BI TP PHNG TRèNH Vễ T KHể I PHNG PHP 1: NNG LU THA I-KIN THC: 1/ f ( x) f ( x) = g ( x ) g ( x ) f ( x) = g ( x) 2/ g ( x) f ( x) = g ( x) f ( x) = g ( x ) 3/ f ( x) f ( x ) + g ( x ) = h( x ) g ( x ) f ( x ) + g ( x) + f ( x).g ( x) = h( x) 4/ 2n f ( x) f ( x) = n g ( x) g ( x) (n N * ) f ( x) = g ( x ) 5/ 2n g ( x) f ( x) = g ( x ) (n N * ) 2n f ( x) = g ( x) 6/ n +1 f ( x) = n +1 g ( x) f ( x) = g ( x) (n N * ) 7/ n +1 f ( x) = g ( x ) f ( x) = g n +1 ( x) (n N * ) II-BI TP Bi 1: Giai phng trinh: x +1 = x (1) x x x x=3 x = x + = (x 1) x 3x = HD: (1) Bi 2: Giai phng trinh: HD:Ta cú: x 2x + = x x x x = x = x 2x + = 2x + = x 2 x + = x x 2x = x = Bi 3: Giai phng trinh: HD: Ta cú: x + x = 2x x + x = 2x x + = 2x + x x x x + = x + x + (1 x)(1 x) x x 1 x x x + x=0 x=0 x + = x 3x + (2 x + 1) = x x + x2 + x = x = Bi 4: Giai phng trinh: x x2 = x HD:K: x x (1) PT x ( x 2)( x + 2) = x x + = ( ) x2 =0 x + = ( ) x = x = 17 (2) Kt hp (1) v (2) ta c:x = Bi Giai phng trinh : 3x = x 3+x HD:k: x ú pt a cho tng ng: x3 + 3x + x = 3 10 10 x+ = x = ữ 3 3 Bi Giai phng trinh sau : x + = x x HD:k: x phng trinh tng ng : (1+ 3+ x ) x = x + + = 3x = 9x x = 97 x + + = x 18 Bi Giai phng trinh sau : + 3 x ( x + ) = x + 3 3x ( x + ) HD: pt ( x + 3x ) = x = Bi Giai v biờn luõn phng trinh: HD: Ta cú: x2 = x m x m x m x2 = x m 2 2 x = x 4xm + m 2mx (m + 4) = Nu m = 0: phng trinh vụ nghiờm Nu m 0: m2 + x= 2m iờu kiờn cú nghiờm: x m m2 + 2m m + Nu m > 0: m2 + 2m2 m2 < m + Nu m < 0: m2 + 2m2 m2 m Túm lai: Nu m hoc < m 2: phng trinh cú mụt nghiờm x= m2 + 2m Nu < m hoc m > 2: phng trinh vụ nghiờm Bi Giai v biờn luõn phng trinh vi m l tham s: x2 = x m Bi 10 Giai v biờn luõn theo tham s m phng trinh: x x =m m HD: iờu kiờn: x Nu m < 0: phng trinh vụ nghiờm Nu m = 0: phng trinh tr thnh x ( x 1) = cú hai nghiờm: x1 = 0, x2 = Nu m > 0: phng trinh a cho tng ng vi x m =0 ( x m)( x + m 1) = x = m + Nu < m 1: phng trinh cú hai nghiờm: x1 = m; x2 = (1 m) + Nu m > 1: phng trinh cú mụt nghiờm: x = m III-Bi ỏp dng: Bi 1:Giai cỏc phng trinh sau: 1/ 4/ x + x = 13 + x x2 + = x + 2/ 5/ x + 34 x = x +3 = x 3/ x + 3x = 6/ x + x = 12 x 7/ x x x + x + = 8/ x2 = =0 11/ 2x + = 10/ 5x + 13/ 16 x + 17 = x 23 14/ 19 3x + + x = 9/ = 6x x 12/ x = 15/ 20 2x = 2x Bi 2: Giai phng trinh: a) x2 = x b) x x + = c) x + x + = d) 3+ x + x = e) 3x + x = f) g) x + = 2x + h) 3x + x + = x + i) 3+ x x =1 x 4x = Bi 3: Tim m phng trinh sau cú nghiờm: x + 3x = 2m + x x Bi 4: Cho phng trinh: x2 x = m a) Giai phng trinh m = b) Tim m phng trinh cú nghiờm Bi 5: Cho phng trinh: x + mx = x m a) Giai phng trinh m=3 b) Vi giỏ tr no ca m thi phng trinh cú nghiờm Bi 6: Giai cỏc phng trinh sau: a/ b/ c/ x7 x3 = 2x = 3x x + = d/ e/ x x + x = 17 2 x3 x 27 + x 12 = f) ( x + 3) 10 x = x x 12 g/ h/ i/ x x = x x x + x = x + x + 12 = II PHNG PHP 2: A V PHNG TRèNH TR TUYT I I-KIN THC: S dng hng ng thc sau: f ( x ) = g ( x) ( f ( x ) 0) f ( x) = g ( x ) f ( x) = g ( x) f ( x ) = g ( x ) ( f ( x) < 0) II-BI TP: Bi 1: Giai phng trinh: HD: (1) (x 2) = x x 4x + + x = (1) |x 2| = x Nu x < 2: (1) x = x (vụ nghiờm) Nu x : (1) x = x x = (thoa man) Võy: x = Bi 2: Giai phng trinh: x + + x + + x + 10 x + = x + x + (2) x + x + + x + + + x + 2.3 x + + = x + x + + HD: (2) x x + + 1+ | x + |= 2.| x + | t y = x +1 (*) (y 0) phng trinh(*) a cho tr thnh: y + 1+ | y |= | y 1| Nu y < 1: y + + y = 2y y = (loai) Nu y 3: y + + y = 2y y = Nu y > 3: y + + y = 2y (vụ nghiờm) Vi y = x + = x = (thoa man) Võy: x = Bi 3:Giai phng trinh: x + 2x + x + + 2x = HD:K: x PT x + 2 x + + x + x + = 14 2x + + x + = 14 x = x = 15 Bi 4:Giai phng trinh: (Thoa man) Võy:x = 15 x + x + x x = HD:K: x Pt Nu x 1+ x +1 + x x +1 = x>2 pt x +1+ x 1 = x = x +1+ (Loai) x 1 = Nu x2 pt x + + x = 0x = Võy tõp nghiờm ca phng trinh l: (Luụn ỳng vi x ) S = { x R | x 2} III-Bi ỏp dng: Giai cỏc phng trinh sau: 1/ 2/ x2 + x + = x4 x +4 =3 3/ x2 6x + = 2x 4/ 5/ x2 x + + x2 + 4x + = 6/ x x + x x + = 10 8/ x2 x + + x2 x + = 7/ 9/ 11/ 13/ x2 6x + + 2x2 + 8x + = x2 x + x + x + x x = x + x + + x + 11 x + = x2 + x x2 + x + = 15/ x x + + x = 10 17/ x+ x+ 19/ 21/ x + x + x x = 10/ x x + x x =1 12/ x + 2x + x + + 2x = 14/ 2x + + 2x + 2x 2x = 16/ 1 + x+ = 2 18/ x+3 ( x 1) + x + x x + = x + x + = 5x + x2 x + + 2x = x + x +1 = 20/ x2 4x + = x 22/ x + x = III PHNG PHP 3: T N PH Phng phỏp t n ph thụng thng i vi nhiờu phng trinh vụ vụ t , giai chỳng ta cú thờ t t = f ( x ) v chỳ ý iờu kiờn ca t nu phng trinh ban u tr thnh phng trinh cha mụt bin t quan trng hn ta cú thờ giai c phng trinh ú theo t thi viờc t ph xem nh hon ton Bi Giai phng trinh: x x2 + x + x2 = HD:iu kin: x Nhõn xột x x x + x = 1 t t = x x thi phng trinh cú dang: t + = t = Thay vo tim c x = t Bi Giai phng trinh: x x = x + HD:iờu kiờn: x t t = x + 5(t 0) thi x = t2 Thay vo ta cú phng trinh sau: t 10t + 25 2 (t 5) = t t 22t 8t + 27 = 16 (t + 2t 7)(t 2t 11) = Ta tim c bn nghiờm l: t1,2 = 2; t3,4 = Do t nờn ch nhõn cỏc gỏi tr t1 = + 2, t3 = + T ú tim c cỏc nghiờm ca phng trinh l: x = vaứx = + Cỏch khỏc: Ta cú thờ binh phng hai v ca phng trinh vi iờu kiờn x x Ta c: x ( x 3) ( x 1) = , t ú ta tim c nghiờm tng ng n gian nht l ta t : y = x + v a vờ hờ i xng (Xem phn t n ph a v h) Bi Giai phng trinh sau: x + + x = HD:iờu kiờn: x t y = x 1( y 0) thi phng trinh tr thnh: y + y + = y 10 y y + 20 = ( vi y 5) ( y + y 4)( y y 5) = y = + 21 + 17 (loaùi), y = 2 T ú ta tim c cỏc giỏ tr ca x = 11 17 ( Bi Giai phng trinh sau : x = ( 2004 + x ) x ) HD: K: x t y = x thi phng trinh tr thnh: ( y ) Bi Giai phng trinh sau : x + x x (y + y 1002 ) = y = x = = 3x + x HD:iờu kiờn: x < Chia ca hai v cho x ta nhõn c: x + x 1 = + t t = x , ta giai c x x x Bi Giai phng trinh : x + x x = x + 1 HD: x = khụng phai l nghiờm , Chia ca hai v cho x ta c: x ữ+ x = x t t= x , Ta cú : t + t = t = x = Bi 7.Giai phng trinh: 3x + 21x + 18 + HD:t y = Phng trinh cú dang: 3y + 2y - = x x2 + x + = x2 + x + ; y Vi y = x x = x2 + x + = x = y= y =1 y =1 L nghiờm ca phng trinh a cho Nhn xột : i vi cỏch t n ph nh trờn chỳng ta ch giai quyt c mụt lp bi n gian, ụi phng trinh i vi t lai quỏ khú giai t n ph a v phng trỡnh thun nht bc i vi bin : Chỳng ta a bit cỏch giai phng trinh: u + uv + v = (1) bng cỏch u u Xột v phng trinh tr thnh : ữ + ữ+ = v v v = th trc tip Cỏc trng hp sau cng a vờ c (1) a A ( x ) + bB ( x ) = c A ( x ) B ( x ) u + v = mu + nv Chỳng ta hay thay cỏc biờu thc A(x) , B(x) bi cỏc biờu thc vụ t thi s nhõn c phng trinh vụ t theo dang ny a) Phng trỡnh dng : a A ( x ) + b.B ( x ) = c A ( x ) B ( x ) Nh võy phng trinh Q ( x ) = P ( x ) cú thờ giai bng phng phỏp trờn nu: P ( x ) = A ( x ) B ( x ) Q ( x ) = aA ( x ) + bB ( x ) Xut phỏt t ng thc : x + = ( x + 1) ( x x + 1) x + x + = ( x + x + 1) x = ( x + x + 1) ( x x + 1) ( )( ) x4 + = x2 x + x2 + 2x + x + = ( x x + 1) ( x + x + 1) Hay tao nhng phng trinh vụ t dang trờn vớ d nh: x 2 x + = x + cú mụt phng trinh p , chỳng ta