libro polochic matemática 2 º semestre 2016
TRANSCRIPT
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4. Hmfbokkgrmt` - Nrup` U`k`fbofVgnuid` sgjgstrg - ONGX=5
Jmtgjtofm
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4. Hmfbokkgrmt` Nrup` U`k`fbofVgnuid` sgjgstrg ONGX
Jmtgjtofm
=5
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Jmtgjtofm =5
Vgnuid` sgjgstrg
Xgsgrvmd`s t`d`s k`s dgrgfb`s. ]ugdm ronur`smjgitg pr`bohodm km rgpr`duffoi t`tmk ` pmrfomkdg gstg jmtgromk gdufmtov`, p`r fumkquogr jgdo` ` pr`fgdojogit`, soi km mut`rozmfoi dgk Oistotut`Numtgjmktgf` dg Gdufmfoi Xmdo`aiofm, ONGX. Vgni mrtfuk` 48 dg km F`istotufoi U`ktofm dgNumtgjmkm qug sg rgagrg m km mut`rm.
Oistotut` Numtgjmktgf` dg Gdufmfoi Xmdo`aiofm, ONGX.Gs uim `hrm pr`dufodm p`r gk Dgpmrtmjgit` dg Xgdmffoi y Dosg`, pmrm gk
Oistotut` Numtgjmktgf dg Gdufmfoi Xmdo`aiofm, ONGX.
== mvgiodm =2-47, Foudmd Iugvm, z`im 8Foudmd dg Numtgjmkm.
UHR3 8657 =5=5
F`rrg` gkgftriof`3 ongrEongr.gdu.ntUnoim wgh3 www.ongr.gdu.nt
Gdofoi 85=1Ojprgs` gi ONGX tmkkgrgs nraf`s
Fdon`3 ====554858
OVHI ?>2??8?1=41==
ONGX
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idofg
idofg ............................................................................................................................................................... O
Hogivgiodm y hogivgiod`! .................................................................................................................... =
Vgjmim =2Gfumfo`igs fumdrtofms..................... ..................... ..................... ..................... ..................... ..... =6
Umrm f`jgizmr! Km bost`rom dg k`s soni`s jmtgjtof`s..................... ..................... ..................... ................. =4
Gk juid` dg km jmtgjtofm
=. Gfumfo`igs fumdrtofms ................... ..................... ..................... ..................... ..................... ..................... ...... =7
=.= Fkmsofmfoi dg kms gfumfo`igs fumdrtofms..................... ..................... ..................... ................. =1
8. Xgs`kufoi dg gfumfo`igs fumdrtofms f`jpkgtms................... ..................... ..................... ................. =>
6. Xgs`kufoi dg gfumfo`igs fumdrtofms oif`jpkgtms.................... ..................... ..................... ............. 8=
4. Xgs`kufoi dg pr hkgjms f`i gfumfo`igs fumdrtofms..................... ..................... ..................... ...... 86
Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 87
Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 87
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 81
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 82
Xmz`imjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 8?Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 65
Vgjmim =?Oigfumfo`igs.................... ..................... ........................ ..................... ..................... ..................... .................. 6=
Umrm f`jgizmr! Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom..................... ..................... ................ 68
Gk juid` dg km jmtgjtofm
=. Oigfumfo`igs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 66
=.= Fkmsofmfoi dg kms oigfumfo`igs.................... ..................... ..................... ..................... .................. 66
=.8 Xgs`kufoi dg oigfumfo`igs dg projgr nrmd`.................... ...................... ..................... ............. 64 =.6 Xgs`kufoi dg oigfumfo`igs dg sgnuid` nrmd`.................... ..................... ..................... .......... 6>
Xgsujgi................... ..................... ..................... ..................... ...................... ..................... ..................... ..................... .... 6?
Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 6?
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 45
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 48
Xmz`imjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 46
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 44
OJmtgjtofm idofg
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Vgjmim 85Vostgjms dg gfumfo`igs dg 88.................... ..................... ..................... .... 47
Umrm f`jgizmr! Kms vmromhkgs.................... ..................... ..................... ..................... ..................... ..................... .... 41
Gk juid` dg km jmtgjtofm
=. Vostgjms dg gfumfo`igs koigmkgs dg 88 .................... ..................... ..................... ..................... ............. 4>
=.= Jt`d` dg onumkmfoi................... ..................... ..................... ..................... ..................... .................. 4>
=.8 Jt`d` dg sustotufoi.................. ..................... ..................... ..................... ..................... .................. 42
=.6 Jt`d` dg rgduffoi.................... ..................... ..................... ..................... ..................... .................. 4?
8. Vostgjms dg gfumfo`igs i` koigmkgs dg 88 ..................... ..................... ..................... ..................... ..... 7=
6. Mpkofmfo`igs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 78
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 76
Oivgstonug gi km rgd..................... ..................... ..................... ..................... ..................... ..................... ..................... . 76
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 74
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ 71
Xmz`imjogit` knof` ................... ..................... ..................... ..................... ..................... ..................... ................... 7>Dgsmrr`kkg iugvms bmhokodmdgs..................... ..................... ..................... ..................... ..................... ..................... 72
Vgjmim 8=Vostgjms dg gfumfo`igs dg 66.................... ..................... ..................... .... 7?
Umrm f`jgizmr! @hcgt`s trodojgiso`imkgs.................. ..................... ..................... ..................... ..................... ... 15
Gk juid` dg km jmtgjtofm
=. Vostgjms dg gfumfo`igs koigmkgs dg 66 .................... ..................... ..................... ..................... ............. 1=
=.= Jt`d` dg rgduffoi.................... ..................... ..................... ..................... ..................... .................. 1=
=.8 Jt`d` dg sustotufoi.................. ..................... ..................... ..................... ..................... .................. 16
8. Mpkofmfo`igs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 17
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 1>
Oivgstonug gi km rgd.................... ..................... ..................... ..................... ..................... ..................... ..................... .. 1>
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 12
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ >5
Xmz`imjogit` knof` ................... ..................... ..................... ..................... ..................... ..................... ................... >=
Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ...................... >8
Vgjmim 88Vostgjm dg f``rdgimdms rgftminukmrgs ......................................... >6
Umrm f`jgizmr! Umrmkgk`s y jgrodomi`s................... ..................... ..................... ..................... ..................... ...... >4
Gk juid` dg km jmtgjtofm
=. Vostgjm dg f``rdgimdms rgftminukmrgs.................... ..................... ..................... ..................... ................ >7
8. Dostmifom gitrg puit`s (d)................... ..................... ..................... ..................... ..................... ..................... >1
OO ONGX U`k`fbof
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6. Uuit` jgdo` (J).................... ..................... ..................... ..................... ..................... ..................... ................. >2
4. Ugidogitg (j).................. ..................... ..................... ...................... ..................... ..................... ..................... ... 25
Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 26
Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... 26
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 24
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 21
Xmz`imjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. 2>
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 22
Vgjmim 86Auifo`igs jmtgjtofms O..................... ..................... ..................... ..................... ........... 2?
Umrm f`jgizmr! Xgkmfo`igs f`todomims3 kms auifo`igs..................... ..................... ..................... ..................... ?5
Gk juid` dg km jmtgjtofm
=. Auifoi jmtgjtofm..................... ..................... ..................... ..................... ..................... ..................... .......... ?=
=.= Auifo`igs dg projgr nrmd`................... ..................... ..................... ..................... ..................... ........ ?8 =.8 Nrfm dg auifo`igs dg projgr nrmd .................... ..................... ..................... ..................... ....... ?8
=.6 Dgtgrjoimfoi dgk d`joio` g ojmngi dg auifo`igs................... ..................... ..................... .. ?4
=.4 Mpkofmfo`igs dg kms auifo`igs koigmkgs................... ..................... ..................... ..................... ........ ?7
Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... ?>
Oivgstonug gi km rgd.................. ..................... ...................... ..................... ..................... ..................... ..................... ... ?>
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. ?2
Mnokodmd dg fkfuk` jgitmk.................. ..................... ..................... ..................... ..................... ..................... .......... =55
Xmz`imjogit` knof` .................... ..................... ...................... ..................... ..................... ..................... ................. =5=
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =58
Vgjmim 84Auifo`igs jmtgjtofms OO.................. ..................... ..................... ..................... ........... =56
Umrm f`jgizmr! Gk j`vojogit` pmrmhkof`................... ..................... ..................... ..................... ..................... . =54
Gk juid` dg km jmtgjtofm
=. Auifo`igs fumdrtofms ..................... ..................... ..................... ..................... ..................... ..................... ...... =57
=.= Nrfm dg auifo`igs fumdrtofms.................... ..................... ..................... ..................... .................. =56
=.8 Mpkofmfo`igs................... ..................... ..................... ..................... ..................... ..................... ................. =5?
Xgsujgi.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... ===Oivgstonug gi km rgd................... ..................... ..................... ..................... ..................... ..................... ..................... ... ===
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. ==8
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... ==4
Xmz`imjogit` knof` ..................... ..................... ..................... ..................... ..................... ..................... ................. ==7
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. ==1
OOOJmtgjtofm idofg
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Vgjmim 87Xgpms`3 sgjmims =2 m 84..................... ..................... ...................... ..................... ..... ==>
Gk juid` dg km jmtgjtofm
Gfumfo`igs fumdrtofms ..................... ..................... ..................... ..................... ..................... ..................... ............ ==?
Oigfumfo`igs ................... ..................... ..................... ..................... ..................... ..................... ..................... ............. =8=
Vostgjms dg gfumfo`igs dg 88...................... ..................... ..................... ..................... ..................... .............. =86
Vostgjms dg gfumfo`igs dg 66...................... ..................... ..................... ..................... ..................... .............. =87
Vostgjms dg f``rdgimdms rgftminukmrgs..................... ..................... ..................... ..................... .................... =8>
Auifo`igs jmtgjtofms O..................... ..................... ..................... ..................... ..................... ..................... ........ =8?
Auifo`igs jmtgjtofms OO.................... ..................... ..................... ..................... ..................... ..................... ........ =6=
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ =66
@rogitmfo`igs s`hrg km prughm pmrfomk .............................................................................................................. =64
Tmkkgr jmtgjtof`
Urughm M.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =67
Vgjmim 81Gfumfo`igs dg km rgftm.................... ..................... ..................... ..................... .................... =47
Umrm f`jgizmr! Xgpms` dg km pgidogitg.................. ..................... ..................... ..................... ...................... ...... =41
Gk juid` dg km jmtgjtofm
=. Gfumfo`igs dg km rgftm ................... ..................... ..................... ..................... ..................... ..................... ........ =4>
8. Mpkofmfo`igs.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... =7=
6. Xgftms pmrmkgkms y pgrpgidofukmrgs.................... ..................... ..................... ..................... ..................... ... =78
Xgsujgi.......................................................................................................................................................... =74Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... =74
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =77
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ =72
Xmz`imjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... =7?
Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ..................... . =15
Vgjmim 8>Km forfuiagrgifom y km pmrh`km................... ..................... ..................... ............ =1=
Umrm f`jgizmr! Fb`kq'oc, gk fmkgidmro` smnrmd` dg k`s jmyms..................... ..................... ..................... .... =18
Gk juid` dg km jmtgjtofm
=. Km forfuiagrgifom................... ..................... ..................... ..................... ..................... ..................... ................... =16
8. Km pmrh`km ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... =11
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... =>=
Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... =>=
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =>8
OQ ONGX U`k`fbof
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Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... =>4
Xmz`imjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... =>7
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =>1
Vgjmim 82
Vgffo`igs fiofms.................... ..................... ..................... ..................... ..................... ................ =>>
Umrm f`jgizmr! Aonurms fiofms..................... ..................... ..................... ..................... ..................... ................... =>2
Gk juid` dg km jmtgjtofm
=. Vgffo`igs fiofms................... ..................... ..................... ..................... ..................... ..................... ................. =>?
