# A Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION

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1 Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi, Urmi, Irn Deprmen of Elecricl nd Compuer Engineering, Universi of Tbri, Tbri, Irn Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi, Urmi, Irn Absrc: Polnomils of high degrees ofen pper in mn problems such s opimiion problems. Equions of he fourh degree or so clled qurics re one pe of hese polnomils. Since now here is no n simple mehod o solve he generl forms of quric equions. In his pper we propose novel, simple nd precise nlicl mehod o solve quric equions wihou n consrins. Ke words: Polnomils, Qurics, Equions of he Fourh Degree INTRODUCTION A quric funcion is polnomil of degree four. The generl form of quric funcion is s follows: f ( ) b c d e, 0 () Seing f ( ) 0 resuls in quric equion of he form wrien below: b c d e 0, 0 () The quric is he highes order polnomil equion h cn be solved b rdicls in he generl cse (I. Sewr, 00) (i.e., one where he coefficiens cn ke n vlue). In 50 Lodovico Ferrri discovered he soluion o he quric equion (J.J. O'Connor nd E. F. Roberson). Like ll lgebric soluions of he quric equions, his mehod required he soluion of cubic equion o be found. The generl form of cubic equion is s follows: b c d 0 0, () Gerolmo Crdno proved h he soluion of he generl cubic funcion in equion () is (N. Jcobson, 009): b q q p q q p 7 7 () Where: b p c q 7 b bc d (5) B hving, wo oher soluions for equion (), nd, cn esil be found. The mos imporn poin in ll of he mehods for solving quric equion is he complei of hese soluions. Some mehods ssume severl consrins in order o simpl led o he response. In his pper we hve proposed novel, simple nd precise soluion for solving ll generl forms of quric equions wihou n consrins. In secion we re going o describe our simple generl soluion nd in secion we will provide he soluion wih wo emples. Finll conclusions re summried secion. Corresponding Auhor: Amir Fhi, Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi, Urmi, Irn E-mil: fhi.mir@homil.com

2 Aus. J. Bsic & Appl. Sci., 6(6): -6, 0. The Proposed Simple Mehod o Solve The Qurics: Assume he generl form of he quric shown in equion (6): b c d e 0 0, (6) Le b The equion (6) leds o: p q s 0 (7) In which p, q nd s re defined s below: b p c 8 q s b 8 bc d b b c bd e 56 6 (8) In order o solve he equion (7) ssume equion (9) wih wo vribles (B nd D) wih relionship shown in equion (9): ( B) ( D) (9) Since equion (9) cn be solved esil, we cn r o se he equion (7) equl o he equion (9). Then from equion (9) we hve: ( B B ) ( D D ) Soring he equion bove leds o: (B ) D B D 0 (0) In order o se he equions (0) nd (7) equl, he epressions wrien below should be rue: p B B ( p )/ q D D q / () And o se: s B D () Vrible chnges in equion () should be pplied: () Replcing equion () in equion (7) leds o:

3 Aus. J. Bsic & Appl. Sci., 6(6): -6, 0 p q s 0 p q s ( ) ( ) 0 p c q d s e eb D So: ( c) d B, D s p q eb D [( ) ( ) ] / q s p p 6 p ( p s) q 0 ( ) p( ) ( p s) q 0 () Replcing rbirr one of he roos of equion () in equion () he equion () will be se, hen he equion (0) will be equl o he equion (7). Therefore finding he roos of equion (0) from equion (9), he roos of equion (7) will be found esil.. Emples: Emple : Find he roos of 0. Soluion: B consrucing equion () for his emple, will be found: (5) Replcing in he given equion in he emple leds o: / /8 /6 0 / / /8 0 B ( /)/ D q //8 ( /8) ( /8) Tking squre roo from he equion bove:

