THE ONE METHOD TO FIND ROOTS OF EQUATIONS
KairdenBayetov, HasantaevaNodira, ShaimovaZhazira.
Department of mathematics, Physical-mathematical faculty, Kazakh StateWomen’sTeacher
Training University, Almaty city
ABSTRACT
In this article, we describe a method for solving the fourth-degree equation (the quartic), applies
it in mathematical logic to find sets. A step-by-step detailed solution of two equations is
given.Keywords: Method, equation, solution.
АННОТАЦИЯ В данной статье описывается метод решения уравнения 4-й степени,
применение его в математической логике для нахождения множеств. Приведено
пошаговое подробное решение, двух уравнений.
Ключевые слова: Метод, уравнение, решение.
ТҮЙІНДЕМЕ Бұл мақалада 4-ші дәрежелі теңдеуді шешудің әдістемесі және
математикалық логикада жиынды табу үшін қарастырылады. Екі теңдеудің рет-ретімен
толық шешу жолы көрсетілген.
Түйін сөздер: Әдіс, теңдеу, шешім.
1.Introduction
Lodovico Ferrariis credited with the discovery of the solution to the quartic in 1540, but since
this solution, like all algebraic solutions of the quartic, requires the solution of acubicto be found,
it could not be published immediately.[1]The solution of the quartic was published together with
that of the cubic by Ferrari's mentorGerolamoCardanoin the bookArs Magna.[2]
The Soviet historian I. Y. Depman claimed that even earlier, in 1486, Spanish mathematician
Valmes wasburned at the stakefor claiming to have solved the quartic equation.[3]Inquisitor
GeneralTomás de Torquemadaallegedly told Valmes that it was the will of God that such a
solution be inaccessible to human understanding.[4]HoweverBeckmann, who popularized this
story of Depman in the West, said that it was unreliable and hinted that it may have been
invented as Soviet antireligious propaganda.[5] Beckmann's version of this story has been widely
copied in several books and internet sites, usually without his reservations and sometimes with
fanciful embellishments. Several attempts to find corroborating evidence for this story, or even
for the existence of Valmes, have failed.[6]
The proof that four is the highest degree of a general polynomial for which such solutions can be
found was first given in theAbel–Ruffini theoremin 1824, proving that all attempts at solving the
higher order polynomials would be futile. The notes left byÉvariste Galoisprior to dying in a
duel in 1832 later led to an elegantcomplete theoryof the roots of polynomials, of which this
theorem was one result.[7]
2. Method development
Quartic equations in one variable are equations in which the variable is raised to the 4th power in
at least one term. In general, they are equations of the form
𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒 = 0
(1)
where 𝑎 ≠ 0.
Quartic equations are sometimes also referred to as bi-quadratic equations.
The factorization method can be used to solve quartic equations just like it can be used to solve
quadratic or cubic equations. Instead of applying it once (as in quadratic equations) or twice (as
in the case of cubic equations), we need to apply the method thrice in the case of quartic
equations to solve them. In general, the number of applications of the factorization method is one
less than the degree of the equation that we are trying to solve.
Since the factorization method is familiar to us already, this lesson will cover the basics needed
to set the stage for their application to quartic equations.
Fourth-degree polynomials, equations of the form
𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒 = 0
wherea is not equal to zero, are called quartic equations. If you divide both sides of the equation
by a you can simplify the equation to
𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒 = 0
A quartic equation with real number coefficients can have either four real roots, two real roots
and two complex roots, or four complex roots. Complex roots occur in conjugate pairs. To solve
a general quartic equation, you need to solve associatedcubicandquadraticequations in a multistep process. Certain special quadratics can be solved with simpler methods.
3.Results
Example 1.Solve the quartic equation given a set B, is the set of roots of the equation
x 4  x 3  x 2  x    0 .
B  x1 , x 2 , x3 ...
In the bellow table we given all the coefficient of equation


-12
-2


27
18
Solution:
x 4  x 3  x 2  x    0
First of all the values we need to substitute, we get
x 4  (2) x 3  (12) x 2  18x  27  0
we need to convert our signs before coefficients
x 4  2 x 3  12 x 2  18 x  27  0
We break up the following terms in preparation for the factorization
x 4  9 x 2  2 x 3  18 x  3x 2  27  0
Thus, we have found the correct factorization to proceed forward:
x 2 ( x 2  9)  2 x( x 2  9)  3( x 2  9)  0
We obtain the formula
( x 2  9)( x 2  2 x  3)  0
Using the quadratic equation, the roots of the first equation are -3 and 3, and the roots of the
second are 3 and -1. These four roots are the roots of the original quartic.
x2 9  0
x2  9
x 2  2x  3  0
x3 
2  2 2  4 1 (3)
4
x 1  3
2
2

