# Elementary methods for solving equations of the third degree and fourth degree by ProQuest

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```									  Scientia Magna
Vol. 5 (2009), No. 4, 5-13

Elementary methods for solving equations of the
third degree and fourth degree
Fatmir M. Shatri

Department of Mathematics, University of Prishtina, Kosova
E-mail: fushashkencore@gmail.com

Abstract In this paper we will establish some new methods for solving third and fourth
degree equations.
Keywords Cardano method, Ferrari method, equation.

§1. Introduction
In this paper, using new transformations, we will introduce new elementary methods for
solving third and fourth degree equations, which essentially diﬀer from Cardano’s method and
Ferrari’s method. We will present the importance of these transformations through some ex-
amples.

§2. Equations of the third degree
Let
x3 + px2 + qx + r = 0                                 (2.1)
be an equation of the third degree. Usually, it is solved by the means of Cardano’s method.
Let us start from the following deﬁnition:
Deﬁnition 1. The equation of form

X3 + a = 0                                      (2.2)

is called binomial equation of the third degree.
The following theorem gives the condition to be satisﬁed by the coeﬃcients of equation
(2.1) in order that (2.1) becomes in the form (2.2).
Theorem 2. Necessary and suﬃcient condition in order that equation (2.1) be an equation
of form (2.2) is that its coeﬃcients satisfy the condition

p2 − 3q = 0.                                     (2.3)

Proof of Theorem 2. (Necessary Condition) Substituting x = m + y, in equation (2.1),
where y is a new variable, we have

y 3 + (3m + p)y 2 + (3m2 + 2pm + q)y + m3 + pm2 + qm + r = 0.              (2.4)
6                                         Fatmir M. Shatri                                    No. 4

In order that equation (2.4) be a binomial equation of y, it is necessary that its coeﬃcients
satisfy conditions
3m + p = 0, 3m2 + 2pm + q = 0.                            (2.5)
From the ﬁrst equation of (2.5) we have
p
m=− .                                            (2.6)
3
Substituting the value of m from (2.6) in the second equation of (2.5) we obtain condition (2.3).
(Suﬃcient Condition). Let the coeﬃcients of equation (2.1) satisfy condition (2.3). Then
for equation (2.1) we have
p2
x3 + px2 + x + r = 0.                                    (2.7)
3

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