[...] implying (p^sup 2^ - 3q)m^sup 2^ + (pq - 9r)m + q^sup 2^ - 3pr = 0. The parameter m is determined such that the coefficients of equation (2.10) satisfy condition (2.3). [...] equation (2.11) of m is obtained (the quadratic resolvent of equation (1.1)). 3.19) We calculate the value of parameter m in order that the coefficients of equation (3.19) satisfy condition (3.6. [...] 3m^sup 3^ + 7m^sup 2^ + 4m - 4 = 0.
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: firstname.lastname@example.org 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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