Fight Over Solving the Cubic

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1 Solutio Coetry: Fight Over Solvig the Cui Solutio of Mi Proles: 1. The ui equtio 6 0 shoul hve roots, with t lest oe of the eig rel. The grph of f() 6 0 is: A y usig either SOLVE or TRACE, we isover tht the is root.. For the se the -- se, (the oe rel root with the other two roots eig ople ojugtes). Also, the ft shoul shoul pper twie shoul ot e surprise sie the se is equivlet to the - - se. As to the other two sig optios, the other two ses re etreous roots rete y the lgeri ipultios. For the - se, , while for the - se, O tht if you the two flse roots, you get oule the tul root!. The equtio is 10. The steps re Let u v, whih y sustitutio iplies tht (u-v) (u-v) 10 or u u v uv v u v 10 Choose prout uv 1/ oeffiiet of -ter, or uv (1/)() 1 Sustitute uv 1 ito the ui epressio ivolvig u v to get u -(1)u (1)v v u -v 10, whih reues to u v 10 But, v 1/u iplies tht u 1/u 10 or (u ) 10(u ) 1 0 Sie this e solve s qurti, we hve 10± ± 6 u 5± 6 or Siilrly, usig u /v, we oti v 5 ± Thus, u - v 5 ± 6-5 ± 6 The se ses give the rel root of The - se gives , while the - se gives Agi, ½ of the su of the two flse roots equls the rel root! 6 u 5 ± 6

2 Etesio 1: For the equtio 15 4 or , the steps re: Let u v, whih y sustitutio iplies tht (u-v) -15(u-v) -4 0 or u u v uv v -15u 15v -4 0 Choose prout uv 1/ oeffiiet of -ter, or uv (1/)(-15) -5 Sustitute uv -5 ito the ui epressio ivolvig u v to get u -(-5)u (-5)v v -15u 15v 4 0, whih reues to u v 4 0 But, v -5/u iplies tht u 15/u -4 0 or (u ) 4(u ) 15 0 Sie this e solve s qurti, we hve 4± ± 11 u ± 11 or Siilrly, usig u -5/v, we oti v ± 11 u ± 11 Thus, u - v ± 11 - ± 11 or ± 11 11, whih ivolve tkig ui roots of ople epressios. The grph shows three rel roots, whose vlues re , , 4: Etesio : This tehique ws suggeste y Rfel Boelli, Itli thetii, i his Alger i the 1550s. Assue tht eh ui root portio of the epressio for the solutio is ople uer, i.e. 11 i 11 pqi. The ( 11 ) (i) or 11 i i i, whih i siplifie ottio eoes 11i ( ) ( )i. Settig rel ople prts equl, 11, or y the istriutive lw, ( ) 11 ( ). Tryig soe siple ftor oitios, let 1, whih y sustitutio iplies tht ± 1. Testig oth optios of, 1, -1 i the epressio 11 ( ), we fi tht oly, 1 works 11 i. Usig the se pproh for evlutig 11 pqi, we isover tht p q i. Thus, perhps to our surprise, (i) (-i) 4. Etesio : Trtgli s proess is revele eoes geerl: Epig the left sie of the lgeri epressio, (u-v) uv(u-v) u - u v uv v u v uv u v.

3 Sustitutig uv, u v, u v ito the estlishe ietity, we reisover Trtgli s prole of solvig. Thus, Trtgli s pproh uses the ietity i reverse. Solvig the syste uv u v, we hve v /(u) thus u - [/(u)]. Siplifyig, (u ) u (/) 0, whih y the qurti forul yiels u. Usig the sustitutio u /(y) les siilrily to v. But, sie u-v, we hve estlishe the lssil Cro-Trtgli forul of -. Sustitutig y / ito the geerl ui equtio y y y 0, les to the followig sequee: Whih gives the esire vlues of 7 i the kow equtio, whih oul the e solve y the Cro Trtgli forul. Note: Eves (198, pp ) eserves the thks for this ie evelopet. Ope-Ee Eplortio: Oe stuets eterie tht resole isriit is D the eisio riteri eoes: D>0 iplies oe rel two ople roots D0 iplies either oe rel triple root or two rel roots ( sigle oule) D<0 iplies three rel roots They test their hoie usig the two equtios 6 0 (I rel root, ople roots) 15 4 ( rel roots). Multiple referees evelop this ie of ui

