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THE FORMULA OF FINDING THE ROOTS
OF THE QUARTIC EQUATION
A RESEARCH
Presented to
Mr. Arnel M. Yurfo, MS MATH
Negros Oriental State University
Bayawan-Sta. Catalina Campus
Bayawan City, Philippines
In Partial Fulfillment
of the Requirements for the Subject
SPEC MATH 5 (FUNDAMENTALS OF MATHEMATICS)
Mike Loel P. Balbon
Lee Marie M. Marfiel
Elmo E. Labrador
Rommel A. Binalayo
Jonas S. Alcano
Elmer M. Baluran
Noel Q. Papasin
Sandro M. Herrero
March 2014
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ACKNOWLEDGEMENT
The researchers would like to express their profound gratitude
to the following people for
their invaluable contribution that helped in the completion of
this study:
To their families. They are the greatest team, friends and
companies who welcomed and
surrounded them with incredible love, prayer and support;
To their friends and classmates who have stood with them and
their families without question
and in absolute loyalty;
To Mr. Arnel M. Yurfo, an indescribable mentor who spent
countless hours encouraging and
helping them with this research paper. His examples and
perpetual optimism defied circumstances
and discouragement and gave them confidence that they can do
this.
To www.google.com for such a wonderful web browser.
To those who have loved, prayed and financially supported them
through these years, who have
gone with them in every trial, who have fought for them in
prayers and in strength.
Above all, the LORD ALMIGHTY, for His Divine Love and Mercy that
has kept the researchers
in good health and determination to accomplish their study.
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DEDICATION
To all the people along the roads of the world who have been
kind to us and our families. To
those who have fed us, given us place to sleep, rest, and loved
a stranger.
To Mr. Arnel M. Yurfo, our exemplar, whose encouragement and
motivation bolster us to pursue
our dreams.
Lastly, to all the NUMERICAN SOCIETY members of NORSU-BSC, whose
enthusiasm and
youthfulness serves as our morphine and antidote to stress,
discouragement and pain all
throughout.
- The FAO Group
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ABSTRACT
The main purpose of this study is to define and find the roots
of a quartic
equation with specified coefficients
In the course of history, there are five algorithms already
established to solve the general
quartic equation. But in order to create a formula in solving
special cases such as
, the researchers were able to formulate the FAO theorem.
The solutions presented in this paper are newly introduced by
the researchers based on the
solutions derived by some mathematicians. The researchers come
up with the following results:
a) ; b) ; c) ; d)
Moreover, no one can guarantee computational success to any
particular algorithms.
Excessive complications may be adverse especially polynomials in
the degrees higher than 3.
However, the theorem has been proven and examples are also
presented to support the results.
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TABLE OF CONTENTS
TITLE PAGE..i
ACKNOWLEDGEMENT..ii
DEDICATION...iii
ABSTRACT...iv
TABLE OF CONTENTS....v
LIST OF NOTATIONS..vi
CHAPTER 1. The Problem and its Scope.....1
Introduction.....1
Statement of the Problem....3
Significance of the Study.....3
Objectives........4
Scope and Limitation...4
Review on Related Literature..5
CHAPTER 2. Methodology and Basic Concept........ 9
Research Methodology.,...9
The general quartic formula.....9
CHAPTER 3. Results And Discussions............12
FAO Theorem.........12
Proving FAO Theorem...... 14
Examples... 17
CHAPTER 4. Conclusion and Recommendations...22
REFERENCES...23
CURRICULUM VITAE24
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LIST OF NOTATIONS
Notations
Name
Page
= Equal sign iv,1,2,4,5,7,8,9,10-21
Not equal iv,2,4,9
_ Subtraction, negative iv,1,2,5,7,8,10-21
+ Addition, positive iv,1,2,4,5,8-21
Square root 1,2,7,11,13,16,18,20
Element of 2,4,9
R Real Numbers 2,4,9
Greater than 9,14
imaginary 18,19,20,21
Plus minus 2,7
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CHAPTER 1
THE PROBLEM AND ITS SCOPE
Introduction
Equation is a statement of an equality between two expressions.
It is used in almost all
branches of pure and applied mathematics and in the physical,
biological, and social sciences.
A polynomial equation has the form;
in which the coefficients , ... an are constants, the leading
coefficient is not equal to zero,
and n is a positive integer. The greatest exponent n is the
degree of the equation. Equations of the
first, second, third, fourth, and fifth degrees are often
called, respectively, linear, quadratic, cubic,
biquadratic or quartic, and quintic equations.
