I Tavi
I Tavi I Tavi
I Tavi 38 6 amovxsnaT jgufuri mecadineoba kvadratuli utoloba 1. amovxsnaT utoloba x 2 – 4 > 0. davSaloT mamravlebad utolobis marcxena mxare: (x – 2)(x + 2) > 0 . ra SemTxvevaSi iqneba ori ricxvis namravli dadebiTi? (1) ricxviT wrfeze aRniSnulia x–2>0 (wiTlad) da x–20 (wiTlad) x+2 0) ⇔ (x ∈ (–∞;–2) (2;∞)). 2. amovxsnaT x 2 – 3 ≤ 0 utoloba. (x 2 – 3 ≤ 0) ⇔ ((x − )(x + ) ≤ 0). rogori ori ricxvis namravli iqneba uaryofiTi? savarjiSoebi: 1 amoxseniT utoloba: a) (x–2)(x+5)>0; b) x(x+3)≥0; g) (2x–3,5)(x+3,5)≥0; d) (x+1)(x+4)0; v) (5x–1)(x+2)≤0; z) (x–5)(3x+1) x + 2) ⇔ x∈(−∞;–1) (2;∞)) > >
7 vietas Teorema 1 SeadgineT dayvanili kvadratuli gantoleba, Tu cnobilia, rom misi fesvebia: x 1 =2, x 2 =3. vietis Teorema: Tu ax2 + bx + c = 0 kvadratuli gantolebis diskriminanti D≥0, maSin misi fesvebis jami – b s, xolo fesvebis namravli c a s tolia . a damtkiceba: rogorc cnobilia, roca D≥0 kvadratul gantolebas aqvs ori amonaxseni (an gansxvavebuli an toli). x 1 = , x 2 = x + x = 1 2 + − b + = 2 b − 4ac − b − 2a 2 b − 4ac = = –2b b = – 2a a . xolo x ·x = 1 2 · 2 2 2 ( − b) + b − 4ac = 2 4a = c a . amrigad, roca D≥0 x + x = – 1 2 b a , x1 ·x c = 2 a . marTebulia vietis Teoremis Sebrunebuli Teoremac: mocemuli x 1 da x 2 ricxvebi iseTi x 2 +px+q=0 gantolebis amonaxsnebia, sadac p=–(x 1 +x 2 ) da q=x 1 x 2 . magaliTi 1 amoxseniT gantoleba vietis Teoremis gamoyenebiT: a) x 2 –x–12=0; b) x 2 +8x+15=0. amoxsna: a) –12 davSaloT mamravlebad: –12=1·(–12)=(–1)·12=2·(–6)=(–2)·6=(–3)·4=3·(–4). vietis Teoremis Tanaxmad miRebuli Tanamamravlebidan gantolebis amonaxsnebi iqnebian isini, romelTa jamic 1-is tolia: –3+4=1. e.i. x =–3, x =4. 1 2 fransua vieta (1540-1603) !Tu 2 x +bx+c=0, gantolebas (a=1; b;c∉Z) aqvs racionaluri fesvebi, maSin isini aucileblad mTeli ricxvebia. 39
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I<br />
<strong>Tavi</strong><br />
38<br />
6 amovxsnaT<br />
jgufuri mecadineoba<br />
kvadratuli utoloba<br />
1. amovxsnaT utoloba x 2 – 4 > 0.<br />
davSaloT mamravlebad utolobis marcxena mxare: (x – 2)(x + 2) > 0 .<br />
ra SemTxvevaSi iqneba ori<br />
ricxvis namravli dadebiTi?<br />
(1) ricxviT wrfeze aRniSnulia x–2>0<br />
(wiTlad) da x–20 (wiTlad) x+2 0) ⇔ (x ∈ (–∞;–2) (2;∞)).<br />
2. amovxsnaT x 2 – 3 ≤ 0 utoloba.<br />
(x 2 – 3 ≤ 0) ⇔ ((x − )(x + ) ≤ 0).<br />
rogori ori ricxvis namravli<br />
iqneba uaryofiTi?<br />
savarjiSoebi:<br />
1 amoxseniT utoloba:<br />
a) (x–2)(x+5)>0; b) x(x+3)≥0; g) (2x–3,5)(x+3,5)≥0;<br />
d) (x+1)(x+4)0; v) (5x–1)(x+2)≤0;<br />
z) (x–5)(3x+1) x + 2) ⇔ x∈(−∞;–1) (2;∞))<br />
><br />
>
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