4 isriit, iluig ger.ht, 7. Teher Coetry: Eves (198) Duh (1990) oth tell the fsitig story of the ui fight quite well. However, voi spoilig stuets eplortio of the suggeste proles y elyig your shrig of these referees with stuets. Reig the fter eplorig the proles will provie goo follow-up, eig review of the tsks tehiques ivolve. The iteret soure MthDL yi1491 gives very thorough presettio o the roer history tehiques for solvig the ui. Whe the geerl Cro-Trtgli equtio is estlishe i Etesio Prole #, eourge stuets to re-erive their solutios to the two equtios Also, stuets shoul test their isriit D o these two equtios to ofir the root possiilities. O Cro s ostt urgig o Mrh 5, 159, Trtgli filly ivulge his etho for solvig the equtio, ut i so i the for of rypti poe: Whe the ue the thigs together Are equl to soe isrete uer, Fi two other uers ifferig i this oe. The you will keep this s hit Tht their prout shll lwys e equl Etly to the ue of thir of the thigs. The reier the s geerl rule Of their ue roots sutrte Will e equl to your priipl thig. Ask stuets to try to eipher the poe i ters of Trtgli s solutio tehique. Other trsltios of the poe eist, log with oetry (e.g. yi1496.) Prole #, whe the possiility of four roots pper for ui ppers, provies gret opportuity to eplore the ie of etreous roots. They re ofte rete y lgeri ipultios of either frtiol or ril equtios. Mig (1970) provies goo itroutio to the worl of etreous roots its history. As goo soure of writig projets, stuets eplore y of the followig ies reltive to the solutio of the ui higher egree equtios:

5 Duh (1990) suggests: Cotrry to populr elief, igiry uers etere the rel of thetis ot s tool for solvig qurtis ut s tool for solvig uis. Iee, thetiis oul esily isiss 11 whe it ppere s solutio to 11 0 (for this equtio hs o rel solutios). But they oul ot so esily igore 11 whe it plye pivotl role i yielig the solutio 4 for the previous ui. So it ws uis, ot qurtis, tht gve ople uers their iitil ipetus their ow-uispute legitiy. Do you gree with this li regrig the epte of igir y uers? Copriso resoures re Nhi (1998) Ivestigte further the ui ttle, iluig its otetul history, the tul otest with its questios, the susequet evets, the historil iplitios. A gret ut reltively ukow resoure is Norgr (197). Ivestigte the susequet historil steps i the serh for lose forul for solvig the qurti equtio, the quiti equtios, et. I itio to Cro Ferrri, key thetiis ivolve re Leiiz, Euler, Lgrge, Guss, Ruffii, Ael, Glois. Soe goo tet resoures re Deryshire (006) Kleier (007), plus these sple iteret resoures Ivestigte the geoetril eig of usig y /() to epress the geerl ui equtio y y y 0 ito the for with -ter. Tht is, wht hppes geoetrilly to the grph of the ui? Wht hppes to the roots? C the tehique e etee to qurtis? Two goo resoures re Stuets with strog kgrou i strt lger shoul ivestigte the speil ture of the quiti its solutio itys geoetry. Shur (1997) is the str guie, ut it shoul e opleete y Aitiol Referees: Deryshire, J. (006). Ukow Qutity: A Rel Igiry History of Alger. Joseph Hery Press. Duh, W. (1990). Cro the solutio of the ui. Jourey Through Geius: The Gret Theores i Mthetis. Joh Wiley & Sos. pp

6 Eves, H. (198). A etroriry izrre story i Gret Moets i Mthetis: Before pp Kleier, I. (007). A History of Astrt Alger. Birkhäuser. Nhi, P. (1998). A Igiry Tle : The Story of 1. Prieto Uiversity Press. Mig, K. (1970). A history of etreous solutios. Mthetis Teher. Ferury, pp Norgr, M. (197). Sielights o the Cr-Trtgli otroversy. Ntiol Mthetis Mgzie. Vol. 1#7, pp Shur, J. (1997). Geoetry of the Quiti. Joh Wiley Sos.

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