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International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org Graphical view of complex roots of quadratic Equation Jagtap Sharadkumar, Vice Principal ,Shivaji Mahavidyalaya, Udgir. & Suvarna D. Kamble Asst.Professor Shivaji Mahavidyalaya, Udgir. ABSTRACT: The Complex roots of a Quadratic equation has been investigated analytically while we deal with numbers purely in real domain .The imaginary or complex roots give a deep insight of this roots of quadratic equation .Here we are trying to analyze the roots of quadratic equation graphically. Some examples are examined in two dimensional and three dimensional graphacially. INTRODUCTION: The earliest methods for solving quadratic equations were geometric.Babylonian mathematicians used a simple version of Quadratic formula as far back as 2000 B.C [1] .Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom(2050 BC to 1650 BC), contains the solution to a twoterm quadratic equation.[2]. Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC [3][4]. The Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[5] The equation most closely related to the form we know today was first written down by a Indian mathematician named Brahmagupta [6]. He also described the method of completing the square and recognized that the discriminant must be positive,[6] . European mathematics gained resurgence during the 1500s, and, in 1545, Girolamo Cardano compiled all that was previously known about the quadratic formula and also allowed for the possibility of imaginary solutions[7]. René Descartes, the Father of Modern Mathematics, published the quadratic formula in his book [8]. When we solved the quadratic equation it was necessary to take the the disciminant with complex number available.So Complex numbers were first introduced by G. Cardano(1501-1576) in his Ars Magna chapter 37 as a tool for finding the roots intended to use imaginary numbers, remained unknown[9]. Later, in 1545, Girolamo Cardano (1501-1576) discovered imaginary numbers[10]. and in 1777 Leonhard Euler (1707 β 1783 suggested the symbol π = β1 [11]. For most of the 20th E-mail id:- [email protected] Page 1 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org century, graphing was rarely mentioned as a method for solving quadratic equations in high school or college algebra texts. Students learned to solve quadratic equations by factoring, completing the square, and applying the quadratic formula. Recently, graphical methods have started to appear in textbooks, but they are generally not highly emphasized [12]. I) REPRESENTATION OF COMPLEX NUMBER Complex numbers are numbers with both real and imaginary component. Complex number has general form: π = π + ππ (1) Where π is complex number, a and b are both real numbers and π is β1 . The number a is called the real part and b is called the imaginary part the complex number c .since a complex number b has two components it can be plotted as a point a on a plane (as shown in fig 1.1) Imaginary axis b p π + ππ a real axis Figure 1.1 Representing a complex number in rectangular co-ordinates The horizontal axis of the plane is the real axis and the vertical axis of the plane is the imaginary axis so that any complex number π + ππ can be represented as a single point π units along the real axis and π units along the imaginary axis .A complex number represented this way is said be in rectangular coordinates ,since the real and imaginary axes define the sides of a rectangle. A complex number can be also represented as a vector of the length π§ and angle π pointing from the origin of the plane of the plane p (in fig 1.2) . E-mail id:- [email protected] Page 2 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org Imaginary axis p π+π z π a real axis Figure 1.2 Representing a complex number in polar co-ordinates A complex number represented this way is said to be in polar coordinates as π = π + ππ = π§β π The relationships among the rectangular and polar coordinate terns π, π, π§ πππ π are: π = π§ πππ π (2) π = π§ π πππ (3) π§ = π2 + π2 (4) π π = tanβ1 π (5) The rectangular coordinates to represent complex numbers .ach complex number consist of a pair of real numbers(π, π) .The first number (a) is the real part of the complex number and the second number (b) is the imaginary part of the complex number . If complex number π1 and π2 are defined as π1 = π1 + π1 and π2 = π2 + π2 then the addition, subtraction, multiplication and division of π1 and π2 are defined as: π1 + π2 = π1 + π2 + π1 + π2 π π1 β π2 = π1 β π2 + π1 β π2 π π1 × π2 = π1 π2 β π1 π2 + π1 π2 + π1 π2 π E-mail id:- [email protected] Page 3 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org ππ ππ ππ + ππ ππ ππ ππ β ππ ππ = + π ππ π π π + ππ π π π π + ππ π A complex numbers is created automatically when a complex value is assigned .to create a complex values i or j , both of which are predefined to be β1 II) QUADRATIC EQUATION The availability