=.= Km gkopsg.................... ..................... ..................... ..................... ..................... ..................... ..................... ... =>?
=.8 Km boprh`km................... ..................... ..................... ..................... ..................... ..................... ................. =26
Xgsujgi.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... =22
Oivgstonug gi km rgd... ................... ..................... ..................... ...................... ..................... ..................... .................... =22
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. =2?
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... =?8Xmz`imjogit` knof`................... ..................... ..................... ...................... ..................... ..................... .................... =?6
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. =?4
Vgjmim 8?Tg`rgjm dg Uotn`rms.................. ..................... ..................... ..................... ..................... ..... =?7
Umrm f`jgizmr! Uotn`rms dg Vmj`s................... ..................... ..................... ..................... ..................... ............ =?1
Gk juid` dg km jmtgjtofm
=. Tg`rgjm dg Uotn`rms................... ..................... ..................... ..................... ..................... ..................... ......... =?>
=.= Dgj`strmfoi dgk tg`rgjm.................... ..................... ..................... ..................... ..................... ........ =?>
=.8 Mpkofmfo`igs................... ..................... ..................... ..................... ..................... ..................... ................. 855
Xgsujgi.......................................................................................................................................................... 856Oivgstonug gi km rgd... ................... ..................... ..................... ..................... ..................... ..................... ..................... 856
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 854
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 851
Xmz`imjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... 85>
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 852
Vgjmim 65Tg`rgjm dg Tmkgs, f`inrugifom ysgjgcmizm dg troinuk`s................... ..................... ..................... ..................... ............... 85?
Umrm f`jgizmr! Xmz`igs y pr`p`rfo`igs.................... ..................... ..................... ..................... ..................... .... 8=5
Gk juid` dg km jmtgjtofm
=. Tg`rgjm dg Tmkgs.................... ..................... ..................... ..................... ..................... ..................... ................. 8==
=.= Tg`rgjm dg Tmkgs gi ui troinuk`..................... ..................... ..................... ..................... ............... 8=8
QJmtgjtofm idofg
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8. Troinuk`s f`inrugitgs.................... ..................... ..................... ..................... ..................... ..................... .... 8=4
8.= Frotgro`s dg f`inrugifom dg troinuk`s..................... ..................... ..................... ..................... ..... 8=7
6. Troinuk`s sgjgcmitgs................... ..................... ..................... ..................... ..................... ..................... ........ 8=1
6.= Frotgro`s dg sgjgcmizm dg troinuk`s.................... ..................... ..................... ..................... ......... 8=>
6.8 Mpkofmfo`igs..................... ..................... ..................... ..................... ..................... ..................... ............... 8=>
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 8=?
Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 8=?
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 885
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ 888
Xmz`imjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 886
Dgsmrr`kkg iugvms bmhokodmdgs..................... ..................... ..................... ..................... ..................... ..................... 884
Vgjmim 6=Fugrp`s ng`jtrof`s O..................... ..................... ..................... ..................... .................... 887
Umrm f`jgizmr! Uorjodg dg Km Dmitm..................... ..................... ..................... ..................... ..................... ...... 881
Gk juid` dg km jmtgjtofm
=. Fugrp`s ng`jtrof`s O.................. ..................... ..................... ..................... ..................... ..................... ........ 88>
=.= Fkmsofmfoi dg k`s fugrp`s ng`jtrof`s................... ..................... ..................... ..................... .. 88>
=.8 Urosjms.................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 882
=.6 Uorjodgs.................. ..................... ..................... ..................... ..................... ..................... ..................... .. 86=
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 866
Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 866
Mut`f`itr`k.................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 864
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ 861
Xmz`imjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 86>Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ...................... ..................... ..................... ..................... 862
Vgjmim 68Fugrp`s ng`jtrof`s OO................... ..................... ..................... ..................... .................... 86?
Umrm f`jgizmr! Gk ijgr` po ()..................... ..................... ..................... ..................... ..................... .................. 845
Gk juid` dg km jmtgjtofm
=. Fugrp`s ng`jtrof`s OO..................... ..................... ..................... ..................... ..................... ..................... .... 84=
=.= Gk fokoidr` .................. ..................... ..................... ..................... ..................... ..................... ..................... .. 84=
=.8 Gk f`i` ................... ..................... ..................... ..................... ...................... ..................... ..................... ..... 847
=.6 Km gsagrm.................... ..................... ..................... ..................... ..................... ..................... ..................... .. 842
Xgsujgi..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 875
Oivgstonug gi km rgd... ..................... ..................... ..................... ..................... ..................... ..................... ................... 875
Mut`f`itr`k..................... ..................... ..................... ..................... ..................... ..................... ...................... ................ 87=
Mnokodmd dg fkfuk` jgitmk.................... ..................... ..................... ..................... ..................... ..................... ........ 878
Xmz`imjogit` knof`..................... ..................... ..................... ..................... ..................... ..................... ................... 876
Dgsmrr`kkg iugvms bmhokodmdgs.................... ..................... ..................... ..................... ..................... ..................... . 874
QO ONGX U`k`fbof
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Vgjmim 66Xmz`igs tron`i`jtrofms..................... ..................... ..................... ..................... ........... 877
Umrm f`jgizmr! Bopmrf` dg Iofgm y km tron`i`jgtrm.................... ..................... ..................... ..................... 871
Gk juid` dg km jmtgjtofm
=. Xmz`igs tron`i`jtrofms auidmjgitmkgs..................... ..................... ..................... ..................... ........... 87>
=.= Vgi` ................... ..................... ..................... ..................... ..................... ..................... ..................... ........... 87>
=.8 F`sgi` ..................... ..................... ..................... ..................... ...................... ..................... ..................... ... 87?
=.6 Tmingitg................... ..................... ..................... ..................... ..................... ..................... ..................... ... 815
8. Xmz`igs tron`i`jtrofms rgfpr`fms.................... ..................... ..................... ..................... ..................... . 81=
6. Xmz`igs tron`i`jtrofms oivgrsms.................... ..................... ..................... ..................... ..................... ..... 818
4. Mpkofmfo`igs..................... ..................... ..................... ..................... ...................... ..................... ..................... ... 817
Xgsujgi................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 81>
Oivgstonug gi km rgd... ................... ..................... ..................... ..................... ..................... ..................... ..................... 81>
Mut`f`itr`k.................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 812
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 8>5Xmz`imjogit` knof`................... ..................... ..................... ..................... ..................... ..................... ..................... 8>=
Dgsmrr`kkg iugvms bmhokodmdgs................... ..................... ..................... ..................... ..................... ..................... .. 8>8
Vgjmim 64Xgpms`3 sgjmims 81 m 66.................... ..................... ..................... ..................... ....... 8>7
Gk juid` dg km jmtgjtofm
Gfumfo`igs dg km rgftm ..................... ..................... ..................... ..................... ..................... ..................... ............. 8>>
Km forfuiagrgifom y km pmrh`km.................... ..................... ..................... ..................... ..................... ................... 8>?
Vgffo`igs fiofms................... ..................... ..................... ..................... ..................... ..................... ..................... ... 82=
Tg`rgjm dg Uotn`rms.................. ..................... ..................... ..................... ..................... ..................... ................. 826
Tg`rgjm dg Tmkgs, f`inrugifom y sgjgcmizm dg troinuk`s.................... ..................... ..................... ...... 827
Fugrp`s ng`jtrof`s O.................... ..................... ..................... ..................... ..................... ..................... ............. 82>
Fugrp`s ng`jtrof`s OO..................... ..................... ..................... ..................... ..................... ..................... .......... 82?
Xmz`igs tron`i`jtrofms................... ..................... ..................... ..................... ..................... ..................... .......... 8?=
Mnokodmd dg fkfuk` jgitmk................... ..................... ..................... ..................... ..................... ..................... ......... 8?6
@rogitmfo`igs s`hrg km prughm aoimk ................................................................................................................... 8?4
Tmkkgr jmtgjtof`
Urughm H.................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... 8?7
Hohko`nrmam ................................................................................................................................................... 657
QOOJmtgjtofm idofg
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= 8 6 4
6
4
7
8
=
5
x
y
=Jmtgjtofm Hogivgiodm y hogivgiod`!
Hogivgiodm y hogivgiod`!
F`itoiumj`s f`i gitusomsj` gk furs` dg jmtgjtofm dg 4. hmfbokkgrmt`. Gstg kohr` f`rrgs-
p`idg mk sgnuid` sgjgstrgdgk nrup` U`k`fbofy f`istm dg dogfosogtg sgjmims, gi kms qugsg dgsmrr`kkmri foif` f`jpgtgifoms jmrfmdms p`r gk Furrfuk` Imfo`imk Hmsg (FIH).
Gk furs` gst gstrufturmd` dg tmk a`rjm qug k`s f`itgiod`s s`i nrmdumkgs y d`sofmd`s pmrm
a`rtmkgfgr y giroqugfgr kms rmjms sonuogitgs3
Mrotjtofm3 pgrjotg rgmkozmr `pgrmfo`igs mrotjtofms f`i k`s ijgr`s rgmkgs.
knghrm3 myudm m kggr, `pgrmr y rgprgsgitmr oia`rjmfoi gi gk kginumcg sojhkof` dg kmjmtgjtofm.
Ng`jgtrm3 rgaogrg kms pr`pogdmdgs dg k`s fugrp`s ng`jtrof`s f`j` rgms y v`kjgigsdg p`kogdr`s. Tmjhoi rgkmfo`im kms pr`pogdmdgs gitrg rgftms.
Tron`i`jgtrm3 gstm rmjm dg km jmtgjtofm sg gifmrnm dgk gstudo` dg k`s kmd`s y inu-k`s dg troinuk`s rgftinuk`s, m trmvs dg kms rmz`igs tron`i`jtrofms.
Mitgs dg gitrmr mk dgtmkkg dg kms f`jpgtgifoms pmrm gstg furs`, `hsgrvgj`s km p`rtmdm.
Gk juid` dg km p`rtmdm i`s rgfugrdm qug s`j`s pmrtg dgk pkmigtm Togrrm y qug gi k, km
jmtgjtofm gs uim bgrrmjogitm qug i`s myudm m dgsfuhror, mprgidgr y dosarutmr kms mrtgs y
km fukturm.
K`s ijgr`s jmyms i`s rgfugrdmi qug pgrtgigfgj`s m ui pughk` juktotiof` g oitgrfukturmk,
f`i ui f`i`fojogit` jmtgjtof` qug dghgj`s vmk`rmr y f`isgrvmr.
Km jmi` dmid` ui fkof i`s oivotm m gxpk`rmr,
dgsfuhror y f`jpmrtor f`i`fojogit`s f`i gkrgst` dgk juid` m trmvs dg oitgrigt.
Km ojmngi dg ui nrup` dg jmi`s mkrgdgd`r
dg ui f`rmzi i`s bmhkm dg gstmhkgfgr rgkm-
fo`igs f`i kms pgrs`ims qug i`s r`dgmi.
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8 ONGX U`k`fbof
Fj` mkfmizmr kms f`jpgtgifoms jmtgjtofms0
I`s gia`fmrgj`s gi foif` f`jpgtgifoms gspgffms. Umrm smhgr so kms bm mkfmizmd`, gk Furrfuk`
Imfo`imk Hmsg pr`p`ig oidofmd`rgs dg k`nr`, gst`s frotgro`s s`i f`j` ui tgrjjgtr` qug jodg
su dgsgjpg` gi fmdm f`jpgtgifom. Orgj`s mvmizmid` pms` m pms`.
