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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.

The quadratic formula, in the contemporary elementary algebra, is the solution of the quadratic equation. Of course, there exists a considerable amount of other mathematical methods to solve the quadratic equation, eschewing using the classical quadratic formula, such as graphing factoring, completing the square, etc. However, the quadratic formula is the most suitable and effective method of solving the quadratic equations in all but cases because it allows us to solve the quadratic equation, avoiding over sophisticated computations or building graphs. In truth, it is practically the simplest way to solve the standard quadratic equation, which requires only elementary mathematical skills. The general quadratic equation comes in the form ax2 + bx + c = 0; for real numbers a, b, and c, with a ± 0. In this case, ‘x’ represents the unknown. The quadratic formula uses the ‘a’, ‘b’, and ‘c’ from the formula: ‘ax2 + bx + c’, where ‘a’, ‘b’, and ‘c’ are real numbers. In other words, these numbers can be considered as the ‘numerical coefficients’ of the standard quadratic equation. The quadratic formula is formally stated as the following equation: ax2 + bx + c = 0, with a ≠ 0. In fact, according to the statement that postulates that it is derived from the mathematical process of completing the square, the quadratic formula for all the solutions of this equation takes the form: x = (-b/2a) ± (b2 – 4ac)1/2/2a). Since a ≠ 0, we are free to divide both sides of the previously formulated equation: ax2 + bx + c = 0, by ‘a’ before finding the solutions. Therefore, we get the final equation: x = (-b ± (b2 – 4ac)1/2)/2. A root of the quadratic equation is a solution for the quadratic formula. From the point of view of geometry, these roots represent the ‘x’ values at which the specific parabola in the rectangular coordinate system, which is given as y = ax2 + bx + c, crosses the x-axis. In other words, the quadratic equation also provides us with the axis of symmetry of any parabola. Therefore, this equation can be used for the determination of the problem of how many zeros to expect the parabola to have.

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

Naturally, the quadratic equations might seem like a non-applicable conception, which is virtually useless in various spheres of mundane life. Of course, a deep and profound knowledge of these algebraic principles is absolutely essential in different assignments that are closely connected with certain algebraic topics, such as writing college application essays for those students who have linked their lives with algebra and/or statistics. However, the lion’s share of students continues to consider that the quadratic equations are too complicated for use in daily life. Luckily, it is a very sophisticated objective to find an opinion, which is farther from the truth. Thereby, let us demonstrate the usefulness of this knowledge in everyday life. For example, even the simplest task of computation the area of the room involves quadratic equations. Let us suppose that a person needs to determine the volume of the paint, which is necessary for painting the wall. We can use the standard quadratic equation; express the volume of the paint as ‘x’ and the linear dimensions of the room as natural coefficients. Therefore, all we have to do is to solve a standard quadratic equation and obtain correct results that can significantly save our time. Naturally, it is only a primitive example of the case in which the understanding of the significant nuances of the quadratic equations is essential. Nowadays, the quadratic equations are, basically, the fundament of all modern topology, classical linear algebra and even modern trigonometry. They are widely used in virtually all scientific disciplines from pure mathematical sciences, such as functional analysis and topology, to various disciplines that are traditionally referred to as humanitarian fields of knowledge, such as modern management, accounting, linguistics, anthropology and even psychology. Therefore, one has to be ready to be faced with the necessity to apply previously obtained mathematical knowledge in practice, while composing different coursework examples, even those that are not directly connected with modern algebra. Undoubtedly, if the student is able to give correct answers to the elementary mathematical questions, such as how to find the discriminant or what is the quadratic formula, her chances to finish the work in time, eschewing undesirable deadlines, and receive the highest marks are significantly increased.

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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