I Tavi
I Tavi I Tavi
I Tavi x 1 da x 2 ricxvebi arian x 2 –(x 1 +x 2 )x+x 1 x 2 =0 gantolebis amonaxsnebi 40 b) 15=1·15=(–1)(–15)=3·5=(–3)(–5) 1+15=16; –1–15=–16; 3+5=8; –3–5=–8. radgan (–3)(–5)=15 da –3–5=–8, amitom x 2 +8x+15=0 gantolebis amonaxsnebia: x 1 = –3, x 2 = –5. magaliTi 2 SeadgineT kvadratuli gantoleba, Tu misi fesvebia –1 da 5. amoxsna: vietis Teoremis Sebrunebuli Teoremis mixedviT SesaZlebelia SevadginoT dayvanili x2 +px+q=0 gantoleba, sadac p=–(x +x ) da 1 2 q = x x . 1 2 e.i. p = –4 da q = –5. saZiebeli gantolebaa x2 – 4x – 5 = 0. SeavseT gamotovebuli adgilebi: 1 x 2 +5x–7=0 gantolebisTvis x 1 +x 2 = ? da x 1 x 2 = ? . 2 x 2 –7x–1=0 gantolebisTvis x 1 +x 2 = ? da x 1 x 2 = ? . 3 1 da –1 arian x 2 – ? x+ ? =0 gantolebis fesvebi. 4 x 2 +2x–5=0 gantolebis fesvebi ? niSnianebia. savarjiSoebi: 1 gantolebis amouxsnelad ipoveT misi fesvebis jami da namravli (Tuki isini arseboben). a) x 2 –12x+16=0; b) 5x 2 –7x+2=0; g) 3x 2 –2x–1=0; d) 3x 2 +51x–12=0; e) –1,5x 2 –2x+2=0; v) 4x 2 –7=0; z) 7x 2 – 7 x=0; T) ( 3 +1)x 2 –2=0; i) ( 2 –1)x 2 –2x=0. 2 vietis Teoremis gamoyenebiT SeamowmeT aris Tu ara mocemuli simravle Sesabamisi kvadratuli gantolebis amonaxsenTa simravle: a) x2 +7x+12=0 {3;4}; b) m2 g) 8t +3m+2=0 {–2;–1}; 2 –10t+3=0 { 1 3 ; 2 4 }; d) 4x2 +8x–5=0 { 1 5 ; – 2 2 }. 3 SeadgineT kvadratuli gantoleba, romlis amonaxsnebia x da x : 1 2 a) x =1, x =–3; 1 2 d) x = 1 b) x =2, x =4; 1 2 g) x =0,3, x =1,5; 1 2 1 2 , x2 =2 5 ; e) x1 = x2 = – 2 3 ; v) x1 = 3, x 2 = 2 3 ; z) x =2– 1 5, x =2+ 2 5; T) x =x =3 5 –1; 1 2 i) x =1– 1 2 , x =2+ 2 2 .
4 SeadgineT kvadratuli gantoleba, Tu misi fesvebis saSualo ariTmetikuli A-s, xolo saSualo geometriuli G-s tolia: a) A=4, G=2; b) A=16, G=12; g) A=100, G=100. 5 amoxseniT kvadratuli gantoleba vietis Teoremis gamoyenebiT: a) x 2 +5x+6=0; b) x 2 –10x+16=0; g) t 2 +4t–21=0; d) x 2 –3x+2=0; e) x 2 –2x–3=0; v) y 2 –9y+14=0. 6 ipoveT ricxvTa yvela wyvili, romelTa a) jamia 8, xolo namravli 15; b) jamia 7, namravli ki — 10. 7 ipoveT p ricxvi, Tu gantolebis erT-erTi amonaxseni mocemuli ricxvia: a) x 2 –3x+p=0, x 1 =2; b) x 2 +px+4=0, x 1 =–2; g) 2x 2 +px–1=0, x 1 =1; d) x 2 –px+15=0, x 1 =3. 8 gantolebis amouxsnelad daadgineT misi amonaxsnebis niSnebi: a) 3x 2 –2x–1=0; b) 2x 2 +5x+1=0; g) x 2 –4x+3=0; d) y 2 –7x–11=0; e) 0,5y 2 –12y+4=0; v) 5m 2 +8m+1=0. 9 gansazRvreT 5x 2 –4x–7=0 gantolebis fesvebis niSnebi. daadgineT, romeli fesvis modulia meti. 10* m-is ra mniSvnelobebisTvis eqneba gantolebas sxvadasxva niSniani fesvebi? gansazRvreT im fesvis niSani, romlis modulic metia: a) x 2 –7x+m=0; b) x 2 +3x+m=0. 