Qmym cid`sg qu sgffo`igs dgk kohr` bmfgi p`sohkg qug ustgd dgsmrr`kkg fmdm f`jpgtgifom qug
prgsgitmj`s m f`itoiumfoi.
F`jpgtgifom = Vgffoi dgk kohr`Vgjmim
=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64
Ur`dufg pmtr`igs
mrotjtof`s,
mknghrmof`s y
ng`jtrof`s
mpkofmid` pr`pog-dmdgs, rgkmfo`igs,
nurms ng`j-
trofms, sjh`k`s
y sgmkgs dg
agijgi`s imtu-
rmkgs, s`fomkgs y
fukturmkgs.
Umrmf`jgizmr! R R R R R R R R R R
Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R R R R R
Xgsujgi
Oivgstonug gikm rgd R R R R R R R R R R R R R
Mut`f`itr`k R R R R R R R R R R R R R R R
Mnokodmd dgfkfuk` jgitmk R R R R R R R R R R R R R
Xmz`imjogit`knof` R R R R R R R R R R R
Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R R R R
F`jpgtgifom 8 Vgffoi dgk kohr` Vgjmim=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64
Xgsugkvg
sotumfo`igs,
pr`hkgjms dg
fmrftgr a`rjmk
qug dgjmidmi gk
d`joio` dgk pgi-
smjogit` knof`
jmtgjtof` y
kms pgrmfo`igs
jmtgjtofmsdg mrotjtofm y
knghrm gi k`s
f`icuit`s iu-
jrof`s rgmkgs y
f`jpkgc`s.
Umrmf`jgizmr! R R R
Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R R R R R R
Xgsujgi
Oivgstonug gikm rgd R R R R R R R R R R
Mut`f`itr`k R R R R R R R R R R R R R R
Mnokodmd dgfkfuk` jgitmk R R R R R R R R R R R R R R R R R
Xmz`imjogit`knof` R R R R R R R R R R R
Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R R R R R R R
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6Jmtgjtofm Hogivgiodm y hogivgiod`!
F`jpgtgifom 6 Vgffoi dgk kohr`Vgjmim
=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64Mpkofm f`i`-
fojogit`s s`hrg
auifo`igs, jm-
trofgs, ng`jgtrm
y vgft`rgs, gi
sotumfo`igs qug
pr`jugvgi gk
jgc`rmjogit` y
trmisa`rjmfoi
dgk jgdo` imturmk,
s`fomk y fukturmkdg su f`itgxt`.
Umrmf`jgizmr! R R R R
Gk juid` dgkm jmtgjtofm R R R R R R R R R R R R
Xgsujgi
Oivgstonug gikm rgd R R R R R R R R R R
Mut`f`itr`k R R R R R R R R R R
Mnokodmd dgfkfuk` jgitmk R
Xmz`imjogit`knof` R R R R R
Dgsmrr`kkg iugvmsbmhokodmdgs R R R R
F`jpgtgifom 4 Vgffoi dgk kohr`Vgjmim
=2=?858=88868487818>828?656=686664
Wtokozm tfiofms
dg sufgso`igs
y sgrogs pmrmoitgrprgtmr
bgfb`s s`fomkgs,
gf`ijof`s y
ng`nrf`s.
Umrmf`jgizmr! R R R R
Gk juid` dg
km jmtgjtofm R R
Xgsujgi
Oivgstonug gikm rgd
Mut`f`itr`k R R
Mnokodmd dgfkfuk` jgitmk
Xmz`imjogit`knof` R R R R R R R
Dgsmrr`kkg iugvmsbmhokodmdgs R R R R R R
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F`i su gsaugrz`, p`dr f`iquostmr gstms f`jpgtgifoms m trmvs dg trgs top`s dg f`itgiod`s3
Dgfkmrmtov`s3 k`s f`itgiod`s dgfkmrmtov`s kg mp`rtmri gk f`i`fojogit` dg k`s dos -toit`s mspgft`s tgrof`s, f`ifgptumkgs y fogitaof`s dgk rgm.
Ur`fgdojgitmkgs3 f`j` oidofm su i`jhrg, k`s f`itgiod`s pr`fgdojgitmkgs sgdgsmrr`kkmri m hmsg dg gcgrfofo`s, pr`fgdojogit`s ` mikosos dg pr`hkgjms gi k`s
fumkgs pugdm dgj`strmr gk d`joio` y km pugstm gi prftofm dg k`s f`i`fojogit`s
dgfkmrmtov`s.
Mftotudoimkgs3 k`s f`itgiod`s mftotudoimkgs s`i t`d` mqugkk` qug togig qug vgr f`isu mftotud mitg gk gstudo` y mitg km vodm, gi ngigrmk. Gi gk kohr` gif`itrmr mftovodm-
dgs y rgakgxo`igs s`hrg sotumfo`igs f`todomims d`idg oitgrvogig km jmtgjtofm.
F`jpgtgifom 7 Vgffoi dgk kohr`Vgjmim
=2 =? 85 8= 88 86 84 87 81 8> 82 8? 65 6= 68 66 64Gjpkgm kms tg`rms
dg ng`jgtrm y
tron`i`jgtrm
pmrm oitgrprgtmr
oia`rjmfo`igs y
gkmh`rmr oia`rjgs
s`hrg sotumfo`igs
rgmkgs.
Umrmf`jgizmr! R R R R R R R R
Gk juid` dgkm jmtgjtofm R R R R R R R R R R R
Xgsujgi
Oivgstonug gikm rgd R R R R R R R R
Mut`f`itr`k R R R R R
Mnokodmd dgfkfuk` jgitmk
R
Xmz`imjogit`knof` R R R
Dgsmrr`kkg iugvmsbmhokodmdgs R R
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7Jmtgjtofm Hogivgiodm y hogivgiod`!
U`rtmdm
Km p`rtmdm dg fmdm sgjmim, f`j` km dgk pgrodof`,
oidofm gk ijgr` dg km sgjmim y miuifom gk ttuk`
dgk tgjm qug gstudomr.
Dghmc` dgk ttuk` vgr gstg mpmrtmd`. Kg sgrvor dg
rutm pmrm smhgr qu gif`itrmr3 kgfturm, f`itgiod`s
y mftovodmdgs. Vogjprg sg prgsgitmi fumtr` sgffo`-
igs gi gk josj` `rdgi.
K`nr`s dg km sgjmimM f`itoiumfoi mpmrgfgi k`s k`nr`s ` rgsuk-
tmd`s qug mkfmizmr mk imkozmr gk gstudo` dg
fmdm sgjmim.
Km kostm tgrjoim f`i uim kigm pmrm qug ustgd
gsfrohm su k`nr` pgrs`imk.
F`i`zfmj`s iugstr` kohr`!
Gstg kohr` f`jogizm f`i ui idofg dg f`itgiod`s ngigrmkgs y tgrjoim f`i kms fkmvgs ` s`kufo`igs dgk`s gcgrfofo`s.
Wsmr kms fkmvgs f`i rgsp`ismhokodmd kg pgrjotor dgsmrr`kkmr mut`i`jm gi su mprgidozmcg. Fmdm sgjmim
f`itogig fumtr` sgffo`igs proifopmkgs. Bmnmj`s ui rgf`rrod`3
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=5 ONGX U`k`fbof
Kohr`, fkmsg rmdomk y frfuk` dg gstudo`
Vu gquop` dg trmhmc`!
Gk kohr`, f`i sgr uim hugim bgrrmjogitm, i` k` gs t`d`. Umrm qug ustgd mkfmifg gk iovgk dg
f`jpgtgifom dgsgmd`,iugstr` sostgjm p`ig m su dosp`sofoi3 gk kohr`, km fkmsg rmdomk y km
oivotmfoi m pmrtofopmr gi ui frfuk` dg gstudo`.
Km fkmsg rmdomk togig f`j` auifoi proifopmkgxpkofmr y amfokotmr km f`jprgisoi dg k`s tgjms
trmtmd`s gi gk kohr`.
Uugdg gsfufbmr km fkmsg rmdomk gi uim gjos`rm
dg su k`fmkodmd, dgsfmrnmrkms gi iugstrm p-
noim www.ongr.gdu.nt ` mdquororkms gi FDgi km
f``rdoimfoi rgno`imk.
Gk frfuk` dg gstudo` gs gk kunmrpmrm f`jpmrtor y mprgidgr cuit`s.
Mpr`vgfbg gst`s rgfurs`s y mpygsg gi pgrs`ims dg su f`juiodmd pmrm rgs`kvgr sus dudms.
Gk kohr`fujpkg fumtr` auifo`igs3
m. Tgxt`, gi gk qug gifugitrm km oia`rjmfoi y gk
dgsmrr`kk` dg k`s f`itgiod`s m gstudomr.
h. Uozmrri, pmrm qug durmitg km fkmsg rmdomk
suhrmyg odgms ojp`rtmitgs ` rgmkofg dostoitms
mftovodmdgs.
f. Fumdgri` dg trmhmc`, f`i gcgrfofo`s pmrm prmf-
tofmr k` mprgidod`.
d. Bgrrmjogitm dg mut`gvmkumfoi, fumid` rg-
sugkvg su mut`f`itr`k fmdm sgjmim.
Amktm fmjhomrp`rtmdotm
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==Jmtgjtofm Hogivgiodm y hogivgiod`!
Iugstrm jgt`d`k`nm pms` m pms`
Umrm amfokotmr su mprgidozmcg y mpr`vgfbmr js y jgc`r gk gstudo` fmdm sgjmim, sonm gst`s
pms`s. I` sg smktg ioinui`!
Kgm gk f`itgiod` dg km sgjmim
Kggr gk f`itgiod` i`s pgrjotg tgigr uim odgm ngigrmk dgk tgjm3 qu sm-
hgj`s, f`i qu k` rgkmfo`imj`s, gtf. Gstg projgr f`itmft` tmjhoi i`s
bmr fmgr gi km fugitm dgk gsaugrz` m rgmkozmr pmrm mprgidgr k` iugv` y i`s
p`idr "gi `idm" pmrm km fkmsg rmdomk.
=
Gsfufbg km fkmsg rmdomkF`i k`s 7 sgitod`s
Km fkmsg rmdomk gs iugstrm jmgstrm. Dg mb qug gk pr`nrmjm sg kkmjg "Gk
Jmgstr` gi Fmsm". Kms jmgstrms y jmgstr`s k`fut`rgs gxpkofmi gk f`itgio-
d`, pr`p`igi gcgrfofo`s y `tr`s gcgjpk`s pmrm mjpkomr gk tgjm.
8
Dgspus dg km fkmsg rmdomk, su trmhmc` pgrs`imkGstudo` y mut`f`itr`k
Aoimkozmdm km fkmsg rmdomk gs gk j`jgit` dg su trmhmc` pgrs`imk. Dostrohuym
su togjp`3 gs jgc`r ui p`f` fmdm dm, qug t`d` km vspgrm.
6
F`isuktg sus dudmsWi gstudomitg oitgkongitg smhg fuid` pgdor myudm
F`isuktg k`s tgjms qug i` kg bmi qugdmd` fkmr`s gi `tr`s kohr`s, gi oitgr-
igt, f`i amjokomrgs ` mjon`s. Vgnur` qug gif`itrmr pgrs`ims dospugstms
m myudmrkg.
4
Umrtofopg gi ui frfuk` dg gstudo`
Mprgidgr cuit`sUinmsg dg mfugrd` f`i tr`s gstudomitgs y rnmiofgi ui frfuk` dg gstu-
do`. V`kofotgi km myudm dg mknuim pgrs`im v`kuitmrom dg km f`juiodmd. Gs`
kgs myudmr m rgs`kvgr dudms y rga`rzmr k` mprgidod`. Mdgjs, tgidri km
`p`rtuiodmd dg oitgrfmjhomr mprgidozmcgs, odgms y sgitojogit`s.
