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19-Discriminant - = 11 22 = = = = = 1 1 1 1 =1 1 1 =1 1 = 1...

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± ± ± ± ± ± 8SHE]²%PXIVREXMZI 'PEWWM¿GEXMSR 1IXLSHW 6YPI &EWIH± -RWXERGI &EWIH 1IXLSHW ERH 2IEVIWX 2IMKLFSVW± (MWGVMQMRERX %REP]WMW² JSV GSRXMRYSYW I\TPEREXSV] ZEVMEFPIW SRP]±
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± ± ± ± ± ± (MWGVMQMREXMSR JSV 'SRXMRYSYW )\TPEREXSV] :EVMEFPIW (MWGVMQMRERX JYRGXMSRW EVI XLI IWWIRGI SJ XLI SYXTYX JVSQ E HMWGVMQMRERX EREP]WMW± (MWGVMQMRERX JYRGXMSRW EVI XLI PMRIEV GSQFMREXMSRW SJ XLI WXERHEVHMWIH MRHITIRHIRX ZEVMEFPIW [LMGL ]MIPH XLI FMKKIWX QIER HMJJIVIRGIW FIX[IIR XLI KVSYTW± -J XLI VIWTSRWI MW E HMGLSXSQ]³SRP] X[S GPEWWIW XS FI TVIHMGXIH´ XLIVI MW SRI HMWGVMQMRERX JYRGXMSRµ MJ XLI VITSRWI ZEVMEFPI LEW O PIZIPW³MI XLIVI EVI O GPEWWIW XS TVIHMGX´¶ YT XS O·¸ HMWGVMQMRERX JYRGXMSRW GER FI I\XVEGXIH¶ ERH [I GER XIWX LS[ QER] EVI [SVXL I\XVEGXMRK±
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± ± ± ± ± ± (MWGVMQMRERX *YRGXMSRW 7YGGIWWMZI HMWGVMQMRERX JYRGXMSRW EVI SVXLSKSREP XS SRI ERSXLIV¶ PMOI TVMRGMTEP GSQTSRIRXW¶ FYX XLI] EVI RSX XLI WEQI EW XLI TVMRGMTEP GSQTSRIRXW ]SY [SYPH SFXEMR MJ ]SY NYWX HMH E TVMRGMTEP GSQTSRIRXW EREP]WMW SR XLI MRHITIRHIRX ZEVMEFPIW¶ FIGEYWI XLI] EVI GSRWXVYGXIH XS QE\MQMWI XLI HMJJIVIRGIW FIX[IIR XLI ZEPYIW SJ XLI VIWTSRWI¶ RSX XLI XSXEP ZEVMERGI¶ FYX XLI ZEVMERGI FIX[IIR GPEWWIW± 8LI MRMXMEP MRTYX HEXE HS RSX LEZI XS FI GIRXIVIH SV WXERHEVHM^IH FIJSVI XLI EREP]WMW EW MW XLI GEWI MR TVMRGMTEP GSQTSRIRXW¶ XLI SYXGSQI SJ XLI ¿REP HMWGVMQMRERX EREP]WMW [MPP RSX FI EJJIGXIH F] XLI WGEPMRK±
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± ± ± ± ± ± (MWGVMQMRERX *YRGXMSRW % HMWGVMQMRERX JYRGXMSR¶ EPWS GEPPIH E GERSRMGEP VSSX¶ MW E PEXIRX ZEVMEFPI [LMGL MW GVIEXIH EW E PMRIEV GSQFMREXMSR SJ HMWGVMQMREXMRK ³MRHITIRHIRX´ ZEVMEFPIW¶ WYGL XLEX 0 = F 1 \ 1 + F 2 \ 2 + ... + F T \ T + G¶ [LIVI XLI F¹W EVI HMWGVMQMRERX GSIJ¿GMIRXW¶ XLI \¹W EVI HMWGVMQMREXMRK ZEVMEFPIW¶ ERH G MW E GSRWXERX± 8LMW MW WMQMPEV XS QYPXMTPI VIKVIWWMSR¶ FYX XLI F¹W EVI HMWGVMQMRERX GSIJ¿GMIRXW [LMGL QE\MQM^I XLI HMWXERGI FIX[IIR XLI QIERW SJ XLI GVMXIVMSR ³HITIRHIRX´ ZEVMEFPI± 2SXI XLEX XLI JSVIKSMRK EWWYQIW XLI HMWGVMQMRERX JYRGXMSR MW IWXMQEXIH YWMRK SVHMREV] PIEWX·WUYEVIW¶ XLI XVEHMXMSREP QIXLSH¶ FYX XLIVI MW EPWS E ZIVWMSR MRZSPZMRK QE\MQYQ PMOIPMLSSH IWXMQEXMSR±
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± ± ± ± ± ± 0IEWX 7UYEVIW 1IXLSH SJ IWXMQEXMSR SJ (MWGVMQMRERX *YRGXMSRW 8LI ZEVMERGI GSZEVMERGIW QEXVM\ GER FI HIGSQTSWIH MRXS X[S TEVXW² SRI MW XLI ZEVMERGI [MXLMR IEGL GPEWW ERH XLI SXLIV XLI ZEVMEFMPMX] FIX[IIR GPEWIW¶ SV [I GER HIGSQTSWI XLI WYQ SJ WUYEVIW ERH GVSWW TVSHYGXW ³XLI WEQI YT XS E GSRWXERX JEGXSV´ 8 = & + ; 8 = < ( - R - 4 1 R ) < & = < ( 4K - 4 1 R ) < FIX[IIR·GPEWW ; = < ( - R - 4K ) < [MXLMR - R MW XLI MHIRXMX] QEXVM\± 4¸R MW XLI SVXLSKSREP TVSNIGXMSR MR XLI WTEGI 1 R ± ³M±I± 4 1 R = 1 R 1 R / R ´± 7YGL XLEX ( - R - 4 1 R ) < MW XLI QEXVM\ SJ GIRXIVIH GEWIW± 4K MW XLI QEXVM\ TVSNIGXMRK SRXS XLI WYFWTEGI KIRIVEXIH F] XLI GSPYQRW SJ XLI FMREV] HMWGVMQMREXMRK QEXVM\ +± 8LMW QEXVM\ LEW K GSPYQRW ERH E SRI SR VS[ M ERH GSPYQR N MJ SFWIVZEXSR M FIPSRKW XS KVSYT N E 8E = E &E + E ;E±
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± ± ± ± ± ± 8LI ¿VWX HMWGVMQMRERX JYRGXMSR ³SV ZEVMEFPI SV E\MW´ MW XLI PMRIEV GSQFMREXMSR SJ XLI SVMKMREP ZEVMEFPIW XLEX QE\MQMWIW² E &E / E 8E
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