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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
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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 – 1783 suggested the symbol 𝑖 = −1 [11]. For most of the 20th
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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
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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:
𝑐 = 𝑎 + 𝑖𝑏
(1)
Where 𝑐 is complex number, a and b are both real numbers and 𝑖 is −1 . 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
𝑎 + 𝑖𝑏
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 𝑎 + 𝑖𝑏 can be represented as a single point 𝑎 units along the real axis
and 𝑏 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 𝑧 and angle 𝜃 pointing from the
origin of the plane of the plane p (in fig 1.2) .
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Imaginary axis
p
𝑎+𝑏
z
𝜃
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
𝑐 = 𝑎 + 𝑖𝑏 = 𝑧∠𝜃
The relationships among the rectangular and polar coordinate terns 𝑎, 𝑏, 𝑧 𝑎𝑛𝑑 𝜃 are:
𝑎 = 𝑧 𝑐𝑜𝑠𝜃
(2)
𝑏 = 𝑧 𝑠𝑖𝑛𝜃
(3)
𝑧 = 𝑎2 + 𝑏2
(4)
𝑏
𝜃 = tan−1 𝑎
(5) The rectangular
coordinates to represent complex numbers .ach complex number consist of a pair of real
numbers(𝑎, 𝑏) .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 𝑐1 and 𝑐2 are defined as 𝑐1 = 𝑎1 + 𝑏1 and 𝑐2 = 𝑎2 + 𝑏2 then the addition,
subtraction, multiplication and division of 𝑐1 and 𝑐2 are defined as:
𝑐1 + 𝑐2 = 𝑎1 + 𝑎2 + 𝑏1 + 𝑏2 𝑖
𝑐1 − 𝑐2 = 𝑎1 − 𝑎2 + 𝑏1 − 𝑏2 𝑖
𝑐1 × 𝑐2 = 𝑎1 𝑎2 − 𝑏1 𝑏2 + 𝑎1 𝑏2 + 𝑏1 𝑎2 𝑖
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𝒄𝟏 𝒂𝟏 𝒂𝟐 + 𝒃𝟏 𝒃𝟐 𝒃𝟏 𝒂𝟐 − 𝒂𝟏 𝒃𝟐
=
+
𝒊
𝒄𝟐
𝒂 𝟐 𝟐 + 𝒃𝟐 𝟐
𝒂 𝟐 𝟐 + 𝒃𝟐 𝟐
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 −1
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 :
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 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,
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
We will get one of the following three possible solution sets.
1. Two real distinct (i.e. not equal) solutions.
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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 don’t 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 we’ll get. To see how this works let’s start off by recalling
the quadratic formula.
𝑥=
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
The quantity 𝑏2 − 4𝑎𝑐 in the quadratic formula is called the discriminant. It is the value of the
discriminant that will determine which solution set we will get. Let’s go through the cases one at a
time
1. Two real distinct solutions. We will get this solution set if 𝑏2 − 4𝑎𝑐 > 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 −𝑏 plus or minus a real
number. This means that the numerator will be two different real numbers. Dividing either
one by 2a won’t 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 𝑏2 − 4𝑎𝑐 = 0. Here we will be taking the
square root of zero, which is zero. However, this means that the “plus or minus” part of the
numerator will be zero and so the numerator in the quadratic formula will be −𝑏 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 𝑏2 − 4𝑎𝑐 < 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
“plus or minus” in front of the square root.
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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: 𝑥 2 + 3𝑥 + 2
Roots: -1,-2
Real and equal roots. Example: 𝑥 2 + 2𝑥 + 1
Roots: -1,-1
Complex roots. Example: 𝑥 2 + 2𝑥 + 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
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 → 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑦
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
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 I’ve drawn for the three
different examples we already seen.
Figure 1 : The graph of equation 𝑥 2 + 3𝑥 + 2 and its roots -1,-2
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Figure 2 : The graph of equation : 𝑥 2 + 2𝑥 + 1 and its roots -1,-1
Figure 3 : The graph of equation : : 𝑥 2 + 2𝑥 + 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: 𝑦 = 𝑥 2 + 2𝑥 + 1 and its complex roots -1+i and -1-i.
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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
𝑦 𝑡 = 𝑒 −0.2𝑡 (𝑐𝑜𝑠 𝑡 + 𝑖 𝑠𝑖𝑛 𝑡)
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 𝑦 𝑡 = 𝑒 −0.2𝑡 (𝑐𝑜𝑠 𝑡 + 𝑖 𝑠𝑖𝑛 𝑡) 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
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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
𝑦 𝑡 = 𝑒 −0.2𝑡 𝑐𝑜𝑠 2𝑡
𝑦 𝑡 = 𝑒 −0.2𝑡 𝑠𝑖𝑛 2𝑡
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
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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
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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
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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
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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).
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Sons, Inc. ISBN 0-471-54397-7
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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
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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. 439–445. ISBN 0-471-54397-7.
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several methods you may never have seen" (PDF). Retrieved 18 April 2013.
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