I Tavi
I Tavi I Tavi
I Tavi 26 nax. 1 kvadratul gantolebas SesaZlebelia hqondes erTi amonaxseni, ori amonaxseni an arc erTi amonaxseni. amovxsnaT paragrafis dasawyisSi dasmuli amocanis Sesabamisi gantoleba: x 2 + x – 6 = 0 gantolebas mivceT saxe: x 2 = –x + 6 avagoT y = x 2 da y = –x + 6 funqciaTa grafikebi. radgan sigrZe arauaryofiTi ricxviT gamoisaxeba, amitom amocanis amonaxseni iqneba A wertilis abscisa. e. i. x = 2. maSasadame, Tavdapirveli kvadratis gverdis sigrZe 2 m-ia. magaliTi 1 amoxseniT grafikulad x 2 = m (1) gantoleba. amoxsna: avagoT y = x2 da y = m funqciaTa grafikebi. Tu davakvirdebiT 1-el naxazs, advilad davaskvniT, rom erTi amonaxseni, roca m = 0 x 2 = m gantolebas aqvs ori amonaxseni, roca m > 0 arc erTi amonaxseni, roca m < 0 kerZod, Tu m = 0, maSin (1) gantolebis amonaxseni iqneba x = 0. xolo, roca m > 0, maSin (1) gantolebis amonaxsnebi iqneba x = 1 da x = − (axseniT, ratom?). 2 amrigad, m0 x∈∅ x∈{0} x∈{ ; − } magaliTi 2 amoxseniT grafikulad Semdegi gantoleba: 2x2 – 3x + 5 = 0. amoxsna: b) (2x2 – 3x + 5 = 0) ⇔ (x2 + 3 5 x + 2 2 = 0) ⇔ (x2 = 3 5 x – 2 2 ) avagoT y = x2 da y = 3 5 x – funqciaTa grafikebi. radgan 2 2 miRebul grafikebs saerTo wertili ara aqvs, maSasadame gantolebas amonaxseni ar eqneba. e.i. amonaxsenTa simravle iqneba ∅ .
magaliTi 3 amoxseniT gantolebebi: a) x 2 – 5 = 0; b) 3x 2 + 2 = 0; g) (x – 7) 2 = 10. amoxsna: a) (x2 − 5 = 0) ⇔ (x2 = 5) ⇔ (x = ± ); b) (3x2 + 2 = 0) ⇔ (x2 = – 2 3 ) ⇔ (x ∈ ∅); g) ((x − 7) 2 x – 7 = x = 7 + = 10) ⇔ x – 7 = – ) ⇔ x = 7 – e.i. x = 7 + 1 ; x = 7 − 2 . SeavseT gamotovebuli adgilebi: 1. x 2 – 2x – 3 = 0 gantolebis pirveli koeficientia ? , meore _ ? , mesame _ ? . 2. Tu y = x2 da y = kx + b wrfes aqvs a) erTi saerTo wertili; b) ori saerTo wertili; g) arc erTi saerTo wertili Sesabamisad, x2 – kx – b = 0 gantolebas eqneba a) ? amonaxseni; b) ? amonaxseni; g) ? amonaxseni. 3. a) Tu x 2 = 3, maSin x = ? ; b) Tu x 2 = –5, maSin x = ? ; g) Tu x 2 = 16, maSin x = ? . 4. f : x → x2 da g: x → – 1 x – 7 funqciaTa gadakveTis wertilebis 2 abscisebi aris ? = 0 gantolebis amonaxsnebi. savarjiSoebi: 1 gantoleba amoxseniT grafikulad: a) x2 + 3x + 2 = 0; b) x2 + 5x + 4 = 0; g) x2 + 2x + 4 = 0; d) x2 = 1; e) 0,1x2 – 2 = 0; v) 5,4x2 + 3 = 0; z) x2 + 4x + 4 = 0; T) x2 + 3x + 5 = 0; i) 2x2 – 2 = –3x. 2 amoxseniT gantoleba: a) x2 = 36; b) x2 = 0; g) x2 = –4; d) x2 = 0,64; e) x2 = 5 ; v) –3x2 = –48; z) 0,2x2 + 3 = 0; T) 2x2 = 40; i) x2 = ; k) x2 + 3 = 12; l) x2 – 0,69 = 1; m) 2 – x2 = 9; n) (x2 – 5) 2 = 49; o) (2x + 1) 2 = 17; p) (1 – 3x) 2 = 18; J) (x+2)(x–2)=12; r) (2x – 7)(2x + 7) = 11; s) (3x + 2) 2 – 12x + 21 = 0; t) (x+1) 2 + x = 3x–1; u) 4x2 –(2–5x2 ) = 1 2 x2 ; f) 3x2 – 5x = 6(5x2 – 2) – 5x. ) Tu m>0, maSin (x2 = m) ⇔ ⇔ x = x = – ) aseve, ((x + a) 2 = m) ⇔ ⇔ x + a = x + a = – ) yuradReba! gantolebis grafikulad amoxsnisas vRebulobT mis miaxloebiT amonaxsens! 27
- Page 3: nana jafariZe maia wilosani nani
- Page 6 and 7: IV Tavi . . . . . . . . . . . . . .