Xgfugrdg qug sogjprg pugdg mfudor m su tut`r msonimd`.
7
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=8 ONGX U`k`fbof
Fj` mpr`vgfbmr jgc`r su gstudo`0
Husqug ui kunmr fj`d` y f`i hugim okujoimfoi. Gs ojp`rtmitg qug sg
mkgcg dgk ruod` y dg kms dostrmffo`igs.
Gkocm ui b`rmro` pmrm trmhmcmr y gstudomr. Km f`istmifom y km dosfopkoim s`i
sus jgc`rgs f`jpmgrms dg gstudo`.
Kgm f`i mtgifoi kms oistruffo`igs dg k`s gcgrfofo`s mitgs dg rgs`kvgrk`s.
F`isuktg sus dudms f`i `trms pgrs`ims dg su f`juiodmd qug pugdmi
myudmrkg.
ONGX
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=2=
8 6 4
6
4
7
8
=
5
x
y
=:o
x8+
y8:
r8
Gfumfo`igs fumdrtofms
]u gif`itrmr gstm sgjmim0
Km bost`rom dg k`s soni`s jmtgjtof`s
Gfumfo`igs fumdrtofms
Gfumfo`igs dg projgr nrmd`
Gfumfo`igs fumdrtofms y pr`hkgjms dg mpkofmfoi
Gstm sgjmim k`nrmr3
F`i`fgr km bost`rom dg k`s soni`s jmtgjtof`s.
Odgitoaofmr gk nrmd` dg kms gfumfo`igs.
Xgf`i`fgr gfumfo`igs fumdrtofms f`jpkgtms g oif`jpkgtms.
Xgs`kvgr gfumfo`igs dg sgnuid` nrmd` p`r amft`rozmfoi y p`r arjukm.
Xgs`kvgr f`i mnokodmd gfumfo`igs dg projgr nrmd`.
Bmkkmr km vmromfoi dgk rgm dg fumdroktgr`s.
=6Jmtgjtofm Vgjmim =2
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Umrm f`jgizmr!Km bost`rom dg k`s soni`s jmtgjtof`s
Vg p`drm pgismr qug k`s soni`s utokozmd`s gi kms doagrgitgs `pgrmfo`igs jmtg-
jtofms bmi sod` k`s josj`s dg sogjprg, pgr` k` fogrt` gs qug bmi vmromd` f`igk pms` dgk togjp`. M pmrtor dgk sonk` RQ, p`f` m p`f`, sg augr`i sustotuygid` kms
pmkmhrms, qug oidofmhmi kms `pgrmfo`igs, p`r k`s soni`s.
F`i`zfmj`s km bost`rom dg mknui`s soni`s jmtgjtof`s.
+ Vujm y rgstmVg dofg qug k`s otmkomi`s utokozmhmi uim py uim jpmrm oidofmr
km sujm y km rgstm (pkus y joius, gi kmti). Gstms kgtrms sg gko-
joimr`i fumid` gk jmtgjtof` mkgji C`bmiigs ^odjmii
oitr`duc` k`s soni`s + y , qug utokozmj`s gi km mftumkodmd.
Ur`duft` ` juktopkofmfoi^okkomj @unbtrgd pr`pus` uim fruz pmrm dgsonimr km jukto-
pkofmfoi. U`stgro`rjgitg, gk jmtgjtof` N`ttarogd ^okbgkjKgohioz dgfodo pr`p`igr gk puit` sojpkg, p`rqug km fruz sg
f`iauidm f`i kmxdg kms gfumfo`igs.
mh m/h
DovosoiKm dovosoi gs km `pgrmfoi jmtgjtofm qug sg pugdg rgprg-
sgitmr f`i js soni`s. Gitrg gkk`s gst km hmrrm b`roz`itmk,
soni` dg `rongi rmhg. Kugn`, gk jmtgjtof` Munustus Dg
J`rnmi km trmisa`rj gi `hkofum. U`stgro`rjgitg C`bmiiBgoirofb Xmbi oivgit gk soni` "". Kgohioz oitr`duc` k`s d`spuit`s "3"< k mf`isgcmhm su us` pmrm rgmkozmr km `pgrmfoi
gi uim s`km kigm y pmrm tgigr uim rgkmfoi f`i gk soni` dg
juktopkofmfoi dg ui s`k` puit .
m
h m
3
h
:Onumk qugGstg soni` sg dghg m X`hgrt Xgf`rdg, qug gjpgz m utoko-
zmrk` gi =77>. k dgfm3 I` bmy d`s f`sms qug pugdmi sgr
js onumkgs qug d`s rgftms pmrmkgkms.
Gk soni` dg km onumkdmd onumk qug sgr dg jufbm utokodmd gi kms gfumfo`igs qug
gstudomrgj`s gstm sgjmim.
M trmhmcmr!
Kg oivotmj`s m imvgnmr p`r oitgrigt g oivgstonmr s`hrg k`s dgjs soni`s jmtgjtof`s y `trms bost`roms
dg oitgrs. Kg sungroj`s gstm dorgffoi3 bttp3//n``.nk/=8gn_B
=4ONGX U`k`fbof
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=. Gfumfo`igs fumdrtofmsOiofogj`s iugstr` furs` dg jmtgjtofm rgf`rdmid` kms gfumfo`igs koigmkgs ` dg
projgr nrmd`, qug ym bgj`s trmhmcmd` gi k`s furs`s mitgro`rgs.
Wim gfumfoi koigmkgs km gxprgsoi mknghrmofm qug gst a`rjmdm p`r d`s jogj-hr`s, gi k`s qug pugdgi bmhgr `pgrmfo`igs mknghrmofms, sgpmrmd`s p`r gk soni`
onumk qug (:) y km vmromhkg ` oifniotm gst gkgvmdm m km p`tgifom ui` (x), p`r gkk`
sg kg f`i`fg tmjhoi f`j` gfumfoi dg projgr nrmd`.
Ugr` i` t`dms kms gfumfo`igs s`i dg projgr nrmd`, f`i`zfmj`s qu dgtgrjoim
gk nrmd` dg uim gfumfoi.
Gk nrmd` dg uim gfumfoi k` gstmhkgfg gk j`i`jo` ` trjoi` dg jmy`r nrmd`
mhs`kut`< gs dgfor, km vmromhkg gkgvmdm mk gxp`igitg jmy`r. Gi kms gfumfo`igs
sonuogitgs vgrgj`s fj` dgtgrjoimr gk nrmd` y su i`jhrg.
7x 6 + 4x: 1Gfumfoi dg projgr nrmd` ` koigmk, (gkgxp`igitg jmy`r dg km oifniotmxgs =).
8x8 >x + 6 : 5 Gfumfoi dg sgnuid` nrmd` ` fumdrtofm.
6x6 8x8+ =4x: 5 Gfumfoi dg tgrfgr nrmd` ` fhofm.
x4 =6x+ 61 : 5 Gfumfoi dg fumrt` nrmd`` hofumdrmdm.
Gstm sgjmim gstudomrgj`s km rgs`kufoi dg kms gfumfo`igs fumdrtofms ` dgsgnuid` nrmd`.
Wim gfumfoi fumdrtofmdg uim vmromhkg gs ui p`koi`jo` a`rjmd` p`r km sujmmknghrmofm dg trjoi`s fuy` nrmd` jxoj` gs d`s, gs dgfor, km oifniotm gst
gkgvmdm mk fumdrmd` ` su gxp`igitg gs d`s (x8).
Km gxprgsoi ngigrmk qug rgprgsgitm uim gfumfoi fumdrtofm gs3
mx8+hx+ f: 5
D`idg3x3 gs km vmromhkg.
mx83 gs gk trjoi` fumdrtof`, d`idg gk f`gfogitg mi` pugdg sgr fgr`.
hx3 gs gk trjoi` koigmk ` dg projgr nrmd`.
f3 gs gk trjoi` oidgpgidogitg.
Kms gfumfo`igs fumdrtofms pugdgi tgigr3 d`s s`kufo`igs, uim s`kufoi, ` i`
tgigr s`kufoi dgitr` dg k`s ijgr`s rgmkgs.
=7Jmtgjtofm Vgjmim =2
Gk juid` dg km jmtgjtofm
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=.= Fkmsoaofmfoi dg kms gfumfo`igs fumdrtofms
Kms gfumfo`igs fumdrtofms pugdgi sgr f`jpkgtms ` oif`jpkgtms. Qgmj`s kms
fmrmftgrstofms dg fmdm uim.
m. Gfumfo`igs fumdrtofms f`jpkgtms
Wim gfumfoi dg sgnuid` nrmd` sg f`isodgrm f`jpkgtm fumid` k`s f`gfogitgsiujrof`s m, hy gk trjoi` oidgpgidogitg f s`i dostoit`s dg fgr`, p`r k` tmit`
f`itogigi k`s trgs trjoi`s. Mknui`s gcgjpk`s dg gfumfo`igs fumdrtofms
f`jpkgtms s`i3
x8 8x+ 2 : 5 8x8+ 4x 1 : 5
F`gfogitg m: =
F`gfogitg h: 8
Trjoi` oidgpgidogitg f : 2
F`gfogitg m: 8
F`gfogitg h: 4
Trjoi` oidgpgidogitg f: 1
h. Gfumfo`igs fumdrtofms oif`jpkgtmsGi uim gfumfoi fumdrtofm i`pugdg amktmr gk trjoi` dg sgnuid` nrmd` mx8p`rqug dgcmrm dg sgr fumdrtofm, pgr` s pugdg amktmr gk trjoi` dg projgr nrmd`
hx, gk trjoi` oidgpgidogitg f` mjh`s. Gi fumkquogrm dg k`s fms`s sg trmtm dg
uim gfumfoi fumdrtofm oif`jpkgtm.
Km gxprgsoi jmtgjtofm qug gcgjpkofm fmdm uim dg gstms gfumfo`igs gs3
Fumid` amktm gk trjoi` koigmk hx.
Gcgjpk`3 4x8 ? : 5
Fumid` amktm gk trjoi` oidgpgidogitg f.
Gcgjpk`3 8x8+ 1x: 5
Fumid` amktm gk trjoi` koigmk hxy gk trjoi`
oidgpgidogitg f.
Gcgjpk`3 ?x8: 5
mx8+f: 5
mx8
+hx: 5
mx8: 5
Gcgrfofo` =Odgitoaoqug k`s vmk`rgs dg k`s f`gaofogitgs ijgrof`s y dgk trjoi` oidgpgidogitg gi kms gfumfo`igs
fumdrtofms sonuogitgs y gsfrhmk`s gi km f`kujim f`rrgsp`idogitg. Togig ui gcgjpk`.
Gfumfoi F`gafogitgm h f
x8 ? : 5 = 5 ?
x8 2x: 5
4x8+ 8x+ = : 5
84x8 17x + 8= : 5
8x8 4 > x+ = : 5
=1ONGX U`k`fbof
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8. Xgs`kufoi dg gfumfo`igs fumdrtofms f`jpkgtmsGi gstm sgffoi gstudomrgj`s km rgs`kufoi dg gfumfo`igs fumdrtofms f`jpkgtms.
K`s jt`d`s qug utokozmrgj`s sgri3 p`r amft`rozmfoi y p`r arjukm ngigrmk.
Wi pms` ojp`rtmitg pmrm rgs`kvgr gfumfo`igs fumdrtofms f`jpkgtms gs `rdgimr
km gfumfoi, dg tmk a`rjm qug t`d`s k`s trjoi`s qugdgi gi gk jogjhr` ozquogr-
d`, dg prgagrgifom, y gk kmd` dgrgfb` sg onumkm m 5< gs dgfor, kkgvmrkm m km a`rjm
mx8+ hx + f :5
m. V`kufoi p`r amft`rozmfoi
Umrm rgs`kvgr gfumfo`igs fumdrtofms p`r amft`rozmfoi, dghgj`s ms`fomrkms f`i
gk troi`jo` dg km a`rjmx8 + hx + f` mx8+ hx + fy gxprgsmrkm f`j` ui pr`duft`
dg sus amft`rgs.