of complex number often simplifies the calculations required to solve problems. (When complex numbers available, the square root of a negative number presents no difficulties, so we can greatly simplify these calculations. solve for the root of a quadratic equation, Use complex number so that based on the value of the discriminant) A quadratic equation is a polynomial in which the first term is of the second degree. This means that is it squared. The standard form of the quadratic equation is : ππ₯ 2 + ππ₯ + π = 0 There are different ways to find the roots of a quadratic equation. The roots are the solutions to an equation. There are two solutions to every quadratic equation; however, they may or may not be real numbers. One way to find the solutions to this type of equation is by factoring. This is not always the best way to solve a quadratic equation, especially if the solutions are not whole numbers III) The Quadratic Formula One simple was to solve a quadratic equation is by using the quadratic formula. With just some simple math, you will have the solution quickly, whether the solution is a whole number, an irrational number or even an imaginary number. The quadratic formula Quadratic equation in standard form, ππ₯ 2 + ππ₯ + π = 0 We will get one of the following three possible solution sets. 1. Two real distinct (i.e. not equal) solutions. E-mail id:- [email protected] Page 4 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org 2. A double root. this arises when we can factor the equation into a perfect square. 3. Two complex solutions. These are the only possibilities for solving quadratic equations in standard form. however, if we start with rational expression in the equation we may get different solution sets because we may need avoid one of the possible solutions so we donβt get division by zero errors. Now, it turns out that all we need to do is look at the quadratic equation (in standard form ) to determine which of the three cases that weβll get. To see how this works letβs start off by recalling the quadratic formula. π₯= βπ ± π2 β 4ππ 2π The quantity π2 β 4ππ in the quadratic formula is called the discriminant. It is the value of the discriminant that will determine which solution set we will get. Letβs go through the cases one at a time 1. Two real distinct solutions. We will get this solution set if π2 β 4ππ > 0 . In this case we will be taking the square root of a positive number and so the square root will be a real number. Therefore the numerator in the quadratic formula will be βπ plus or minus a real number. This means that the numerator will be two different real numbers. Dividing either one by 2a wonβt change the fact that they are real, nor will it change the fact that they are different. 2. A double root. We will get this solution set if π2 β 4ππ = 0. Here we will be taking the square root of zero, which is zero. However, this means that the βplus or minusβ part of the numerator will be zero and so the numerator in the quadratic formula will be βπ In other words, we will get a single real number out of the quadratic formula, which is what we get when we get a double root. 3. Two complex solutions. We will get this solution set if π2 β 4ππ < 0 . If the discriminant is negative we will be taking the square root of negative numbers in the quadratic formula which means that we will get complex solutions. Also, we will get two since they have a βplus or minusβ in front of the square root. E-mail id:- [email protected] Page 5 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org The Quadratic formula is complete by itself and can be used to find out roots of all types quadratic equations. There are three different types of roots possible. We will see it by examples. Real and distinct roots. Example: π₯ 2 + 3π₯ + 2 Roots: -1,-2 Real and equal roots. Example: π₯ 2 + 2π₯ + 1 Roots: -1,-1 Complex roots. Example: π₯ 2 + 2π₯ + 2 Roots: -1+i, -1-i In every case, we can see, quadratic equation has exactly two roots. Even though you get a single root in case 2, we consider it as two same roots. IV) Graphical view of quadratic equations A graph consists of an independent variable (say x) and a dependent variable (say y) and graph simply represent the relationship between y and x. Since y depends on x, whenever x changes, y also changes. Thus we can say y is a function of x (i.e., y = f(x) ) and the graph between x and y is simply the function f(x). Our quadratic equation can also be considered as a function if we write it as ππ₯ 2 + ππ₯ + π = 0 β ππ₯ 2 + ππ₯ + π = π¦ π¦ = ππ₯ 2 + ππ₯ + π whenever y = 0, this equation turns to a quadratic equation. which is in the form y = f(x). The above form can be easily plotted using the graph Here is the plot Iβve drawn for the three different examples we already seen. Figure 1 : The graph of equation π₯ 2 + 3π₯ + 2 and its roots -1,-2 E-mail id:- [email protected] Page 6 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org Figure 2 : The graph of equation : π₯ 2 + 2π₯ + 1 and its roots -1,-1 Figure 3 : The graph of equation : : π₯ 2 + 2π₯ + 2 and its roots -1+i, -1-i From the above graphs, it can be seen that, quadratic functions are always U-shaped curves in X-Y plane. The imaginary number i means that the number is in another dimension. So we need to introduce one more dimension to the graph. So our X-Y 2D graph becomes X-Y-Z 3D graph. The below 3D graph ,the curve: π¦ = π₯ 2 + 2π₯ + 1 and its complex roots -1+i and -1-i. E-mail id:- [email protected] Page 7 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org Figure 4 : The three dimensional graph its roots -1+i, -1-i The red marked points are the roots, the roots are not even in the plane of the curve it is a complex plane in 3d similarly the curve in 2d is a surface in 3d.So in 3d graph roots are the points where surface crosses the complex plane. Example: Complex data has both real and imaginary components, and graphing complex data with real part. For example, consider the function π¦ π‘ = π β0.2π‘ (πππ π‘ + π π ππ π‘) If this function is plotted , only the real data will plotted the imaginary part will be ignored. This shown in figure 5 1 0.8 0.6 y(t) 0.4 0.2 0 -0.2 -0.4 -0.6 0 2 4 6 8 10 12 14 t Figure 5: Representation of the function π¦ π‘ = π β0.2π‘ (πππ π‘ + π π ππ π‘) of complex function vs time If both the real and imaginary part of the function then Both parts can be plotted as a function of time on the same axes .shown below 1 real imaginary 0.8 0.6 y(t) 0.4 0.2 0 -0.2 -0.4 -0.6 0 2 4 6 8 10 12 14 t Figure 6: A Graph of a real and imaginary parts of y(t) verses time Alternatively, the real part of the function can be plotted versus the imaginary part. If a single complex argument is supplied to the function, it automatically generates a real part versus the imaginary part. The result is shown in below figure E-mail id:- [email protected] Page 8 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org 1 real imaginary 0.8 Imaginary Part 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.4 -0.2 0 0.2 Re a l Pa rt 0.4 0.6 0.8 1 Finally, the function can be plotted as a polar plot showing magnitude versus angle. The result is shown in figure. 90 1 120 60 0.8 0.6 150 30 0.4 0.2 180 0 210 330 240 300 270 Three dimensional line graphs An example of three dimensional line graph consider the following functions π¦ π‘ = π β0.2π‘ πππ 2π‘ π¦ π‘ = π β0.2π‘ π ππ 2π‘ These functions might represent the decaying oscillations of mechanical systems in two dimensions so x and y together represent the location of the system at any given time. Here x and y are both functions of the same independent variable t.we could create a series of (x,y) points and plot them using the two dimensional as well as in three dimensional as below E-mail id:- [email protected] Page 9 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org 1 0.8 0.6 0.4 y 0.2 0 -0.2 -0.4 -0.6 -0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x Figure 7: Two dimensional line plot 10 8 t 6 4 2 0 1 0.5 1 0.5 0 0 -0.5 y -0.5 -1 -1 x Figure 7: Three dimensional line plot Conclusion: Complex numbers occur in many problems in science and engineering. The approach outlined here gives a view to graphical representation of complex roots and general understanding of the functions. References: 1. Irving,Ron (2013).Beyond the Quadratic Formula.MAA. p. 34. ISBN 978-0-88385-783 0. 2. The Cambridge Ancient History Part 2 Early History of the Middle East. Cambridge University Press. 1971. p. 530. ISBN 978-0-521-07791-0 3. Aitken,Wayne. "A Chinese Classic: The Nine Chapters" (PDF). Mathematics Department, California State University. Retrieved 28 April 2013. 4. Smith,David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 380. ISBN 978-0-486-20430-7 E-mail id:- [email protected] Page 10 International Journal of Research in Engineering, IT and Social Sciences ISSN 2250-0558, Impact Factor: 5.850, Volume 5 Issue 12, December 2015 Website: www.indusedu.org 5. David Eugene Smith (1958). "History of mathematics". Courier Dover Publications. p.134. ISBN 0-486-20429-4 6. Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006). 7. Boyer, Carl B.; Uta C. Merzbach, rev. editor (1991). A History of Mathematics. John Wiley & Sons, Inc. ISBN 0-471-54397-7 8. Smith David E. and Lantham M. L. (1954). The Geometry of René Descartes (Reprint of 1925 ed.). Courier Dover Publications. ISBN 0-486-60068-8. 9. Cardano, Gerolamo (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3 10. Westfall, Richard S. "Cardano, Girolamo". The Galileo Project. rice.edu. Archived from the original on 2012-07-19. Retrieved 2012-07-19. 11. Boyer, Carl B.; Uta C. Merzbach (1991). A History of Mathematics. John Wiley & Sons. pp. 439β445. ISBN 0-471-54397-7. 12. Ballew, Pat. "Solving Quadratic Equations β By analytic and graphic methods; Including several methods you may never have seen" (PDF). Retrieved 18 April 2013. E-mail id:- [email protected] Page 11