4 Aus. J. Bsic & Appl. Sci., 6(6): -6, 0 /8 ( /8) /8 ( /8), / 0 ( () ) / ( i) / So wo roos of he given equion in his emple re: i,, So: In he oher hnd we hve: /8 ( /8) /0, ( ())// So oher wo roos of he given equion in he emple re:,,, Noe h, is double roo. Therefore he roos of given equion is found simpl s follows: i i, Noe h for he oher five roos of equion (5) sme resuls will be obined. Emple : Find he roos of i i (58 ) 8 0. Soluion: Firs he coefficien of is se o ero b he vrible chnge of: Wih his vrible chnging he given equion chnges o: i (6) In his equion we hve: s 0, q i, p 7 In order o find he equion () should be consruced. Afer consrucing he equion () nd finding hen he given equion will chnge o he desired equion (9). The form equion () for his emple is:

5 Aus. J. Bsic & Appl. Sci., 6(6): -6, ( ) 5( ) (7) Afer solving his equion of degree 6 b simplifing is form ino n equion of degree we hve: i (This is one of he soluions for equion (7) ) i B mking his vrible chnge ino he equion (6) we hve: In his equion: 7 B ( ) D 7 ( ) ( 7) So: i 5i i i And he soluions for he given equion will be found s following: i 5i i i Conclusions: The mos imporn poin in ll of he mehods for solving quric equion is he complei of hese soluions. Some mehods ssume severl consrins in order o simpl led o he response. In his pper we hve proposed novel, simple nd precise soluion for solving ll generl forms of quric equions wihou n consrins. Wih he proposed iniied mehod ll forms of he qurics cn be solved esil. Noe h here is no n pproimion in he resuled nswers of his mehod, nd ll of he nswers re precise. In his mehod quric is rnsformed o n equion of degree wo h cn esil be solved. In he proposed mehod he coefficiens of he resuled equion of degree wo re esil found from solving n equion of degree hree. To prove he efficienc nd simplici of he proposed mehod n emple quric is given in he hird secion of he pper nd i is solved wih he proposed iniied mehod. 5

6 Aus. J. Bsic & Appl. Sci., 6(6): -6, 0 REFERENCES Abrmowi, M. nd I.A. Segun, 97. (Eds.). "Soluions of Quric Equions.".8. in Hndbook of Mhemicl Funcions wih Formuls, Grphs, nd Mhemicl Tbles, 9h prining. New York: Dover, pp: 7-8. Berger, M., in Geomer I. New York: Springer-Verlg. Beer, W.H., 987. CRC Sndrd Mhemicl Tbles, 8h ed. Boc Ron, FL: CRC Press, p:. Beer, W.H., 987b. Hndbook of Mhemicl Sciences, 6h ed. Boc Ron, FL: CRC Press. Birkhoff, G. nd S.A. Mc Lne, 996. Surve of Modern Algebr, 5h ed. New York: Mcmilln, pp: Borwein, P. nd T. Erdéli, 995. "Quric Equions."..E.e in Polnomils nd Polnomil Inequliies. New York: Springer-Verlg, p:. Boer, C.B. nd U.C.A. Merbch, 99. Hisor of Mhemics, nd ed. New York: Wile, pp: Ehrlich, G in Fundmenl Conceps of Absrc Algebr. Boson, MA: PWS-Ken. Fucee, W.M., 996. "A Geomeric Inerpreion of he Soluion of he Generl Quric Polnomil." Amer. Mh. Monhl, 0: Geller, W., S. Gowld, M. Hellwich, H. Käsner nd H. Künsner, 989. (Eds.). VNR Concise Encclopedi of Mhemics, nd ed. New York: Vn Nosrnd Reinhold. Hewinkel, M., 988. (Mnging Ed.). Encclopedi of Mhemics: An Upded nd Annoed Trnslion of he Sovie "Mhemicl Encclopedi." Dordrech, Neherlnds: Reidel. Sewr, I., 00. "Glois heor," ed: Chpmn & Hll/CRC Mhemics. O'Connor, J.J. nd E.F. Roberson, "Lodovico Ferrri," in The McTuor Hisor of Mhemics rchive, ed. School of Mhemics nd Sisics, Universi of S Andrews Scolnd. MhPges "Reducing Qurics o Cubics." hp:// Jcobson, N., "Bsic lgebr,", nd ed: Dover, 009. Smih, D.E.A., 99. Source Book in Mhemics. New York: Dover. vn der Werden, B.L., in Algebr, Vol.. New York: Springer-Verlg. 6

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