квадрат тэгшитгэл

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квадрат тэгшитгэл

  1. 1. Êâàäðàò òýãøèòãýë áîäîõDef; ax 2 + bx + c = 0 õýëáýðèéí òýãøèòãýëèéã êâàäðàò òýãøèòãýë ãýæ íýðëýíý. ¯¿íä: õ –õóâüñàã÷, a,b,c -ºãºãäñºíòîîíóóä ( a ≠ 0 )Def; Õýðýâ êâàäðàò òýãøèòãýëèéí b,c –êîýôôèöèåíò¿¿äèéí ÿäàæ íýã íü 0 –òýé òýíö¿¿ áîë ò¿¿íèéã ã¿éöýä áèø êâàäðàòòýãøèòãýë ãýíý. ÿéöýä áèø êâàäðàò òýãøèòãýë äàðààõ õýëáýð¿¿äòýé áàéæ áîëîõ áà ò¿¿íèéã õýðõýí áîäîõ òàëààð àâ÷¿çüå. 1. ax 2 = 0 áîë x = 0 2. ax 2 + bx = 0 áîë ò¿¿íèéã áîäîõäîî åðºíõèé ¿ðæèãäýõ¿¿í õààëòíààñ ãàðãààä ¿ðæâýð òóñ á¿ðèéã 0 –òýé  x1 = 0 x = 0 x = 0 òýíö¿¿ëæ áîäíî. ª.õ x( ax + b ) = 0 ⇒  ⇒ ⇒ b ãýñýí 2 øèéäòýé ax + b = 0 ax = −b  x2 = −   a  c  x1 = − c a c c 3. ax 2 + c = 0 áîë ò¿¿íèéã áîäîõäîî ax = −c ⇒ x = − ⇒  ýíä − ≥ 0 áàéíà. Õýðýâ − < 0 áîë 2 2 a  c a a  x2 = −  a êâàäðàò òýãøèòãýë øèéäã¿é.Æèøýý íü: 47 − x(3 x + 4) = 2(17 − 2 x) − 62 òýãøèòãýë áîä.Áîäîëò: 47 − 3 x 2 − 4 x = 34 − 4 x − 62 ⇒ 47 − 3x 2 − 4 x − 34 + 4 x + 62 = 0  x1 = 25 = 5− 3x 2 + 75 = 0 ⇒ −3 x 2 = −75 ⇒ x 2 = 25 ⇒   x2 = − 25 = −5 Def; Õýðýâ ax 2 + bx + c = 0 êâàäðàò òýãøèòãýëèéí a,b,c –êîýôôèöèåíò¿¿ä á¿ãä 0 –ýýñ ÿëãààòàé áîë ò¿¿íèéã ã¿éöýäêâàäðàò òýãøèòãýë ãýíý. Îäîî ã¿éöýä êâàäðàò òýãøèòãýëèéã õýðõýí áîäîõ òàëààð àâ÷ ¿çüå. Áîäîõäîî: 1. D = b 2 − 4ac ¿¿íèéã êâàäðàò òýãøèòãýëèéí äèñêðèìèíàíò ãýæ íýðëýíý. Õýðýâ a. D > 0 áîë 2 øèéäòýé b. D = 0 áîë äàâõàöñàí øèéäòýé áóþó 1 øèéäòýé c. D < 0 áîë øèéäã¿é áóþó öààø áîäîõ øààðäëàãàã¿é  −b + D −b± D  x1 = 2a 2. x1, 2 = ⇒ ãýæ áîäíî. 2a  −b− D  x2 =  2aÆèøýý íü: x 2 + 2 x − 8 = 0 òýãøèòãýë áîä.Áîäîëò: Ýíäýýñ êâàäðàò òýãøèòãýëèéí x 2 –èéí ºìíºõ êîýô áóþó a=1, õ –èéí ºìíºõ êîýô áóþó b=2, ñóë ãèø¿¿í áóþó c=-8áàéíà.