K`s pms`s pmrm rgs`kvgr kms gfumfo`igs fumdrtofms f`jpkgtms s`i3
m) @rdgimr km gfumfoi g onumkmrkm m 5.
h) Amft`rozmr gk troi`jo` g onumkmrk` m 5.
f) T jmr fmdm amft`r p`r sgpmrmd` g onumkmrk`s m 5.
d) Dgspgcmr km vmromhkg.
g) U`r ktoj`, f`jpr`hmr gk ` k`s rgsuktmd`s gi km gfumfoi `ronoimk.
Xgs`kvmj`s3 x8+ 7x :1
@rdgimj`s km gfumfoi y km onumkmj`s m 5.
Amft`rozmj`s gk troi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y k`
onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`s kms
s`kufo`igs.
Umrm f`jpr`hmr qug kms rgspugstms s`i f`rrgftms, gi gstg gcgrfofo` sustotuorgj`s
km vmromhkg dg km gfumfoi `ronoimk p`r k`s vmk`rgs dgx=yx
8.
Vox: 6
x8+ 7x: 1
(6)8+ 7(6) : 1
? =7 :
1
1 : 1
Vox: 8
x8+ 7x: 1
(8)8+ 7(8) : 1
4 =5 :
1
1 : 1
Dg gstm jmigrm qugdm f`jpr`hmd` qug mjhms rgspugstms s`i f`rrgftms.
x8+ 7x +1: 5
(x + 6)(x + 8) : 5
x + 6 : 5 x + 8 : 5
x=: 6 x
8: 8
Umrm rgs`kvgr gfum-fo`igs fumdrtofmsf`i mnokodmd, tginmgi fugitm k`s fms`sdg amft`rozmfoi qugvoj`s gi kms sgjmims=6 y =4.
=>Jmtgjtofm Vgjmim =2
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Xgs`kvmj`s3
@rdgimj`s km gfumfoi y km onumkmj`s m 5.
Amft`rozmj`s gk troi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y
k` onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`s
kms s`kufo`igs.
Xgs`kvmj`s3
Amft`rozmj`s gk troi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y
k` onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`skms s`kufo`igs.
x8: 2x =1
6x8+ >x+ 8 : 5
Gcgrfofo` 8Xgsugkvm, gi gstg gspmfo`, kms gfumfo`igs fumdrtofms p`r amft`rozmfoi. Kugn`, gi su fumdgri`
f`jprughg fmdm rgsuktmd`.
x8 2x+ =1 : 5
(x 4)(x 4) : 5
x 4 : 5 x 4 : 5
x=: 4 x
8: 4
(6x+ =)(x+ 8) : 5
6x+ = : 5 x+ 8 : 5
6x: = x: 8
x=:
=
6 x
8: 8
=) x8 8x: 2 8) x8+ 87 : =5x
6) x8: 2x =7 4) x8 =1 : 1x
7) 8x8 >x+ 6 : 5 1) 8x8 4x 1 : 5
=2ONGX U`k`fbof
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h. V`kufoi p`r arjukm ngigrmk
Gi gk mpmrtmd` mitgro`r mprgidoj`s m rgs`kvgr gfumfo`igs dg sgnuid` nrmd`
p`r dgsf`jp`sofoi gi amft`rgs, pgr` bmy gfumfo`igs fumdrtofms gi kms qug gsg
pr`fgdojogit` i` gs dg afok mpkofmfoi. Umrm gkk`, gxostg km arjukm fumdrtofm `
ngigrmk qug myudm m rgs`kvgr fumkquogr top` dg gfumfoi fumdrtofm.
Xgf`rdgj`s km arjukm fumdrtofm arjukm dg Qogtm y k`s gkgjgit`s qug km a`rjmi.
x:h h
8 4mf
8m
D`idg3
x3 rgprgsgitm kms d`s s`kufo`igs dg km gfumfoi (x=yx
8).
h3 gs gk f`gfogitg dgk trjoi` koigmk hx.
3 gstg soni` a`rjmd` p`r k`s soni`s js y jgi`s sorvg pmrm `htgigr kms
d`s s`kufo`igs.
43 gs uim f`istmitg qug juktopkofm mk f`gfogitg my mk trjoi` f.
m3 gs gk f`gfogitg dgk trjoi` fumdrtof` mx8.
f3 gs gk trjoi` oidgpgidogitg.
83 gs uim f`istmitg gi gk dgi`joimd`r qug juktopkofm mk f`gfogitg m.
Umrm utokozmr km arjukm, odgitofmj`s k`s vmk`rgs dg k`s f`gfogitgs iujrof`s y dgk
trjoi` oidgpgidogitg gi km gfumfoi ronoimk. Kugn`, k`s sustotuoj`s gi km arjukm
dg mfugrd` f`i kms f`istmitgs y soni`s oidofmd`s. Bmnmj`s gk projgr gcgjpk`.
Xgs`kvmj`s3 x8
:
4x + =8
@rdgimj`s km gfumfoi y rgkmfo`imj`s
fmdm kotgrmk f`i su vmk`r.
Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs gi
km arjukm.
Xgmkozmj`s kms `pgrmfo`igs oidofmdms
dgitr` y augrm dgk rmdofmk.
Gxprgsmj`s gk rgsuktmd` f`i x= d`idg
`pgrmj`s f`i gk soni` + y x8 d`idg
`pgrmj`s f`i gk soni`.
@htgigj`s k`s vmk`rgs dgx=yx
8.
x8+ 4x =8 : 5
m : =, h : 4, f: =8
x:(4) (4)8 4(=)(=8)
8(=)
x:
4 =1 +42
8 :
4 14
8
x=:
4 + 28
x8:
4 28
x=:
4
8: 8 x
8:
=8
8: 1
Xgfugrdg km kgy dg
soni`s< gi km projgrmsustotufoi(4) :4ydgitr` dgk rmdofmk4(=)(=8) : +42
=?Jmtgjtofm Vgjmim =2
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Xgs`kvmj`s3
Xgkmfo`imj`s fmdm kotgrmk f`i su vmk`r.
Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs
gi km arjukm.
Xgmkozmj`s kms `pgrmfo`igs oidofmdms
dgitr` y augrm dgk rmdofmk.
Gxprgsmj`s gk rgsuktmd` f`ix=d`idg
`pgrmj`s f`i gk soni` +yx8d`idg
`pgrmj`s f`i gk soni`.
@htgigj`s k`s vmk`rgs dgx=yx
8.
Xgs`kvmj`s3
Xgkmfo`imj`s fmdm kotgrmk f`i su vmk`r.
Vustotuoj`s k`s vmk`rgs dg kms kotgrmkgs
gi km arjukm.
Xgmkozmj`s kms `pgrmfo`igs oidofmdms
dgitr` y augrm dgk rmdofmk.
Gxprgsmj`s gk rgsuktmd` f`ix=d`idg`pgrmj`s f`i gk soni` +yx
8d`idg
`pgrmj`s f`i gk soni`.
@htgigj`s k`s vmk`rgs dgx=yx
8.
Gcgrfofo` 6
Xgsugkvm gi su fumdgri` kms gfumfo`igs fumdrtofms f`jpkgtms. Wtokofg km arjukm ngigrmk yf`jprughg fmdm rgsuktmd`.
8x8 7x+ = : 5
m : 8, h : 7, f: =
x:
(7) (7)8 4(8)(=)
8(8)
x: 7 87 24
: 7 =>4
x=:
7 + =>4
x8:
7 =>4
x=:
7 + =>
48.82
x8:
7 =>
45.88
1x8+ 86x =2 : 5
m : 1, h : 86, f: =2
x:(86) (86)8 4(1)(=2)
8(1)
x:86 78? + 468
=8:
86 ?1=
=8
x=:
86 + 6=
=8 x8:
86 6=
=8
x=:
2
=8:
8
6 x
8:
74
=8:
?
8
=) x8 8x: 84
8) 6x8+ 7x: 5
6) 7x8 6x 8 : 5
4) x8 4x 8= : 5
7) 6x8 4x =7 : 5
1) ?x8: 6x+ 85
>) 8x8+ 61 : =>x
2) 4x8+ ? + =8x: 5
?) 1x8+ =5 : =?x
=5) 7x8+ 84x+ =1 : 5
==) ?x8+ 8?x+ 1 : 5
=8) =5x8+ ?x ? : 5
85ONGX U`k`fbof
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6. Xgs`kufoi dg gfumfo`igs fumdrtofms oif`jpkgtms
m. Gfumfo`igs dg km a`rjm mx8+ hx: 5
F`j` docoj`s mitgro`rjgitg, kms gfumfo`igs fumdrtofms oif`jpkgtms s`i
mqugkkms qug kgs amktm gk trjoi` hx` f. Mb`rm gstudomrgj`s kms qug kgs amktm gk
trjoi` f. K`s pms`s pmrm rgs`kvgr gstg top` gfumfo`igs s`i3
m) Amft`rozmr p`r amft`r f`ji.
h) Onumkmr fmdm amft`r m5.
f) Xgs`kvgr kms d`s gfumfo`igs dg projgr nrmd` ` koigmkgs.
Xgs`kvmj`s3x8+ 8x: 5
Amft`rozmj`s gk hoi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y k` onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`s kms s`kufo`igs.
Xgs`kvmj`s3 6x8 7x: 5
Amft`rozmj`s gk hoi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y k` onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`s kms s`kufo`igs.
Xgs`kvmj`s3 2x8+ 8x: 5
Amft`rozmj`s gk hoi`jo`.
T jmj`s fmdm amft`r p`r sgpmrmd` y k` onumkmj`s m 5.
Dgspgcmj`s km vmromhkg y `htgigj`s kms s`kufo`igs.
Gcgrfofo` 4Xgsugkvm gi su fumdgri` kms gfumfo`igs fumdrtofms oif`jpkgtms y f`jprughg fmdm rgsuktmd` gi km
gfumfoi `ronoimk.
=) x8 8x: 5
8) 6x8+ 7x: 5
6) 8x8 6x: 5
4) x8: 8x
7) 8x8: 7x
1) ?x8: 6x
>) 8x8: =>x
2) 1x8: =2x
?) 4x8+ =8x: 5
x(x+ 8) : 5
x: 5 x+ 8 : 5
x=: 5 x
8: 8
x(6x 7) : 5
x : 5 6x 7 : 5
6x : 7
x=: 5 x
8: 7
6
8x(4x + =) : 5
8x : 5 4x + = : 5
x : 58
4x :=
x=: 5 x
8:
=
4
8=Jmtgjtofm Vgjmim =2
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Xgs`kvmj`s3x8 4 : 5
@rdgimj`s km gfumfoi.
Gxtrmgj`s rmz fumdrmdm gi mjh`s jogjhr`s.
Gsfrohoj`s kms s`kufo`igs.
Xgs`kvmj`s3 ?x8 87 : 5
@rdgimj`s km gfumfoi.
Gxtrmgj`s rmz fumdrmdm gi mjh`s jogjhr`s.
Dgspgcmj`s km vmromhkg y gsfrohoj`s kms
s`kufo`igs.
Xgs`kvmj`s3x8 =5 : 5
@rdgimj`s km gfumfoi.
Gxtrmgj`s rmz fumdrmdm gi mjh`s jogjhr`s.
Gsfrohoj`s kms s`kufo`igs.
