  ## Introduction

Equations are at the forefront in high school algebra course. If to look through any algebra textbook, one can see that their study takes more time than any other topic. Indeed, equations are not only of great theoretical value, but also serve for purely practical purposes. The vast number of tasks of spatial forms and quantitative relations of the real world is reduced to the solution of various types of equations. Mastering the ways of solving them, one can find the answers to various questions of science and technology (transport, agriculture, industry, communications, etc.).

## The History of the Development

Some algebraic methods for solving linear and quadratic equations have been known as long as 4,000 years ago in ancient Babylon. The need to solve the equation of not only of the first, but also of the second power in ancient times was caused by problems related to the location of land and the land works of a military nature, as well as with the development of astronomy and mathematics itself. Applying modern algebraic entry, it is possible to state that in their cuneiform texts both partial and full quadratic equations can be found. Typically, solutions of equations contained in Babylonian texts coincide substantially with the modern methods, but it is unknown how the Babylonians came to this rule. Almost all have been found so far in cuneiform texts show only the problem solving with the decision set out in the form of recipes, no indication as to how they were found. Despite the high level of development of algebra in Babylon, in the cuneiform texts there is no concept of a negative number and general methods for solving a quadratic equation.

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Arabs have withdrawn some methods of solving quadratic equations and the equations of higher degrees as well. A famous Arab mathematician, Al-Khwarizmi, in his book The Compendious Book on Calculation by Completion and Balancing described many different ways to solve equations. He introduced six general types of problems and equations to solve them. The peculiarity was that Al-Khwarizmi used the complex radicals to find the roots (solutions) to equations. The solution of such equations is needed in the division of the inheritance. In addition, it is important to mention that after Al-Khwarizmi has used the term “al-Jabr” in Latin script, European scientists began to acknowledge the science of solving quadratic and linear equations he created and, over the time, it transformed into a modern algebra. Quadratic equations were also solved in ancient India. Tasks for quadratic equations were found in such early works as in the astronomical treatise “Aryabhata”, compiled in 499 by Indian mathematician and astronomer Aryabhata. Another Indian scholar, Brahmagupta (VII century), set out the general rule for solving quadratic equations, reduced to a single conical shape: ax2 + bx = c, where a> 0.

In this equation, coefficients except factor a may be negative. Rule Brahmagupta essentially coincides with modern. In addition, he noted that quadratic equations have two possible solutions, one of which may be negative. In addition to its work on issues of common linear and quadratic equations, Brahmagupta went even further. He considered the solution of quadratic equations with two unknowns that was not even considered in the west until the thousand years later, in 1657, Pierre Fermat considered similar problem.

## Examples of Quadratic Equations Use

Human distant ancestors solved various equations as square and equations of higher degrees. These equations are solved in many different and distant from each other countries. The need for equations has been great. The equations were used in construction, military affairs, and in everyday situations. A huge part of life’s problems is reduced to the solution of various equations, and most of these equations are quadratic.

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There are many interesting facts about quadratic equations. First, for example, is the fact that the first nation to use quadratic equations was India, in the 9th century B.C. Second, despite quadratic equations were known 4,000 years ago, the formulas for the solution of quadratic equations in Europe were first set out in 1202 by Italian mathematician Leonardo Fibonacci. Moreover, the general rule for solving quadratic equations has been formulated in Europe only 1544. Third, a common school student knows two-three ways for quadratic equation solution. However, there are around 15 ways to solve a quadratic equation, and the application of this or that way depend on the result and range of use of the equation. Fourth, the quickest and convenient way to solve a quadratic equation is the Viete Theorem. François Viete once was considered the only mathematician in France, who could solve quadratic equations.

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