  2. 2. D = b 2 − 4ac = ( 2 ) − 4 ⋅ 1 ⋅ ( − 8) = 4 + 32 = 36 . Ýíä äèñêðèìèíàíò íü 0 –ýýñ èõ ó÷èð 2 øèéäòýé áàéíà. Òýäãýýðèéã 2  −2+6 4 − b + D − 2 ± 36 − 2 ± 6  x1 = 2 = 2 = 2îëú¸ x1, 2 = = = ⇒ áîëíî. 2a 2 ⋅1 2  x = − 2 − 6 = − 8 = −4  2  2 2Æèøýý íü: 2 x 2 − 9 x − 5 = 0 òýãøèòãýë áîä.Áîäîëò: Ýíäýýñ êâàäðàò òýãøèòãýëèéí x 2 –èéí ºìíºõ êîýô áóþó a=2, õ –èéí ºìíºõ êîýô áóþó b=-9, ñóë ãèø¿¿í áóþó c=-5áàéíà.D = b 2 − 4ac = ( − 9 ) − 4 ⋅ 2 ⋅ ( − 5) = 81 + 40 = 121 áàéíà. Ýíä äèñêðèìèíàíò íü 0 –ýýñ èõ ó÷èð 2 øèéäòýé áàéíà. 2Òýäãýýðèéã îëú¸.  9 + 11 20 x = = =5 − b + D 9 ± 121 9 ± 11  1 4 4x1, 2 = = = ⇒ áîëíî. 2a 2⋅2 4  x = 9 − 11 = − 2 = − 1  2  4 4 2Ñàíàìæ: Êâàäðàò òýãøèòãýë áîäîõ ÿâöàä –b îëîõäîî b êîýôôèöèåíòèéã ýñðýã òýìäýãòýé òîîãîîð àâ÷ áîäíî.Áèå äààæ áîäîõ áîäëîãóóä:I.Äàðààõ ã¿éöýä áèø êâàäðàò òýãøèòãýë¿¿äèéã áîä.3x 2 − 4 x = 0 − 5x2 + 6x = 0 6b 2 − b = 0 4x2 − 9 = 0 − x2 + 3 = 0 1y2 − =0 6 x 2 + 24 = 0 7 a − 14a 2 = 0 1 − 4x2 = 0 9( 2 x − 1)( 2 x + 1) = x( 2 x + 3) ( 3x + 2) 2 = ( x + 2)( x − 3) ( x + 3)( 3x − 2) = ( 4 x + 5)( 2 x − 3)4 x 2 + 6 x = 9 x 2 − 15 x x( x − 15) = 3(108 − 5 x ) 8 .5 x − 3 x 2 = 3 .5 x + 2 x 2 5x2 + 9 4x2 − 9 13x 2 − 3 9 x 2 − 5( x − 7 )( x + 3) + ( x − 1)( x + 5) = 102 − =3 + =3 6 5 5 4II.Äàðààõ ã¿éöýä êâàäðàò òýãøèòãýë¿¿äèéã áîä.x2 − 4x + 3 = 0 x 2 + 3 x − 10 = 0 x 2 + 9 x + 14 = 0 x 2 − 2 x − 35 = 0x 2 − 5x − 6 = 0 x 2 + 8 x − 20 = 0 x2 − 6x + 8 = 0 x2 + x − 6 = 0x2 + 4x + 3 = 0 2 x 2 − 9 x + 10 = 0 x 2 + 14 x + 50 = 0 5 x 2 + 14 x − 3 = 03 x 2 − 14 x + 16 = 0 x 2 − 22 x − 23 = 0 x 2 − 10 x − 24 = 0 15 y 2 − 22 y − 37 = 010 x 2 − x + 1 = 0 4 x 2 − 8x + 3 = 0 5 x 2 + 3x − 8 = 0 5x 2 = 9x + 2x 2 = 3 x + 40 14 = x 2 + 5 x z − 5 = z 2 − 25 0 .7 x 2 = 1 .3 x + 2( x + 4) 2 = 3x + 40 ( 2 x − 3) 2 = 11x − 19 4( x + 3) = ( x − 5) 2 2 ( 3x − 1)( x + 2) = 20
  3. 3. 3 − y y − 2 ( y − 2) 2x2 + 1 x2 + 3 x + 4 x2 − 4 2x + 3 −x=2 − =5 − =1 = + 2 6 3 8 5 5 4 3( x − 1) 2 − x + 4 = 2 x − 2 ( x + 2)( x − 5) − 11x + 12 = 2 − x − 2 5 6 3 3 10 3III.Íýìýëò ìàòåðèàë: Ñóðàõ áè÷ãèéí ¹188-193; ¹205-214

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