Xgs`kvmj`s3x8: =2
Gxtrmgj`s rmz fumdrmdm gi mjh`s jogjhr`s.
Gsfrohoj`s kms s`kufo`igs.
x8: 4
x8 : 4
x: 8
x=: 8 x8: 8
?x8: 87
?x8 : 87
6x: 7
x=:
7
6 x
8:
7
6
x8: =5
x8
: =5x: =5
x=: =56.=1
x8: =56.=1
x8 : =2
x: =2
x=: 6 8 4.84
x8: 6 8 4.84
Gcgrfofo` 7Xgsugkvm gi su fumdgri` kms gfumfo`igs fumdrtofms oif`jpkgtms y f`jprughg fmdm rgsuktmd` gi km
gfumfoi `ronoimk.
=) x8 14 : 5
8) x8 =55 : 5
6) 4x8: =1
4) ?x8: 87
7) x8 1 : 5
1) 1x8 84 : 5
h. Gfumfo`igs dg km a`rjm mx8+ f: 5
F`j` pugdg vgr gi gstms gfumfo`igs amktm gk trjoi` koigmk hx. K`s pms`s pmrm
rgs`kvgr gstg top` dg gfumfo`igs s`i3
m) Gsfrohor gk trjoi` fumdrtof` gi gk projgr jogjhr y gk trjoi`
oidgpgidogitg gi gk sgnuid` jogjhr`.h) Gxtrmgr rmz fumdrmdm gi mjh`s jogjhr`s, mitgp`iogid` gk
soni` mk rmdofmk dgk trjoi` oidgpgidogitg.
f) Gsfrohor kms d`s s`kufo`igs.
88ONGX U`k`fbof
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4. Xgs`kufoi dg pr`hkgjms f`i gfumfo`igs fumdrtofmsJufbms sotumfo`igs dg km vodm domrom y dg kms doagrgitgs fogifoms bmfgi us` dg
kms gfumfo`igs dg sgnuid` nrmd` pmrm kkgnmr m uim s`kufoi. M f`itoiumfoi
gjpkgmrgj`s gstg top` dg gfumfo`igs pmrm rgs`kvgr mknuims mpkofmfo`igs.
Xgs`kvmj`s3
Wim dg kms fmrms dg ui prosjm rgftminukmr jodg 4fgitjgtr`s js gi km hmsg
qug gi su mkturm y gk rgm gs dg =?8 fgitjgtr`s fumdrmd`s. Fukgs s`i kms
dojgiso`igs dg gsm fmrm0
Dg sgr p`sohkg rgmkozmj`s uim dohuc` qug dgsfrohm k`s
dmt`s dgk pr`hkgjm.
Gsfrohoj`s k`s dmt`s dgk pr`hkgjm gi uim tmhkm.
Gsfrohoj`s km arjukm dgk rgm pmrm ui rgftinuk`.
Vustotuoj`s k`s vmk`rgs gi km arjukm.
Xgmkozmj`s kms `pgrmfo`igs.
@rdgigj`s km gfumfoi dg sgnuid` nrmd`.
Vustotuoj`s k`s vmk`rgs gi km arjukm.
@htgigj`s k`s rgsuktmd`s.
F`j` kms jgdodms dg km aonurm i` pugdgi sgr ignmtovms, dgsfmrtmj`s gkrgsuktmd`x8: =1, s`k` t`jmj`s gk vmk`r p`sotov`x
=: =8y k` sustotuoj`s gi
kms vmromhkgs oiofomkgs.
b :x: =8 fj< h :x+ 4 : =8 + 4 : =1 fj
Kms dojgiso`igs dg uim dg kms fmrms dgk prosjm s`i =8 fjdg mkt` y =1 fjdg mifb`.
Mkturm Hmsg rgm
b:x h:x+ 4 M : =?8 fj8
x
x + 4
=?8 fj8
M:hb
=?8 : (x+ 4)(x)
=?8: x8+ 4x
x8+ 4x =?8 : 5
x:
(4) (4)8 4(=)(=?8)
8(=)
x:
4 =1 + >12
8:
4 >24
8
x=:
4 + 828
x8:
4 828
x=:
84
8:=8 x
8:
68
8: =1
86Jmtgjtofm Vgjmim =2
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Gxprgsmj`s k`s dmt`s dgk pr`hkgjm
gi uim tmhkm.
Gsfrohoj`s km gfumfoi.
Dgsmrr`kkmj`s gk fumdrmd`.
Onumkmj`s km gfumfoi m 5.
Dovodoj`s km gfumfoi gitrg 8.
Amft`rozmj`s.
Onumkmj`s fmdm amft`r m 5.
@htgigj`s k`s rgsuktmd`s.
Frostoim togig =5m`s y Mkhgrt` =7.
Dg sgr p`sohkg rgmkozmj`s ui dohuc`qug dgsfrohm k`s dmt`s dgk pr`hkgjm.
Gsfrohoj`s km arjukm dgk v`kujgi
pmrm ui fokoidr` forfukmr rgft`.
Dgspgcmj`s km vmromhkg qug i`s
oitgrgsm.
Vustotuoj`s k`s vmk`rgs gi km arjukm.
Xgmkozmj`s kms `pgrmfo`igs.
@htgigj`s k`s rgsuktmd`s.
Xgs`kvmj`s3
Mkhgrt` gs 7m`s jmy`r qug su bgrjmim Frostoim. Km sujm dg kms gdmdgs mk
fumdrmd` gs 687m`s. Fuk gs km gdmd dg fmdm ui`0
Xgs`kvmj`s3
Wi amhrofmitg dg vms`s forfukmrgs rgft`s dgsgm gkmh`rmr ui vms` dg =5 fjdg mkt`
y f`i uim fmpmfodmd >27 fj6. ]u rmdo` oitgro`r dghg tgigr fmdm vms`0
Gdmd dgFrostoim
Gdmd dgMkhgrt`
Vujm dg kms gdmdgs mkfumdrmd`
x x+ 7 x8 + (x + 7)8
x8+ (x+ 7)8: 687
x8+x8+ =5x+ 87: 687
8x8+ =5x 655 : 5
x8+ 7x =75 : 5
(x+ =7)(x =5) : 5
x+ =7 : 5 x =5 : 5
x=: =7 x
8: =5
Gk vms` dghg tgigr ui rmdo` oitgri` dg 7fgitjgtr`s.
b
r
Q : r8b
Q
b: r
8
>27
(6.=4)(=5): r
8
>27
6=.4: r8
87: r8
7 : r 7 : r
84ONGX U`k`fbof
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=. Gi kms gfumfo`igs fumdrtofmskm oifniotm gst gkgvmdm mk fumdrmd`. Gstms gfumfo`igs pugdgitgigr3 d`s s`kufo`igs, uim s`kufoi ` i` tgigr s`kufoi dgitr` dg k`s ijgr`s rgmkgs.
Km gxprgsoi ngigrmk qug rgprgsgitm uim gfumfoi fumdrtofm gs3
mx8+hx+ f: 5
=.= Fkmsoaofmfoi dg kms gfumfo`igs fumdrtofms3
m. Gfumfo`igs fumdrtofms f`jpkgtms3uim gfumfoi dg sgnuid` nrmd` sg f`isodgrm f`jpkgtmfumid` k`s f`gaofogitgs iujrof`s m, hy gk trjoi` oidgpgidogitg fs`i dostoit`s dg fgr`.
h. Gfumfo`igs fumdrtofms oif`jpkgtms3 gi uim gfumfoi fumdrtofm i` pugdg amktmr gktrjoi` mx8, pgr` s pugdg amktmr gk trjoi` dg projgr nrmd` hx, gk trjoi` oidgpgidogitg
f` mjh`s. Gi fumkquogrm dg k`s fms`s sg trmtm dg uim gfumfoi fumdrtofm oif`jpkgtm.
8. Kms gfumfo`igs fumdrtofms f`jpkgtmssg pugdgi rgs`kvgr p`r amft`rozmfoi y p`r arjukm ngigrmk.
m. V`kufoi p`r amft`rozmfoi3 pmrm rgs`kvgr gfumfo`igs fumdrtofms p`r amft`rozmfoidghgj`s ms`fomrkms f`i gk troi`jo` dg km a`rjmx8 + hx + f` mx8+ hx+ fy gxprgsmrkm f`j`
ui pr`duft` dg sus amft`rgs.
h. V`kufoi p`r arjukm ngigrmk3 km arjukm ngigrmk ` dg Qogtm i`s myudm m rgs`kvgrfumkquogr top` dg gfumfoi fumdrtofm sustotuygid` k`s vmk`rgs dg k`s f`gaofogitgs y dgk
trjoi` oidgpgidogitg gi km arjukm.
x:h h
8 4mf
8m
6. Xgs`kufoi dg gfumfo`igs fumdrtofms oif`jpkgtms< dg mfugrd` mk trjoi` qug kgs amktg, pugdgi
sgr dg km a`rjm3
m. mx8+ hx: 5 sg rgsugkvgi p`r amft`r f`ji.
h. mx8+ f: 5 sg rgsugkvgi `rdgimid` km gfumfoi y gxtrmygid` rmz fumdrmdm gi mjh`s
jogjhr`s.
Xgsujgi
Kg oivotmj`s m qug vosotg k`s sonuogitgs soto`s dg oitgrigt. Gi gkk`s p`dr rga`rzmr sus f`i`fojogit`s
s`hrg kms gfumfo`igs fumdrtofms y rgmkozmr js gcgrfofo`s.
bttp3//n``.nk/Km]zsB
bttp3//n``.nk/m2RHCh
Oivgstonug gi km rgd...
87Jmtgjtofm Vgjmim =2
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Mftovodmd =. Dgjugstrg k` mprgidod`M. @hsgrvg f`i mtgifoi kms gfumfo`igs, dgtgrjoig so s`i f`jpkgtms ` oif`jpkgtms y gsfrohm gk vmk`r
dg k`s f`gaofogitgs y dgk trjoi` oidgpgidogitg. Togig ui gcgjpk`.
5) x8+ 8x 2 : 5
m:
h:
f :
=) =5x8+ =1x 2 : 5
m:
h:
f :
8) 6x8+ 8x: 5
m:
h:
f :
f`jpkgtm
=
8
2 m
6) x8 =5x+ 84 : 5
m :
h:
f :
4) 1x8 2 : 5
m :
h:
f :
7) x8+ ==x+ 84 : 5
m :
h:
f :
H. Jmrqug f`i uim "R" gk fumdr` dg km `pfoi qug rgsp`idg f`rrgftmjgitg m fmdm prgnuitm.
=) Dg kms pfo`igs sonuogitgs, fuk gs uim gfumfoi fumdrtofm0 8x+ ? : =1
7 6 7 : =5
x4 8x8+ 7 : 5
6x8+ 8x 2 : 5
8) Fuk gs gk sgnuid` trjoi` gi km gfumfoix8 =>x+ >8 : 50 5
x8
>8
=>x
6) Fuk gs gk trjoi` oidgpgidogitg gi km gfumfoi 8x8+ 1x ? : 50 8
1?
1x
4) Fuk gs km amft`rozmfoi f`rrgftm dgx8+ 6x 740 (x+ ?)(x 1)
(x+ 1)(x ?)
(x+ ?)(x+ 1)
(x ?)(x 1)
Mut`f`itr`k
81ONGX U`k`fbof
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Mftovodmd 8. Urmftoqug k` mprgidod`M. Xgsugkvm p`r amft`rozmfoi kms gfumfo`igs fumdrtofms f`jpkgtms y gsfrohm k`s rgsuktmd`s dgx
=yx
8.