24
2
x3  3
x4  1
Answer B   3,1,3
Example 2. Solve the quartic equation given a set B, is the set of roots of the equation
x 4  x 3  x 2  x    0 .
B  x1 , x 2 , x3 ...
In the bellow table we given all the coefficient of equation

0


-17
36

-20
x 4  x 3  x 2  x    0
x 4  0  x 3  17 x 2  36 x  20
Step 1. Divide the first term of x 4  0  x 3  17 x 2  36 x  20 by first term of x 1 and we write
x4
the result
 x3
x
x 4  0  x 3  17 x 2  36 x  20 x  1
Step 2. Multiply x 1 by x 3 and subtract the result from x 4  0  x 3  17 x 2  36 x  20 , remainder
we will write to down
x 4  0  x 3  17 x 2  36 x  20 x  1
x4  x3
x3
x 3  17x 2
Step 3. Divide x 3  17x 2 by x . The result is the second term of the quotient
x3
 x 2 and
x
multiply x 1 by x 2 and subtract from x 3  x 2
x 4  0  x 3  17 x 2  36 x  20 x  1
x4  x3
x3 + x2
x 3  17x 2
x3  x2
 16 x 2  36 x
Step 4. Divide  16 x 2  36 x by x .
The result is the third term of the quotient
subtract from  16 x 2  16 x
x 4  0  x 3  17 x 2  36 x  20
x4  x3
 16 x 2
 16 x and multiply x 1 by  16x and
x
x 1
x 3 + x 2  16 x
x 3  17x 2
x3  x2
 16 x 2  36 x
 16 x 2  16 x
20x  20
Step 5. Divide 20x  20 by x .
The result is the fourth term of the quotient
from 20x  20
x 4  0  x 3  17 x 2  36 x  20
x4  x3
20 x
 20 and multiply x 1 by 20 and subtract
x
x 1
x 3 + x 2  16 x +20
x 3  17x 2
x3  x2
 16 x 2  36 x
 16 x 2  16 x
20x  20 20x  20
0
Here, x 3 + x 2  16 x +20 is the quotient and
x 4  0  x 3  17 x 2  36 x  20 =( x 1 )( x 3 + x 2  16 x +20)
Since every cubic equation has at least one real root, you can find a suitable value of p to resolve
the quartic. After you plug in the value of p, you take the square root of both sides to create two
quadratic equations. This gives you a total of four solutions
The remainder is zero so x 4  17 x 2  36 x  20 is divisible by x 1
x 3 + x 2  16 x +20
x 3  2x 2
x2
x 2  3x  10
3 x 2  16 x
3x 2  6 x
 10 x  20
 10 x  20
0
x 2  3x  10  0
This can either be factorized or solved using the quadratic formula. Either way, we get the two
roots as x = -5, x = 2.
Hope you will take the time to practice some of these methods so that you become completely
proficient at these techniques.
4.Conclusion
The above examples combined have given us a good introduction to the use of factorization to
solve equations of practically any degree. In this examples, we also saw how to use the
factorization method to divide a given equation by a given root to get the residual equation.
5.Referenses
1.O'Connor, John J.;Robertson, Edmund F.,"Lodovico Ferrari",MacTutor History of
Mathematics archive,University of St Andrews.
2.Cardano, Gerolamo(1993) [1545],Ars magna or The Rules of Algebra, Dover,ISBN0-48667811-3
3.Depman (1954),Rasskazy o matematike(in Russian), Leningrad: Gosdetizdat
4.P. Beckmann (1971).A history of π. Macmillan. p. 80.
5.Beckmann (1971).A history of π. Macmillan. p. 191.
6.P. Zoll (1989). "Letter to the Editor".American Mathematical Monthly.96(8): 709–
710.JSTOR2324719.
7.Stewart, Ian,Galois Theory, Third Edition(Chapman & Hall/CRC Mathematics, 2004)
8.O'Connor, John J.;Robertson, Edmund F.,"Abu Ali al-Hasanibn al-Haytham",MacTutor
History of Mathematics archive,University of St Andrews.
9.MacKay, R. J.; Oldford, R. W. (August 2000), "Scientific Method, Statistical Method and the
Speed of Light",Statistical Science,15(3): 254–78,doi:10.1214/ss/1009212817,MR1847825
10.Neumann, Peter M.(1998), "Reflections on Reflection in a Spherical Mirror",American
Mathematical Monthly,105(6): 523–528,doi:10.2307/2589403,JSTOR2589403
11.Aude, H. T. R. (1949), "Notes on Quartic Curves",American Mathematical Monthly,56(3):
165,doi:10.2307/2305030,JSTOR2305030
12.Rees, E. L. (1922). "Graphical Discussion of the Roots of a Quartic Equation".The American
Mathematical Monthly.29(2): 51–55.doi:10.2307/2972804.JSTOR2972804.