# What is the quadratic formula : general definition of the term

In fact, the answer to the question: ‘what is the quadratic formula ?’ is quite simple. However, students often do not understand the principles according to which the quadratic equations are solved. Thus, it is highly advisable to gain confidence that one has a profound and comprehensive understanding of the algebraic conception that determines the practical ways of solving these equations. In order to eschew all potential misconceptions that can significantly complicate the student’s understanding, let us examine what is the quadratic formula and what are its mathematical properties.

## What is the quadratic formula : the geometrical importance of the term

In fact, the most significant geometrical application of the quadratic formula is that it allows us to define the points in the Cartesian coordinate system along the x-axis in which there exist a crossing with the parabola. According to the fundamental geometrical properties of various linear curves that are given in a standard thesis theme, which supplies the audience with information about typical functions and their graphical embodiments, a parabola can be defined as any curve, which is described by a second-order polynomial. Therefore, for all parabolas regardless of their specific properties the following equation is correct: p2x = a2x2 + a1x + a0. In this equation, ‘p2’ represents the polynomial of order 2. The indexes a0, a1, and a2 are the constant coefficients. Thus, the quadratic formula is divided into two different terms:

• The definition of the axis of symmetry of parabola can be given as the (-b/2a) term. In fact, it is the first part of our equation. Its definition permits us to define the specific axis of symmetry of the given parabola and calculating points of minimum and maximum.
• The second term, which is expressed as (b2 – 4ac)1/2)/2a, is the measure of distance the zeros are away from the axis of symmetry. From the standpoint of geometry, in this specific issue, the ‘+’ sign represents the distance away to the right, whereas the ‘-’ sign represents the distance away to the left. Therefore, the axis of symmetry would be the ‘x’ value of the zero in a case in which the distance term was to decrease to zero. Thus, we can suppose that there exists only one possible solution to the given quadratic equation.

Therefore, let us summarize information about the most essential properties of the term ‘quadratic formula’ in order to devise the most appropriate answer to the question: what is the quadratic formula in accordance with the modern lab report format. The quadratic formula is an algebraic solution to the standard quadratic equation (ax2 + bx + c = 0), which supplies us with the possibility to represent the solutions of these equations in the rectangular coordinate system.

### What is the quadratic formula : different applications of this mathematical conception in contemporary fields of study

#### What is the quadratic formula : the history of the quadratic equations from Babylon to the present day

Doubtlessly, all ancient civilizations developed a body of mathematical knowledge, in part to serve diverse practical needs of architecture, measurement, commerce, warfare and/or construction. Naturally, thanks to Euclid, the scientific contribution that was made by the ancient Greeks may be the most familiar to the audience. Nevertheless, it is a serious misconception to consider that Egyptian, Babylonian, Chinese and Indian civilizations did not create their own mathematical systems. Thereby, the history of the quadratic equation can be traced back to Ancient Babylon. Various historical works and original literary sources demonstrate that the ancient Babylonians understood how to solve standard quadratic equations. Furthermore, they knew methods for solving both full and partial quadratic equations. Moreover, the mathematical rules that define the principles of solving quadratic equations are analogical to the modern algebraic principles. However, original Babylonian texts do not contain reasoning by which these rules were obtained. Thus, we can only guess in which way these equations were created at the first time.

In truth, in the minds of scientists of the time, the search for solutions of mathematical equations was closely associated with religious issues and doctrines. Metaphorically, this is the point of mental space in which mathematical researches meet religious studies. For example, for Pythagoras, the task of finding the equations that exhaustively describe the observable universe was a purely metaphysical question. He and his followers believed that the universe could be described as the reflection of the primal equations, which complement each other in the universal harmony. Additionally, he considered that music is also built in accordance with the laws of mathematics. Obviously, compared with the task of finding the universal harmony of numbers and sounds, various purely practical methods of the use of the acquired geometrical knowledge seemed extremely insignificant. It is not a gross mistake to suppose that for ancient scientist different questions, which are considered as purely practical nowadays, such as ‘what is the quadratic formula ?’ or ‘how to find the area of a truncated trapezoid?’ were filled with a mystical sense. Various examples of different algebraic problems that can be solved using quadratic equations are found in the treatise on astronomy Ariabhattiam, written by the Indian astronomer and mathematician Aryabhata in 499 AD. One of the conclusions of the first known formula of the square root of the quadratic equation belongs to an Indian scientist Brahmagupta. It dates back to 598. Brahmagupta was the first mathematics who stated a universal rule of solutions of the quadratic equations, expressed in the canonical form: ax2 + bx + c = 0. Moreover, he assumed that all the coefficients in this equation except ‘a’ can be negative. In fact, this mathematical conception is identical with the modern definition of the quadratic formula.

Of course, early mathematical concepts created by Greek and Indian authors greatly influenced the development of mathematical thought during the Middle Ages. Hereby, these conceptions were used by al-Khwarizmi – the Persian scientist who developed the algebraic methods of solving the quadratic equations in the IX century. The next important step in the evolution of various algebraic and analytic methods of solving these equations was overcome only in 1594. Hence, Simon Stevin created a quadratic formula, which covers all cases and types of the quadratic equations. Hereby, he is one of the first mathematicians who answered the question: ‘what is the quadratic formula ?’ from the point of view of modern algebra and geometry. Only in 1637, René Descartes published his magnum opus – La Géométrie, which comprises the quadratic formula in the form we know today. However, the first appearance of the general solution of a standard quadratic equation in the modern mathematical literature appeared in a paper written by Henry Heaton that was published in 1896.

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