Other important types of equations are algebraic, as in = = 7;
trigonometric, as
in sin ; logarithmic, as in ; and exponential, as
in
. Diophantine equations are equations in one or more unknowns,
usually with
integral coefficients, for which integral solutions are
sought.
A root is a value of x that when plugged into the polynomial
equation yields ;
a polynomial equation is solved when all the roots of the
equation have been found.
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We are all familiar with a linear equation or an equation of the
first degree and has only
one root. The single root of the linear equation is is
Likewise, the quadratic, or second-degree, equation ax2 + bx + c
= 0 has two roots, given by the
formula
Processes on solving the two equations above can be used with
complex numbers and
trigonometry to derive the formula in solving cubic and quartic
equations.
Quartic equations often referred to as quartics, in the form
where a is nonzero. However, they are the highest degree
polynomials which can be solved analytically, by radicals, with
no repetitive techniques.
According to history there were five and only five algorithms so
far has been discovered
on solving the general quartic equation. However, there are some
special cases which solutions
are yet to be discovered. Still emphasis has been placed on
finding the real roots of the quartic
equation in the form .
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Statement of the Problem
The main purpose of this study is to define and find the roots
of special case quartic
equation with specified coefficients
Significance of the Study
The most evident role of this study is to have a more profound
and correct understanding
on quartic equation. The general quartic equation has been
subject to great controversy on who
was able to derive the formula first. However, its sense and
significance remains limited and
obscure.
Finding the complete algorithm for the roots of the said
equation, this research will give
enough background information to make the key ideas on solving
accessible to non-specialists
and even to mathematically oriented readers who are not
professional mathematicians.
One of the applications of this study is in the field of
computer science. Quartic often arise
in computer graphics. It is a proven fact that in this field
proper algorithm is important to make
successful computer programs starting from the basic to the most
complicated ones. The results of
this study will help computer geniuses in creating programs with
minimal errors resulting to more
efficient and effective programs that will foster more learning
and eventually will aide in the total
development.
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Moreover, quartic often appears in problems involving
optimization. Optimization, also
known as mathematical programming is a collection of
mathematical principles and methods used
for solving quantitative problems in many disciplines, including
physics, biology,
engineering, economics, and business. The subject grew from a
realization that quantitative
problems in manifestly different disciplines have important
mathematical elements in common.
Because of this commonality, many problems can be formulated and
solved by using the
unified set of ideas and methods presented by this paper that
make up the field of optimization.
Furthermore, the results of this research may be used as basis
for more related studies and
may encourage other researchers to ponder more on this
topic.
Objectives
After this Mathematical research, the researchers are expected
to find out the special
formula of finding the roots of quartic equation , where
, and a, b, c, d, e .
Scope and Limitation
This study focuses only in creating the formula of a special
case quartic equation
with specified coefficient Thus,
it also covers some related studies of the said problem so that
we could come up to the exact
formula in finding its roots.
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Review on Related Literature
One of the landmarks in the history of mathematics is the proof
of the nonexistence of
solutions based solely on radicals and basic arithmetic
operations namely addition, subtraction,
multiplication, and division for solutions of general algebraic
equations especially of degrees
higher than four. Equations in general has been the most
interesting part of the history of
mathematics.
Prior to the 17th century, the theory of equations was
handicapped by the failure of
mathematicians to recognize negative or complex numbers as roots
of equations. Only ancient
Indian mathematicians, such as Brahmagupta, recognized negative
roots, and outside of India and
China negative coefficients of the polynomials were not
recognized. Instead of one type of
quadratic equation, as given above, there would be six different
types, depending on which
coefficients were negative.
In 1629 the French mathematician Albert Girard recognized both
negative and complex
roots of equations and so was able to complete the partial
insight of Franois Vite into the
relation between the roots of an algebraic equation and its
coefficients. Vite discovered that if a
and b are the roots of , then and
More generally, Vite showed that if the coefficient of the first
term of the equation
is unity, then the coefficient of the second term with its sign
changed equals the sum
of all the roots; the coefficient of the third term equals the
sum of all the products formed from the
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multiplication of two of the roots at a time; and the
coefficient of the fourth term with its sign
changed equals the sum of all the products formed from three of
the roots at a time. If the degree
of the equation is even, the last coefficient equals the product
of all the roots; if odd, the
coefficient with its sign changed equals the product of all the
roots. Vite also contributed
important numerical methods for approximating the roots of
equations.
