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58 $1.3. iracionaluri gantolebis amoxsna mizani. problemis gadawyvetisas iracionaluri gantolebis Sedgenisa da amoxs- nis Cvevebis daufleba; iracionaluri gantolebis amoxsnisas tolfasi gantolebis miRebis gaazreba, gareSe fesvis aRmoCenis unaris ganviTareba, koordinatTa meTo- dis gamoyeneba. winapirobebi. sibrtyeze koordinatebisa da or wertils Soris manZilis cod- na; samkuTxedis utolobis, wrfis gantolebis, wrfis RerZebTan gadakveTis povnis codna; gantolebaTa tolfasobis codna. moswavleTa motivaciis konteqstSi gakveTili amocanis dasmiT iwyeba. am amo- canis amoxsnas mivyavarT iseTi gantolebis amoxsnis Ziebaze, romelSic x cvladi kvadratuli fesvis qveSaa. moswavleTa aqtivobebi dakavSirebulia sxvadasxva tipis iracionaluri gan- tolebebis amoxsnebTan, zogi gantolebis amoxsnisas warmoiSveba gareSe fesvi, yu- radRebas vamaxvilebT gareSe fesvis warmoSobis mizezebis axsnaze _ kvadratSi ay- vanisas SeiZleba ar miviRoT mocemuli gantolebis tolfasi gantoleba. am faqtis ilustracia SeiZleba martiv magaliTze _ x–2=0, x 2
fesvi _ x=2 da x=–2, maTgan pirveli gantolebis (x=2 gantolebis) fesvi mxolod
eSe fesvi ar warmoiSveba _ √
√ 6 gantoleba misi orive mxaris kvadratSi ayvaniT miRebuli x=6 gantolebis tolfasia. aqve gavamaxviloT yuradReba imaze, rom kvad- ratSi ayvanisas miRebuli gantoleba mocemuli gantolebis Sedegia _ mocemuli 7. mocemuli utolobis tolfasi utolobaa:
3x 2 +2x+2 (3x+2)(x–1) < 0,
2
(3x+2)(x–1)<0, pasuxi: ( – 23; 1 ) . 8. kvadratis gverdi aRvniSnoT x-iT. markuTxedis gverdebi iqneba x+2 da x–1. pirobiT, (x+2)(x–1)>x 2 , saidanac x>2. x-is umciresi mTeli mniSvnelobaa 3. pasuxi: x=3. Sefasebis sqema: testuris tipis amocanebidan (1-6) TiToeulis swori pasuxi Sefasdes 1 quliT. me-7 da me-8 amocanebidan TiToeulis srulyofili amoxsna Sefasdes 2 quliT, Tumca SeiZleba am amocanebis nawilobrivi Sefasebac. Tu magaliTad, me-7 amocanis amoxsnisas moswavlem utolobis orive mxare gaamravla (3x+2)(x–1)-ze da amiT gadavida kvadratuli utolobis amoxsnaze, maSin Semdgomi swori msjeloba SeiZleba Sefasdes 0,5 quliT. Tu moswavlem gaerTmniSvnelianebiT miRebuli wiladuri gamosaxulebis mniSvnelis nulebic SeinarCuna da sxva Secdoma ar mosvlia, SeiZleba SevafasoT 1,5 quliT.
59 gantolebis yvela fesvi miRebulis fesvebia da mocemuli gantolebis fesvebi am fesvebSi unda veZioT, amitom sakmarisia CavataroT Semowmeba da aRmovaCinoT gar- eSe fesvebi, Tu isini arsebobs. miTiTebebi:
SeiZleba amovxsnaT gantoleba da Semdeg SevarCioT gareSe fesvi: x=-2 akmayofilebs mocemul gantolebas, _ is fesvia. x=2 mocemul gantolebas ar akmayofilebs _ is gareSe fesvia. 8 g) orive mxaris kvadratSi ayvaniT miviRebT x 2 -7x+9=0 kvadratul gantolebas, romlis fesvebia x 1 = √ 13 7+ 2 , x 2 = √ 13 7– 2 . cxadia, √ 13 7+ 2 >3 da amitom mocemuli gantolebis marjvena mxare uaryofiTia, x 1 gareSe fesvia. pasuxi: √ 13 7– 2 .
d) kvadratSi ayvanis Semdeg miRebuli kvadratuli gantolebis fesvebia: x 1 = 8 √ 97 15+ da x 2 =
√ 97 15– . x 2
x 2 gareSe fesvia. pasuxi: 8 √ 97 15+ .
b) II gantolebis fesvebia 1+ √ 2 da 1– √ 2 .