- Page 8 and 9: 6 rogor visargebloT wigniT wignze m
- Page 10 and 11: I Tavi 8 f funqciaa h ar aris funqc
- Page 12 and 13: I Tavi gasaxseneblad! p(A) = m n ,
- Page 14 and 15: I Tavi -4 12 4 a) savarjiSoebi: 1 r
- Page 16 and 17: I Tavi 14 4) rodis moawyves turiste
- Page 18 and 19: I Tavi 16 2 amovicnoT wrfivi funqci
- Page 20 and 21: I Tavi 18 SeavseT gamotovebuli adgi
- Page 22 and 23: I Tavi 20 proeqti: gazomvebidan fun
- Page 24 and 25: I Tavi 22 y 10 -3 -2 -1 0 1 2 3 9 8
- Page 26 and 27: I Tavi 24 15% 1,5 < n ≤ 2 20% 1 <
- Page 30 and 31: I Tavi 28 3 dawereT kvadratuli gant
- Page 32 and 33: I Tavi davazustoT! roca D = 0, kvad
- Page 34 and 35: I Tavi 32 amovxsnaT formuliT (3x 2
- Page 36 and 37: I Tavi 34 z) 10x 2 −120 + 6x = 98
- Page 38 and 39: I Tavi navis sakuTari siCqarea navi
- Page 40 and 41: I Tavi 38 6 amovxsnaT jgufuri mecad
- Page 42 and 43: I Tavi x 1 da x 2 ricxvebi arian x
- Page 44 and 45: I Tavi 42 17 amoxseniT gantoleba: a
- Page 46 and 47: I Tavi 44 yuradReba: Tu D=0, maSin,
- Page 48 and 49: I Tavi gavixsenoT! x = a, maSin a 2
- Page 50 and 51: I Tavi 48 I Tavis damatebiTi savarj
- Page 52 and 53: I Tavi 50 K y = x 2 A B -3 -2 -1 K
- Page 54 and 55: I Tavi 52 39 A punqtidan B punqtisa
- Page 56 and 57: I Tavi narevSi gaxsnili nivTierebis
- Page 58: 56 I TavSi Seswavlili masalis mokle
magaliTi 3<br />
amoxseniT gantolebebi: a) x 2 – 5 = 0; b) 3x 2 + 2 = 0; g) (x – 7) 2 = 10.<br />
amoxsna:<br />
a) (x2 − 5 = 0) ⇔ (x2 = 5) ⇔ (x = ± );<br />
b) (3x2 + 2 = 0) ⇔ (x2 = – 2<br />
3 ) ⇔ (x ∈ ∅);<br />
g) ((x − 7) 2 x – 7 = x = 7 +<br />
= 10) ⇔<br />
x – 7 = – ) ⇔<br />
x = 7 –<br />
e.i. x = 7 + 1 ; x = 7 − 2 .<br />
SeavseT gamotovebuli adgilebi:<br />
1. x 2 – 2x – 3 = 0 gantolebis pirveli koeficientia ? , meore<br />
_ ? , mesame _ ? .<br />
2. Tu y = x2 da y = kx + b wrfes aqvs a) erTi saerTo wertili;<br />
b) ori saerTo wertili; g) arc erTi saerTo wertili<br />
Sesabamisad, x2 – kx – b = 0 gantolebas eqneba a) ? amonaxseni;<br />
b) ? amonaxseni; g) ? amonaxseni.<br />
3. a) Tu x 2 = 3, maSin x = ? ; b) Tu x 2 = –5, maSin x = ? ; g) Tu x 2 = 16,<br />
maSin x = ? .<br />
4. f : x → x2 da g: x → – 1<br />
x – 7 funqciaTa gadakveTis wertilebis<br />
2<br />
abscisebi aris ? = 0 gantolebis amonaxsnebi.<br />
savarjiSoebi:<br />
1 gantoleba amoxseniT grafikulad:<br />
a) x2 + 3x + 2 = 0; b) x2 + 5x + 4 = 0; g) x2 + 2x + 4 = 0;<br />
d) x2 = 1; e) 0,1x2 – 2 = 0; v) 5,4x2 + 3 = 0;<br />
z) x2 + 4x + 4 = 0; T) x2 + 3x + 5 = 0; i) 2x2 – 2 = –3x.<br />
2 amoxseniT gantoleba:<br />
a) x2 = 36; b) x2 = 0; g) x2 = –4;<br />
d) x2 = 0,64; e) x2 = 5 ; v) –3x2 = –48;<br />
z) 0,2x2 + 3 = 0; T) 2x2 = 40; i) x2 = ;<br />
k) x2 + 3 = 12; l) x2 – 0,69 = 1; m) 2 – x2 = 9;<br />
n) (x2 – 5) 2 = 49; o) (2x + 1) 2 = 17; p) (1 – 3x) 2 = 18;<br />
J) (x+2)(x–2)=12; r) (2x – 7)(2x + 7) = 11; s) (3x + 2) 2 – 12x + 21 = 0;<br />
t) (x+1) 2 + x = 3x–1; u) 4x2 –(2–5x2 ) = 1<br />
2 x2 ; f) 3x2 – 5x = 6(5x2 – 2) – 5x.<br />
)<br />
Tu m>0, maSin<br />
(x2 = m) ⇔<br />
⇔ x =<br />
x = – )<br />
aseve,<br />
((x + a) 2 = m) ⇔<br />
⇔<br />
x + a =<br />
x + a = –<br />
)<br />
yuradReba!<br />
gantolebis grafikulad<br />
amoxsnisas<br />
vRebulobT<br />
mis miaxloebiT<br />
amonaxsens!<br />
27
Change language