In 1635 French philosopher and mathematician Ren Descartes
published a text on the
theory of equations, including his rule of signs for the number
of positive and negative roots of an
equation. A few decades later, English mathematician and
physicist Isaac Newton gave an
iterative method of finding roots of equations, it is known
today as the Newton-Raphson method.
At the end of the 18th century, German mathematician Carl
Friedrich Gauss proved that
every polynomial equation has at least one root. The question
remained, however, whether it is
possible to express that root by an algebraic formula involving
the coefficients of the equation, as
had been done for degrees one to four. A major step toward
answering this question was the idea
of French mathematician and astronomer Joseph Lagrange of
permuting the roots of an equation
to study its solutions. This fruitful idea led, in the work of
Italian mathematician Paolo Ruffini,
Norwegian mathematician Niels Abel, and French mathematician
variste Galois, to a complete
theory of polynomials that among other things, showed that a
polynomial could be solved by
means of a general algebraic formula only if the polynomial has
a degree less than five.
However, in the course of history, algebraic equation in the
fourth degree has been said to
be the highest degree polynomial that can be solve logically by
means of radicals; the solution for
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quadratic equation uses a single square root; the cubic equation
solution uses a square root inside
a cube root; although quartic equations uses the same means but
more complicated than the cubic.
The emphasis of this paper has been placed on finding the real
roots of the quartic equation
A concrete solution on quartic was discovered during work on a
problem proposed to
Cardan in 1540. After an unsuccessful attempt at solving this
equation, Cardan turned it over to
his follower Ludovico Ferrari (15221565). Ferrari, using the
rules for solving the cubic or the so
called resolvent cubic, eventually succeeded where his master
had failed. At least, Cardan had the
pleasure of incorporating the result in the Ars Magna, with due
credit given to Ferrari.
The central role of the resolvent cubic in the solution of the
quartic was appreciated by
Leonard Euler (17071783). Eulers quartic solution first appeared
as a brief section (S 5) in a
paper on roots of equations [1, 2], and was later expanded into
a chapter entitled Of a new method
of resolving equations of the fourth degree (S 773783) in his
Elements of algebra [3, 4].
Eulers quartic solution was an important advance, in which he
showed that each of the
roots of a reduced quartic can be represented as the sum of
three square roots, say
where the are the roots of a resolvent cubic.
As the cubic formula is significantly more complex than the
quadratic formula, the quartic
formula is significantly more complex than the cubic formula.
Wikipedia's article on quartic
equations has a lengthy process by which to get the solutions,
but does not give an explicit
formula.
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In fact, there are five algorithms being formulated for solving
general quartic equations.
However, none of these five would completely suit the needs of
stable computations.
The five algorithms are listed below:
1. Descartes-Euler-Cardanos one
(
) (
)
2. Ferrari-Langranges
3. Neumarks
4. Christianson- Browns
(
)
5. Yacoub-Fraidenraich-Browns
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CHAPTER 2
METHODOLOGY AND BASIC CONCEPT
Research Methodology
This section presents the steps the researcher had done to
formulate the results of the
study. First, they studied the formulas in solving the roots of
quadratic and cubic equations. Then,
solutions to quadratic and cubic equations are used in solving
the general quartic equations. The
researchers then have come up to the following steps in solving
the quartic equation in the form
with specified coefficients .
2.1.1 The General Quartic Formula
1. The general quartic formula:
If the value of a > 1, lets first divide all coefficients by
a so that will be equal to 1.
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Now, lets find the values of f, g and h
2. Then, plug these values to the following cubic equation:
3. Now, lets determine the values of p, q, and r. In getting the
values of this
variables, lets consider three cases.
Case 1. If the roots of the resolvent cubic are non-negative,
then the two nonzero roots
will be used to determine the values of p and q.
Case 2. If the roots of the resolvent cubic are two or three
imaginary numbers, then select
the two imaginary numbers to be used to determine the values of
p and q.
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Case 3. If two or three negative numbers exist in the roots of
the resolvent cubic, then you
will select the two integers to be used to determine the values
of p and q.
Now, substitute these selected roots to the following
formulas,
4. The four roots are:
CHAPTER 3
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RESULTS AND DISCUSSIONS
This chapter presents the main result if this study.