I gantoleba tolfasia |x| 2
√ 2 da x 1 =1+ √ 2 , x 2
√ 2 . gantolebebi ar aris tolfasi. g) da d) savarjiSoebi sasurvelia erTad amoixsnas. SesaZloa am savarjiSoebs moyves msjeloba √
√
√
magaliTebis moyvana. magaliTad, √ (x–5)(x–3) gamosaxulebis gansazRvris area (3; 5), √ x–5 ⋅ √
10 e) gantoleba ase gadavweroT: x 2 –3x+4+3 √ x 2 –3x+4 –10=0 da SemoviRoT aRniSvna √ x 2 –3x+4 =y. miviRebT, y 2 +3y–10=0, y 1
2
2 =2. amrigad, √ x 2 –3x+4 =2, saidanac x=0 an x=3. 11 a) y= √
√ 12–x funqciis gansazRvris are _ mocemul gantolebaSi x cv- ladis dasaSveb mniSvnelobaTa simravle carieli simravlea: Z [ \ ] ] ] ]
Z [
] ] ] ] x≥15 12–x≥0 x≤12, ∅. b) gantolebis marcxena mxare ar aris naklebi 1-ze (ori arauaryofiTi ricxvisa da 1-is jamia), amitom 0-is toli ar gaxdeba. g) marcxena mxare ar aris naklebi √ 3 -ze, marjvena ki 1-ia. d) dasaSvebi mniSvnelobebisTvis 3–x≥0, x≤3. x-is am mniSvnelobebisTvis marcxena mxare arauaryofiTia, marjvena mxare uaryofiTia. 60 e) dasaSveb mniSvnelobaTa simravles gansazRvravs sistema: Z [
] ] ] ] 2x–6≥0 3–x≥0,
v) dasaSveb mniSvnelobaTa simravlea [4; +∞). x-is am mniSvnelobebisTvis √
√
am savarjiSoebze muSaobisas moswvales uviTardeba kvlevisa da analizis unari, alternatiuli gzebidan racionaluri gzis moZiebis unari. garkveulwilad, am ti- pis amocanebiT jamdeba mravali sakiTxis codna _ masSi integrirebulia algebris, analizisa da geometriis sakiTxebi. 12 a) mocemuli gantolebis tolfasi gantolebaa:
|x+4|=(x–4)(x+4), saidanac Z [ \ ] ] ] ]
an Z [ \ ] ] ] ]
x+4=(x-4)(x+4) –(x+4)=(x–4)(x+4). I sistemis amonaxsnebia x=–4 da x=5, II sistemas amonaxsni ara aqvs. d) da e) gantolebebis kvadratSi ayvaniT miiReba x 2
fesvebia x=5 da x=1. d) SemTxvevaSi gareSe fesvia x=1, e) SemTxvevaSi gareSe fesvia x=5. 13 SemoviRoT aRniSvnebi: a) SemTxvevaSi 2x– 1 x+2 = y. b) SemTxvevaSi √ 2x– 1 √ x+2 = y. miviRebT y+ 2y =3 gantolebas, saidanac y=2 an y=1. a) SemTxvevaSi 2x– 1
2x– 1 x+2 = 1, saidanac x 1
2
b) SemTxvevaSi √ 2x– 1 √ x+2 = 2 an
√ 2x– 1 √
= 1, saidanac x 1
2
1 gareSe fesvia. pasuxi: x=3. 16 mizani. koordinatTa meTodis, geometriuli warmodgenebis, algebruli xerxebis gamoyenebis unaris ganviTareba; iracionaluri gantolebis amoxsnis mag- aliTiT sxvadasxva maTematikuri meTodis gamoyenebis unaris ganviTareba. jgufebs gavunawilebT or davalebas: a), g) da b), g). es davalebebi erTgvarad Semajamebeli xasiaTisaa da erToblivi-jgufuri, intensiuri ganxilvisTvis misadagebuli. xaz- gasmiT unda ganvumartoT jgufebs, rom maT moeTxovebaT amoxsnis alternatiuli gzebis demonstrireba. Sefasebac swored am moTxovnebis Sesabamisad moxdeba. Tumca pedagogma uyuradRebod ar unda datovos moswavleTa yoveli racionaluri nabiji da asaxos es SefasebebSi. mniSvnelovania miRebuli Sedegebis prezentaciis xarisxis gaTvaliswinebac, erTmaneTis oponireba da jgufebSi saboloo Sedegis aRqmis done. a) gadavweroT gantolaba ase: √
2 +4 +
√ (x+5) 2 +9=5
√ 2 .
am tolobas SeiZleba aseTi geometriuli warmodgena davukavSiroT: sakoordi- nato sibrtyeze: M(x; 0) wertilidan (abscisaTa RerZis wertilidan) A(0;2) da B(-5;-3) wertilebamde manZilebis jamia 5 √ 2 . 61 MA+MB=5 √ 2 , TviT AB monakveTis sigrZec _ AB=5 √ 2 , amrigad M∈AB. Tu AB wrfis gantolebaa y=kx+b, maSin Z [ \ ] ] ] ] 2=k⋅0+b –3=–5k+b, saidanac b=2, k=1. y=x+2 wrfis abscisaTa RerZTan gadakveTis wertilia (–2;0). pasuxi: x=–2. II xerxi. aviyvanoT mocemuli gantoleba kvadratSi: √
2 +10x+34 2 = ( 5 √ 2 – √ x 2 +4 ) 2 . gamartivebis Semdeg gantoleba CavweroT ase: √ 2(x 2 +4)
=2–x, misi kvadratSi ayvaniT miviRebT: x=–2. b) mocemul tolobas SeiZleba aseTi geometriuli warmodgena davukavSiroT: I da III sakoordinato kuTxeebis biseqtrisaze (y=x wrfeze) mdebare M wertilidan A(1;6) da B(4;2) wertilebamde manZilebis jami 5-ia. vipovoT yvela aseTi wertili. Tu AB wrfis gantolebaa y=kx+b, maSin Z [
] ] ] ] 6=k+b 2=4k+b, saidanac k= – 4 3 , b= 22 3 .
