots, that is ah ²+ c \u003d bx;

-squares and roots are equal to the number, that is ah ² + bx \u003d c;

-roots and numbers are square, i.e. bx + c = ax ²;

The treatise of al-Khwarizmi 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.

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. 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 were included in almost all European textbooks of the 16th-17th centuries. and part of the 18th century.

The general rule for solving quadratic equations reduced to a single canonical form x ² + bx \u003d c, with all possible combinations of signs of the coefficients b and c, was formulated in Europe only in 1544 by M. Stiefel.

Vieta has a general derivation of the formula for solving a quadratic equation, but he also 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 17th century, thanks to the works of Girard, Descartes, Newton and other scientists, did the method of solving quadratic equations take on a modern form.