The solution to the quartic equation is derived from the
General
quartic formula by eliminating all b and c. After numerous
attempts, the researchers come up with
the following formula and solutions.
3.1 FAO Theorem
If , divide all coefficients by so that the value of ,
There exist g and h:
Then,
1. Plug these values to the following cubic equation:
2. Now, lets determine the values of p, q, and r. In getting the
values of this variables,
lets consider three cases.
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Case 1. If the roots of the resolvent cubic are non-negative,
then the two nonzero roots
will be used to determine the values of p and q.
Case 2. If the roots of the resolvent cubic are two or three
imaginary numbers, then select
the two imaginary numbers to be used to determine the values of
p and q.
Case 3. If two or three negative numbers exist in the roots of
the resolvent cubic, then you
will select the two integers to be used to determine the values
of p and q.
Now, substitute these selected roots to the following
formulas
3. The four roots are:
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3.2 Proving FAO Theorem
In proving this formula the researchers used the general quartic
formula. Since, the
researchers assume that b = 0 and c = 0, the researchers can
conclude that if they remove b and c
from the general quartic formula, then they could come up with
the right formula that satisfies the
restriction.
Proof.
1. The general quartic formula:
If the value of a > 1, lets first divide all coefficients by
a so that . Now, lets
proceed in finding the values of f, g and h:
since b = 0 and c = 0, then we can conclude that
Likewise, the value of g is
eliminating all b and c will result to:
Consequently, the value of h is
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Since b and c = 0, then researchers can eliminate all the b and
c from the formula and gives
2. The cubic equation:
Since f = 0,
This then gives us,
3. Determine the values of p, q, r and s.
Case 1. If the roots of the resolvent cubic are non-negative
real numbers, then the two
nonzero roots will be used to determine the values of p and
q.
Case 2. If the roots of the resolvent cubic are two or three
imaginary numbers, then
select the two imaginary numbers to be used to determine the
values of p and q.
Case 3. If two or three negative numbers exist in the roots of
the resolvent cubic, then
you will select the two negative numbers to be used to determine
the values of p and q.
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Since b = 0, then this will give us
4. The four roots are:
Since s = 0, we can remove this variable from the formulas. This
then gives us:
Examples
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Solution:
Lets determine first the values of g and h.
Then, lets plug this values to the cubic equation:
Solving the roots by cubic equation calculator gives:
Now, lets determine the values of p, q and r. Since there are
negative roots which fall in
our 3rd
case. Therefore we are going to use and to determine the values
of p and q.
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Then, the four roots are,
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+
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Solution.
Lets divide first the numerical coefficients of the equation by
3 so that a is equal to 1.
This then gives us
Now, lets find the values of g and h,
Then, plug these values to the cubic equation:
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Solving by cubic equation calculator gives:
Next, lets determine the values of p, q and r.
= 0.117873888i
Then, the four roots are:
0.8446456519
0.8446456519
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0.8446456519
0.8446456519
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CHAPTER 4
CONCLUSION AND RECOMMENDATIONS
The general quartic equation is said to be the highest degree
polynomials which can be
solved logically by means of radicals and complex numbers.
However, the process we have to
undergo is very extensive that nobody could guarantee a
computational success. Excessive
complications may be adverse.
We may guess that the five previously published methods have
been elaborated so as to
attach some algebraic quality to the resolvents coefficients,
but their robustness in actual
computations should be a subject of further analysis. The same
principle is applied to the solution
derived by the researchers.
Furthermore, there are several cases about quartic equations
that are left insoluble. The
researchers thereby recommend for a more comprehensive study
regarding this topic.
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REFERENCES
Herbison-Evans, D. (2005). Solving Quartics and Cubics for
Graphics. Retrieved
January 15, 2014 from
http://linus.it.uts.edu.au/~don/pubs/solving.html.
Strong, T. (1859). Elementary and Higher Algebra. Pratt and
Oakley.
Neumark, S. (1965). Solution of Cubic and Quartic Equations,
Pergamon Press, Oxford.
Brown, K.S. (1967). Reducing Quartics to Cubics. Retrieved
January 15, 2014 from
http://mathpages.com/home/kmath296.htm.
Smith, J. T. (2013).Cubic and Quartic Formulas.
Burton, D. M. (2011). The History of Mathematics: An
Introduction, seventh ed., McGraw-
Hill
Companies, Inc., New York.
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