=) x8+ 7x+ 1 : 5
x=:
x
8:
8) x8+ 8x 2 : 5
x=:
x
8:
6) x8+ 8x+ = : 5
x=:
x
8:
4) x8 2x+ =1 : 5
x=:
x
8:
7) x8+ 1x 8> : 5
x=:
x
8:
1) x8+ 2x+ =7 : 5
x=:
x
8:
>) 6x8+ 7x+ 8 : 5
x=:
x
8:
2) 8x8+ >x 4 : 5
x=:
x
8:
H. Xgsugkvm p`r amft`rozmfoi kms gfumfo`igs fumdrtofms oif`jpkgtms y gsfrohm k`s rgsuktmd`s dgx=yx
8
=) x8 6x: 5
x=:
x
8:
8) 4x8+ 85x: 5
x=:
x
8:
6) 6x8 =8x: 5
x=:
x
8:
4) x8 ? : 5
x=:
x
8:
7) x8 4? : 5
x=:
x
8:
1) x8 =7 : 5
x=:
x
8:
8>Jmtgjtofm Vgjmim =2
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@hsgrvg fmdm gfumfoi dg projgr nrmd`, rgsukvmkm jgitmkjgitg y gsfrohm gk rgsuktmd` gi km kigm
f`rrgsp`idogitg. Kugn`, f`jprughg su rgsuktmd` gi km gfumfoi `ronoimk.
M.=) x+ ? : => x:
8) x+ =7 : =5 x:
6) x+ 87 : 75 x:
4) x+ =8 : 65 x:
7) x+ 47 : 15 x:
1) x+ =5 : ?5 x:
>) x+ =1 : 45 x:
2) x+ =2 : 65 x:
?) x+ =4 : 74 x:
=5) x+ 85 : 68 x:
==) x+ =5 : 77 x:
=8) x+ 87 : 75 x:
=6) x 2 : 4 x:
=4) x 1 : =8 x:
=7) x ? : =5 x:
=1) x 2 : 88 x:
=>) x 6 : =8 x:
=2) x 5 : 47 x:
=?) x =5 : =1 x:
85) x =7 : 47 x:
8=) x 15 : 65 x:
88) x 87 : 87 x:
86) x 85 : 68 x:
84) x 61 : 14 x:
H.
=) ?y: 61 y:
8) 4y : 84 y:
6) 2y : 14 y:
4) 2y: 45 y:
7) 6y : 61 y:
1) 7y: 47 y:
>) 6y :65 y:
2) 8y: 44 y:
?) 7y : 15 y:
=5) 2y : ?1 y:
==) =8y : 61 y:
=8) =7y :15 y:
=6)y
1: 4 y:
=4) y2 : 1 y:
=7)y
>: > y:
=1)y
6: =8 y:
=>)y
7: 4 y:
=2)y
=: =7 y:
=?)y
=8: 6 y:
85) 75y : 87 y:
8=) 42y : 4 y:
88) >7y : =7 y:
86) ?5y : 65 y:
84) 15y : 4 y:
82ONGX U`k`fbof
Mnokodmd dg fkfuk` jgitmk
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M. Xgsugkvm gi su fumdgri` ` gi b`cms mpmrtg kms gfumfo`igs fumdrtofms. Wtokofg gk jt`d` qug dgsgg.
=) 8x8: 1x
8) x8
: =5x
6) 6x8: ?x
4) x8 2x: 5
7) 7x8 ?x: 5
1) 2x8+ 4x: 5
>) 4x8: 87
2) 2=x8: =1
?) x8 4 : 5
=5) =55x8: 14
==) x8 14 : 5
=8) =1x8 ? : 5
=6) x8+ 68 : =8x
=4) 8x8
+ =6x: 84
=7) x8 4x 68 : 5
=1) x8 6x =5 : 5
=>) 1x8x =8 : 5
=2) =5x8+x 6 : 5
=?) 4x8+ =>x: =2
85) 1x8+ =>x+ 7 : 5
8=) x(6x =7) : 48
88) x(x 8) : 67
86) (x+ 6)(=5 x) : 2x
84) (7x + ==) : =8x
H. Ukmitgg uim gfumfoi qug dgsfrohm k` giuifomd` gi fmdm pr`hkgjm. Kugn`, rgsukvmkm y f`jprughgsu rgsuktmd`. Xgmkofg gk gcgrfofo` gi su fumdgri`.
=) Wi rgftinuk` gs 7vgfgs js kmrn` qug gk d`hkg dg su mifb`. Vo su rgm gs ?5 fj8, fuit`
jodg gk kmrn` y gk mifb`0
8) Dgtgrjoig gk rgm dg ui fumdrmd` so su domn`imk jodg 75 fj.
6) Gk rgm dg uim jgsm rgftminukmr gs dg 8.>5 j8. Dgtgrjoig gk kmrn` y gk mifb` dg km jgsm, so gk
mifb` gs 5.>7vgfgs su kmrn`.
4) Km sujm dg d`s ijgr`s gs 7y su pr`duft` gs 24. Bmkkg dofb`s ijgr`s.
7) Dgtgrjoig kms dojgiso`igs dg ui tgrrgi` rgftminukmr qug togig ui pgrjgtr` dg 825 jy su
rgm gs dg 4555 j8.
1) Wi cmrdi rgftminukmr dg 75 j dg kmrn` p`r 64 jdg mifb` gst r`dgmd` p`r ui fmjoi` dg
mrgim uioa`rjg. Bmkkg km mifburm dg dofb` fmjoi` so sg smhg qug su rgm gs745 j
.>) D`s ijgr`s imturmkgs sg doagrgifomi gi d`s uiodmdgs y km sujm dg sus fumdrmd`s gs 725.
Bmkkg k`s ijgr`s.
2) Bmkkg km k`inotud dg k`s kmd`s dg ui troinuk` rgftinuk` f`i ui rgm dg 1 fj8, pgrjgtr` dg =8
fjg bop`tgiusm dg 7 fj.
?) Gi tsgnuid`s km mkturm bgi pogs, s`hrg gk iovgk dgk sugk`, dg ui pr`ygftok gst dmdm p`r km
gfumfoi b: 845t =1t8. Fuit` togjp` kg t`jmr mk pr`ygftok kkgnmr m uim mkturm dg ?55pogs
s`hrg gk iovgk dgk sugk`0
8?Jmtgjtofm Vgjmim =2
Xmz`imjogit` knof`
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Jmrqug f`i ui fbgqug km fmsokkm qug jgc`r oidoqug su rgidojogit`.k`nrmd` gipr`fgs`
i`k`nrmd`
Dgspusdggstudo
mr...
F`i`zf` km bost`rom dg k`s soni`s jmtgjtof`s.
Odgitof` gk nrmd` dg kms gfumfo`igs.
Xgf`i`zf` gfumfo`igs fumdrtofms f`jpkgtms g oif`jpkgtms.
Xgsugkv` gfumfo`igs dg sgnuid` nrmd` p`r amft`rozmfoi y p`r arjukm.
Xgsugkv` f`i mnokodmd gfumfo`igs dg projgr nrmd`.
Bmkk` km vmromfoi dgk rgm dg fumdroktgr`s.
F`jpmrgj`s rgms
Gi gstm sgffoi i`tmrgj`s fj` uim pgqugm vmromfoi gi km jgdodm dg k`s
kmd`s dg ui fumdrmd` bmfg uim nrmi doagrgifom gi gk rgm. Urgstg mtgifoi mk
fms` sonuogitg.
F`isodgrgj`s trgs fumdrmd`s f`i kmd`sx,x+ 8yx+ 7qug sg jugstrmi m f`ito-
iumfoi. Kugn`, rgmkofg k` qug sg kg podg gi fmdm oifos`.
x
=) Gsfrohm uim gfumfoi fumdrtofm pmrm gk rgm dg fmdm fumdrmd`.
M=: M8: M6:
8) Vo gk rgm dgk projgr fumdrmd` gs dg 61 fj8, fuit` jodg fmdm fumdrmd` p`r
kmd` y fukgs s`i kms rgms0
=:
8:
6:
M=: M
8: M
6:
6) Vo gk rgm dgk tgrfgr fumdrmd` gs dg 887 fj8, fuit` jodg fmdm fumdrmd`
p`r kmd` y fuk gs gk rgm dg k`s `tr`s d`s0
=: 8: 6:
M=: M
8: M
6:
x + 8x + 7
65ONGX U`k`fbof
Xgvosg su mprgidozmcg
Dgsmrr`kkg iugvms bmhokodmdgs
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=?=
8 6 4
6
4
7
8
=
5
x
y
=:o
x8+
y8:
r8
Oigfumfo`igs
]u gif`itrmr gstm sgjmim0
Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom
Oigfumfo`igs
Fkmsofmfoi dg uim sgrog dg ijgr`s dg mfugrd` m su rgkmfoi.
Gi hsqugdm dg dgtmkkgs
Gstm sgjmim k`nrmr3
F`jprgidgr y dgsfrohor qu gs uim oigfumfoi.
Doagrgifomr uim oigfumfoi dg uim dgsonumkdmd jmtgjtofm.
Xgs`kvgr f`rrgftmjgitg oigfumfo`igs.
Xgprgsgitmr km s`kufoi dg oigfumfo`igs gi km rgftm iujrofm.
@rdgimr y fkmsoaofmr uim sgrog dg ijgr`s rgspgft` m su rgkmfoi.
Urmftofmr km mnudgzm vosumk gi km hsqugdm dg dgtmkkgs.
6=Jmtgjtofm Vgjmim =?
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Umrm f`jgizmr!
Kms dgsonumkdmdgs jmtgjtofms gi km vodm domrom
Kms dgsonumkdmdgs jmtgjtofms sg gjpkgmi t`d` gk togjp` gi gk juid` qug
i`s r`dgm. Vmhgr oitgrprgtmrkms gs ojp`rtmitg pmrm jmigcmr f`rrgftmjgitg km
oia`rjmfoi qug pgrfohoj`s dg iugstr` git`ri`.
Wstgd sg gifugitrm f`i dgsonumkdmdgs jmtgjtofms gi jufbms `fmso`igs,
pgr` quozs i` k` i`tm p`rqug ym kg s`i amjokomrgs. U`r gcgjpk`, pogisg gi k`s
gsfgimro`s sonuogitgs3
Votumfoi Dgsonumkdmd jmtgjtofm
Mkturm jxojm dg ui fmjoiUmrm qug ui fmjoi pugdm pmsmr dghmc`
dg mkni pugitg, dghg tgigr uim mkturm
jgi`rm 4.=7 jgtr`s.
Tmrcgtm dg frdot`
Gk pmn` jioj` jgisumk dghg sgr jmy`r` onumkmk =5 % dg km dgudm t`tmk.
Mkturm jiojm pmrm ui cugn`Gi mknui`s cugn`s dg kms agroms, pmrm qug
ui io` pugdm suhorsg dghg tgigr uim
mkturmjmy`r ` onumkmk kjotg pgrjotod`.
Kjotg dg vgk`fodmd
Km vgk`fodmd pgrjotodm gi kms fmrrgtgrms
dghg sgr jgi`r ` onumkm >5 lj/b.
Joiut`s dg togjp` dg morg Gi mkni dgtgrjoimd` pkmi, gk togjp`
dg morg gi gk fgkukmr qug f`isujoj`s
mk jgs dghg sgr jgi`r ` onumk m =75joiut`s.
K` i`t0 T`dms kms sotumfo`igs mitgro`rgs s`i dgsonumkdmdgs, y muiqug ustgd
i` sg d fugitm, ym utokozm gk rmz`imjogit` jmtgjtof` pmrm mimkozmr gstms y
`trms sotumfo`igs sojokmrgs. Qgrgj`s gstg tgjm gi km sonuogitg sgffoi.
M trmhmcmr!