11* x2 –15x+26=0 gantolebis amouxsnelad ipoveT 1 + x1 1 , sadac x x 1 2 da x mocemuli gantolebis fesvebia. 2 12* SeadgineT kvadratuli gantoleba, romlis fesvebi 3-jer metia 15x 2 –7x–3=0 gantolebis fesvebze (gantolebis amouxsnelad). 13* q-s ra mniSvnelobisaTvis iqneba x 2 –6x+q=0 gantolebis fesvebis sxvaoba 2-is toli. 14 p-s ra mniSvnelobisaTvis iqneba x 2 +2px–12=0 gantolebis fesvebis Sefardeba –3-is toli. 15 daamtkiceT, rom ax 2 +bx+c=0 gantolebis fesvi, roca c≠0 ar SeiZleba iyos ricxvi 0. 16* daamtkiceT, rom ax 2 +bx+a=0 gantolebis fesvebi (Tuki arseboben) urTierTSebrunebuli ricxvebia. 41
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I<br />
<strong>Tavi</strong><br />
x 1 da x 2 ricxvebi<br />
arian<br />
x 2 –(x 1 +x 2 )x+x 1 x 2 =0<br />
gantolebis<br />
amonaxsnebi<br />
40<br />
b) 15=1·15=(–1)(–15)=3·5=(–3)(–5)<br />
1+15=16; –1–15=–16; 3+5=8; –3–5=–8.<br />
radgan (–3)(–5)=15 da –3–5=–8, amitom x 2 +8x+15=0 gantolebis amonaxsnebia:<br />
x 1 = –3, x 2 = –5.<br />
magaliTi 2<br />
SeadgineT kvadratuli gantoleba, Tu misi fesvebia –1 da 5.<br />
amoxsna:<br />
vietis Teoremis Sebrunebuli Teoremis mixedviT SesaZlebelia<br />
SevadginoT dayvanili x2 +px+q=0 gantoleba, sadac p=–(x +x ) da<br />
1 2<br />
q = x x . 1 2<br />
e.i. p = –4 da q = –5.<br />
saZiebeli gantolebaa x2 – 4x – 5 = 0.<br />
SeavseT gamotovebuli adgilebi:<br />
1 x 2 +5x–7=0 gantolebisTvis x 1 +x 2 = ? da x 1 x 2 = ? .<br />
2 x 2 –7x–1=0 gantolebisTvis x 1 +x 2 = ? da x 1 x 2 = ? .<br />
3 1 da –1 arian x 2 – ? x+ ? =0 gantolebis fesvebi.<br />
4 x 2 +2x–5=0 gantolebis fesvebi ? niSnianebia.<br />
savarjiSoebi:<br />
1 gantolebis amouxsnelad ipoveT misi fesvebis jami da namravli<br />
(Tuki isini arseboben).<br />
a) x 2 –12x+16=0; b) 5x 2 –7x+2=0; g) 3x 2 –2x–1=0;<br />
d) 3x 2 +51x–12=0; e) –1,5x 2 –2x+2=0; v) 4x 2 –7=0;<br />
z) 7x 2 – 7 x=0; T) ( 3 +1)x 2 –2=0; i) ( 2 –1)x 2 –2x=0.<br />
2 vietis Teoremis gamoyenebiT SeamowmeT aris Tu ara mocemuli<br />
simravle Sesabamisi kvadratuli gantolebis amonaxsenTa simravle:<br />
a) x2 +7x+12=0 {3;4}; b) m2 g) 8t<br />
+3m+2=0 {–2;–1};<br />
2 –10t+3=0 { 1 3<br />
;<br />
2 4 }; d) 4x2 +8x–5=0 { 1 5<br />
; –<br />
2 2 }.<br />
3 SeadgineT kvadratuli gantoleba, romlis amonaxsnebia x da x :<br />
1 2<br />
a) x =1, x =–3; 1 2<br />
d) x = 1<br />
b) x =2, x =4; 1 2 g) x =0,3, x =1,5;<br />
1 2 1<br />
2 , x2 =2<br />
5 ; e) x1 = x2 = – 2<br />
3 ; v) x1 = 3, x 2<br />
= 2 3 ;<br />
z) x =2– 1 5, x =2+ 2 5; T) x =x =3 5 –1; 1 2 i) x =1– 1 2 , x =2+ 2 2 .
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