radgan AB=5, MA+MB=5, amitom M aris AB da y=x wrfeebis gadakveTis wertili: Z [ \ ] ] ] ]
y= –
4 3x+ 22 3 , saidanac x=22 7 .
II xerxi. mocemuli gantoleba aviyvanoT kvadratSi.
√ 2x 2 –14x+37 2 = ( 5– √ 2x 2 –12x+20 ) 2
5 √ 2x 2 –12x+20 =x+4, kvlav kvadratSi ayvaniT miiReba: (7x–22) 2
7 .
g) utoloba ase gadavweroT:
√ (x–3) 2 +1+
√ (x–2) 2 +4 ≤
√ 10 .
am utolobas SeiZleba aseTi geometriuli warmodgena davukavSiroT: x RerZze mdebare M(x;0) wertilidan A(3;1) da B(2;–2) wertilebamde manZilebis jami ar aRemateba √ 10 -s. MA+M≤AB, vpoulobT AB monakveTis sigrZes _ AB= √ 10 . am ori pirobidan gamomdinareobs: MA+MB≤ √ 10 . maSin M Zevs AB monakveTze _ M aris AB wrfisa da x RerZis gadakveTis wertili. AB wrfis gantolebaa y=3x-8. es wrfe x RerZs kveTs wertilSi, romlis koordinatebia: y=0, x=83. II xerxi.
√ (x–2) 2 +4≤ √ 10 –
√ (x–3) 2 +1 ,
(*)
fesvqveSa gamosaxulebebi nebismieri x-isTvis dadebiTia. ganvixilavT x-is im mniSvnelobebs, romlebisTvisac √ 10 ≥
√ x 2 –6x+10, anu x 2 –6x≤0, x∈[0;6]. (*) utolobis kvadratSi ayvaniT miviRebT x 2
√ 10(x 2 –6x+10) √ 10(x 2 –6x+10)≤6–x, am utolobis orive mxare arauaryofiTia (gavixsenoT, rom ga- nixileba x∈[0;6]). kvadratSi ayvaniT miviRebT: (3x-8) 2 ≤ 0 saidanac x=83. cxadia, 8 3∈[0;6].
pasuxi: 83. 62 17 I xerxi. mocemuli gantoleba ase gadavw- eroT: √ (x–1) 2 +4+
√ (x–8) 2 +25≤7
√ 2 . movZebnoT abscisaTa RerZze iseTi A(x; 0) wertili, romlidanac B(1; ±2) da C(8; ±5) wertilebamde manZilebis jamia 7 √ 2 . B da C wertilebis ordinatebis niSani calsaxad araa gansazRvruli, radgan CvenTvis maTi kvadratebia cnobili. Tumca, (1; 2) da (1; –2), agreTve (8; 5) da (8; –5) wertilebi OX RerZis mimarT simetriulia da amitom am RerZis nebismieri wertilidan erTi da imave manZilebiTaa daSorebuli. ganvixiloT B(1; 2) da C(8;–5) SemTxveva; maSin
√ (x–1) 2 +4, AC= √ (x–8) 2 +25, BC= √ 7
+7 2
√ 2 .
pirobiT, AB+AC=BC, rac niSnavs, rom A wertili BC monakveTs ekuTvnis. vTqvaT, y=kx+b aris BC wrfis gantoleba. maSin Z
\ ] ] ] ] 2=k+b –5=8k+b, saidanac k=–1, b=3. BC wrfis gantolebaa y=–x+3. am wrfis OX RerZTan gadakveTis wertilia (3; 0). maSasadame, x=3. II xerxi.
√ x 2 –2x+5 + √ x 2 –16x+89 =7 √ 2 .
es gantoleba ase gadavweroT:
√ x 2 –16x+89 =7 √ 2 –
√ x 2 –2x+5 . tolobis orive mxare aviyvanoT kvadratSi. x 2 –16x+89=98–14⋅ √ 2 ⋅
√ x 2 –2x+5 +x 2 –2x+5 –14x–14=–14 √ 2x 2 –4x+10 x+1= √ 2x 2 –4x+10 . Tu x+1³0, maSin
2
2 –4x+10 x 2 –6x+9=0 (x–3) 2
cvladis es mniSvneloba akmayofilebs yvela pirobas. pasuxi: x=3. Yüklə 0,64 Mb. Dostları ilə paylaş: |