Uogisg y gsfrohm uim forfuistmifom js gi km qug utokofg kms dgsonumkdmdgs jmtgjtofms. Gxpkoqug dg
a`rjm hrgvg su rgspugstm.
?5 fj
Lj/b
joi
68ONGX U`k`fbof
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=. Oigfumfo`igsMitgs dg gstudomr kms oigfumfo`igs, dghgj`s tgigr fkmr` qu s`i kms dgsonumkdm-
dgs jmtgjtofms, dg kms fumkgs kgmj`s gi km sgffoi mitgro`r.
Wim dgsonumkdmd jmtgjtofmgs uim rgkmfoi dg `rdgi qug sg dm gitrg d`s
vmk`rgs fumid` gst`s s`i dostoit`s< pmrm oidofmr gsg rdgi sg gjpkgmi k`s soni`s
dg rgkmfoi jmy`r qug (9), jgi`r qug (;), jmy`r ` onumk qug () y jgi`r ` onumk
qug (). Qgmj`s ui`s gcgjpk`s3
2?987, sg kgg3 "2?gs jmy`r qug 87"
61;8, sg kgg3 "61gs jgi`r qug 8"
Tgiogid` fkmr` k` mitgro`r, p`dgj`s pmsmr mk tgjm dg gstm sgjmim, kms oigfumfo`igs.
Wim oigfumfoigs uim dgsonumkdmd jmtgjtofm qug prgsgitm mk jgi`s uimvmromhkg
gi mknui` dg sus jogjhr`s, p`r gs` tmjhoi sg kg f`i`fg f`j` dgsonumkdmd
mknghrmofm. Wi`s gcgjpk`s3
y 4==
x+ 4;2
y 11
=.= Fkmsoaofmfoi dg kms oigfumfo`igs
m. U`r gk soni` qug prgsgitmiVo km oigfumfoi prgsgitm gk soni` dg rgkmfoi "jmy`r qug" ` "jgi`r qug", sg
dgi`joim oigfumfoi dg sgitod` gstroft`, p`rqug i`oifkuyg m sus gxtrgj`s.
x 9=5Gs gstroftm p`rqug gk vmk`r dg x sgpugdg mfgrfmr m =5k` js p`sohkg (=5.=,=5.5=, =5.55=), pgr` i k` oifkuyg.
y;6Gk vmk`r dg ysg pugdg mfgrfmr m6 k`js p`sohkg (8.?, 8.??, 8.???), pgr`i`k` oifkuyg.
Vo km oigfumfoi prgsgitm gk soni` dg rgkmfoi "jmy`r ` onumk qug" ` "jgi`r `
onumk qug", sg dgi`joim oigfumfoi gi sgitod` mjpko`` i` gstroft`, p`rqug s
oifkuyg m sus gxtrgj`s.
y 2Gk vmk`r dgypugdg sgr 2` jgi`r qug2, gs dgfor, sk` oifkuyg.
x7Gk vmk`r dgxpugdg sgr7` jmy`r qug
7, gs dgfor, sk` oifkuyg.
66Jmtgjtofm Vgjmim =?
Gk juid` dg km jmtgjtofm
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h. U`r su nrmd`
Mk onumk qug kms gfumfo`igs, gk nrmd` dg uim oigfumfoi gst dmd` p`r gk
gxp`igitg jmy`r dg km vmromhkg.
x 2 ;1, gs dg projgr nrmd` ` koigmk
y
8+ ? 88, gs dg sgnuid` nrmd` ` fumdrtofm
j6+ 86 965, gs dg tgrfgr nrmd` ` fhofm, gtf.
=.8 Xgs`kufoi dg oigfumfo`igs dg projgr nrmd`
Km s`kufoidg uim oigfumfoi gs gk f`icuit` dg vmk`rgs qug pugdg t`jmr km
vmromhkg y qug vgrofmi km oigfumfoi.
U`dgj`s gxprgsmr km s`kufoi dg d`s a`rjms, p`r jgdo` dg uim rgprgsgitmfoinrfm gi km rgftm iujrofm ` f`j` ui oitgrvmk`.
Gk pr`fgdojogit` pmrm rgs`kvgr uim oigfumfoi dg projgr nrmd` gs juy sojokmr mk
qug gjpkgmj`s pmrm rgs`kvgr uim gfumfoi. U`inm mtgifoi mk gcgjpk` sonuogitg3
Xgs`kvmj`s km sonuogitg oigfumfoi3
Xgmkozmj`s km trmisp`sofoi dg trjoi`s.
Xgdufoj`s trjoi`s sgjgcmitgs y
gif`itrmj`s km rgspugstm.
Mb`rm, rgprgsgitmj`s km rgspugstm f`j` oitgrvmk` y gi a`rjm nrfm gi km rgftm
iujrofm.
(,8Zsg kgg3 km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg oiaiot` ignmtov`
bmstm 8, oifkuygid` gk 8.
K` vg0 Vg trmhmcm dg a`rjm juy pmrgfodm m kms gfumfo`igs.
Tginm gi fugitm qug, mk gxprgsmr km rgspugstm f`j` oitgrvmk`, fumid` ui kjotg `
gxtrgj` sg gxtogidg bmstm oiaiot`, sogjprg utokozmj`s pmritgsos.
8x + = x + 6
8x x6 =
x 8
Mk rgprgsgitmr kms`kufoi, utokozmj`spmritgsos( )so gs uimoigfumfoi gstroftm yf`rfbgtgs\ Zso gs uimoigfumfoi mjpkom.
1 7 4 6 8 = 5 = 8 6 4 7 1
Z
64ONGX U`k`fbof
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Juy ojp`rtmitg!
Fumid` ui ijgr` ignmtov`juktopkofm ` dovodg ui kmd` dg km oigfumfoi y pmsm mk
`tr` bmfogid` km pgrmfoi f`itrmrom, gk soni` dg km dgsonumkdmd fmjhom dg sgitod`.
Xgs`kvmj`s3
Mpkofmj`s km pr`pogdmd dostrohutovm.
Xgmkozmj`s km trmisp`sofoi dg trjoi`s y
`pgrmj`s.
Dgspgcmj`s km vmromhkg y `pgrmj`s. F`j`
gk ijgr` qug pmsm m dovodor gs ignmtov`, gk
soni` dg dgsonumkdmd fmjhom dg sgitod`.
Vojpkoaofmj`s y bmkkmj`s km rgspugstm.
(, =6 ) sg kgg3 km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg oiaiot`ignmtov` bmstm =
6, soi oifkuor m =
6.
1 7 4 6 8 = 5 = 8 6 4 7 1
Tmjhoi sg pugdgi prgsgitmr oigfumfo`igs f`i d`s soni`s dg rgkmfoi, kkmjmdms
d`hkgs ` dgtrgs f`jp`igitgs. Qgmj`s gk pr`fgs` pmrm rgs`kvgrkms.
Xgs`kvmj`s3
Dovodoj`s km gxprgsoi gi d`s pmrtgs. Kmprojgrm, a`rjmdm p`r gk projgr y sgnuid`
f`jp`igitg. Km i`jhrmj`s =. Km sgnuidm,
a`rjmdm p`r gk sgnuid` y tgrfgr f`jp`igitg.
Km i`jhrmj`s 8.
Kms trmhmcmj`s p`r sgpmrmd`. Urojgr`, rgs`kvgj`s =.
Odgitoaofmj`s su s`kufoi f`j` s=.
Kugn`, rgs`kvgj`s 8. Odgitoaofmj`s su s`kufoi
f`j` s8.
F`itoiugj`s km s`kufoi dg gstm oigfumfoi d`hkg gi km pnoim sonuogitg.
6(4x+ =) 9=
=8x 69=
=8x 94
x ; 4=8
x ; =6
(
4 + 6x;7x+ 8 >x+ =
4 + 6x; 7x+ 8 >x+ =
4 + 6x; 7x+ 8
4 8; 7x 6x
8 ; 8x
= ;x(s=)
7x+ 8 >x+ =
7x >x= 8
8x=
x=
8(s8)
=
8
67Jmtgjtofm Vgjmim =?
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Gcgrfofo` =
Xgsugkvm kms oigfumfo`igs sonuogitgs gi su fumdgri`. Xgfugrdg trmzmr km rgprgsgitmfoi nrfm.
=) 7x =8 9 6x 4 8) x 7;8x 1
6) 8(6x + =) 6(8x 8)9 x + 1 4) (x =)8
==;(x 8)8
7)(x+ 6)(x + 8)
6 4 ;
x(x + 8)
61)
6x + =
>
8 4x
6
7x 4
=4+
>x
1
>) 4x+ = 6 7x9=5 >x 2) 4x8x+ = 6x+ 8
Mb`rm, dghgj`s rgmkozmr uim oitgrsgffoigitrg k`s f`icuit`s dg vmk`rgs
qug f`jprgidg fmdm s`kufoi. Km rgspugstm pmrm km oigfumfoi d`hkg sgri
mqugkk`s vmk`rgs qug sg oifkuymi gi mjhms s`kufo`igs(s=y s8).
F`j` p`dgj`s vgr, mjhms s`kufo`igs (s=y s8) sg oitgrsgftmi gi gk oitgrvmk`
(=, ), p`r k` qug gstm sgrm km rgspugstm pmrm km oigfumfoi `ronoimk (4 + 6x;
7x+ 8 >x+ =). Gxprgsmj`s gstg rgsuktmd` f`j`3
(=, )sg kgg3km s`kufoi f`jprgidg t`d`s k`s ijgr`s dgsdg =bmstm oiaiot`
p`sotov`, soi oifkuor gk =.
6 8 = 5 = 8 6
(
(
\
s=
s8
=
8
61ONGX U`k`fbof
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=.6 Xgs`kufoi dg oigfumfo`igs dg sgnuid` nrmd`
Umrm rgs`kvgr oigfumfo`igs dg sgnuid` nrmd` i`s hmsmj`s gi gk pr`fgdojogit`
dg km arjukm fumdrtofm, gk fumk mprgidoj`s km sgjmim mitgro`r.
Xgs`kvmj`s3
Onumkmj`s gk p`koi`jo` m fgr`.
Gjpkgmj`s km arjukm fumdrtofm pmrm
bmkkmr kms rmfgs dg km gfumfoi.
Whofmj`s k`s vmk`rgs dg kms rmfgs gi km
rgftm iujrofm3
4 6 8 = 5 = 8 6 4
I`tmj`s qug sg a`rjmi trgs sgnjgit`s3
(, 78 ), (7
8, 6)y(6, )
Mb`rm, gsf`ngj`s ui vmk`r dg prughm
pmrm fmdm sgnjgit` y k` gvmkumj`s gi
km gfumfoi `ronoimk. F`i gst` bmkkmj`sgk soni`dg fmdm sgnjgit`.
Km s`kufoi gst f`jpugstm p`r k`s sgnjgit s (` gk sgnjgit`) qug tginmi
gk josj` soni` qug gk projgr trjoi` dg km oigfumfoi ronoimk. Gi gstg fms`,
sgrm p`sotov`(8x8).
4 6 8 = 5 = 8 6 4
Km s`kufoi gs gk projgr y tgrfgr sgnjgit`.
Git`ifgs, gxprgsmj`s km rgspugstm gi a`rjm dg oitgrvmk` y gi km rgftm iujrofm3
(, 78 )W (6,) sg kgg3km s`kufoi f`jprgidg dgsdg oiaoiot` ignmtov`bmstm
7
8y dg 6m oiaoiot` p`sotov`.
4 6 8 = 5 = 8 6 4
8(4)8 (4) =7 : +8=
8(5)8 (5) =7 :=7
8(4)8 (4) =7 : +=6
8x8
x =7 9 5
8x8x =7 9 5
x:(=) (=)8 4(8)(=7)
8(8)
x:= ==
4
x=: 6 x