phai chn hờ s a,b,c cho phng trinh bõc hai at + bt c = giai nghiờm p Bi Giai phng trinh : ( x + ) = x3 + HD: t u = x + (u 0) ; v = x x + (v phng trinh tr thnh : ( u + v 2 ) u = 2v = 5uv u = v Bi Giai phng trinh : x 3x + = ) Tim c: x = x + x + (*) 37 4 2 2 HD:D thy: x + x + = ( x + x + 1) x = ( x + x + 1) ( x x + 1) Ta vit ( x + x + 1) + ( x x + 1) = (x + x + 1) ( x x + 1) ng nht v trỏi vi (*) ta c : ( x + x + 1) + ( x x + 1) = t : u = x (x + x + 1) ( x x + 1) 3 + x + u ữ ; v = x2 x + v ữ 4 phng trinh tr thnh :-3u+6v=- uv u = 3v T õy ta s tim c x Bi 3: Giai phng trinh sau : x + x = x3 (*) HD:k: x Nhõn xột : Ta vit ( x 1) + ( x + x + 1) = ( x 1) ( x + x + 1) ng nht v trỏi vi (*) ta c : ( x 1) + ( x + x + 1) = ( x 1) ( x + x + 1) v = 9u t u = x , v = x + x + > , ta c: 3u + 2v = uv v= u Ta c : x = Bi Giai phng trinh : x3 3x + ( x + ) x = HD:Nhõn xột : t y = x + ta bin pt trờn vờ pt thun nht bõc i vi x v y : x = y x 3x + y x = x 3xy + y = Pt cú nghiờm : x = 2, x = y Bi 5:Giai phng trinh: 10 x3 + = ( x + ) HD:K: x Pt 10 x + x x + = 3( x + 2) t u = x + (u , v 0) v = x x + Phng trinh tr thnh:10uv = 3(u2+v2) u = 3v ( 3u v ) ( u 3v ) = v = 3u x = 22 Ta cú : + x + x Du bng v ch x = v x + + v ch x = Võy ta cú phng trinh: 2008 x + + 2008 x = , du bng x +1 + 1+ x x +1 A f ( x ) ú : B f ( x) ụi mụt s phng trinh c tao t ý tng : A = f ( x ) A=B B = f ( x ) Nu ta oỏn trc c nghiờm thi viờc dựng bt ng thc d dng hn, nhng cú nhiờu bi nghiờm l vụ t viờc oỏn nghiờm khụng c, ta dựng bt ng thc ỏnh giỏ c II-BI TP: Bi Giai phng trinh : 2 + x = x+9 x +1 HD:k: x 2 + xữ 2 Ta cú : x +1 Du bng 2 = x +1 ( ) x = x+9 + x +1 + x + x + ữ 1 x= x +1 Bi Giai phng trinh : 13 x x + x + x = 16 HD:k: x ( Bin i pt ta cú : x 13 x + + x ) = 256 p dng bt ng thc Bunhiacopxki: ( 13 13 x + 3 + x ) ( 13 + 27 ) ( 13 13 x + + x ) = 40 ( 16 10 x ) p dng bt ng thc Cụsi: 10 x ( 16 10 x 2 ) 16 ữ = 64 2 x= + x2 x = Du bng 10 x = 16 10 x x = Bi Giai phng trinh: x3` 3x x + 40 4 x + = HD:Ta chng minh : 4 x + x + 13 v x3 3x x + 40 ( x 3) ( x + 3) x + 13 Bi 4: Giai phng trinh: HD:Ta cú :VT2=( Nờn : < VT x + x = x 12 x + 38 x + x )2 (1 + 1).(7- x + x - 5) = Mt khỏc:VP = x2 - 12x + 38 =2 + (x - 6)2 Theo gia thit du ''='' xay v ch khi:x = Võy x = l nghiờm nht ca phng trinh a cho Bi 5: Giai phng trinh: HD:K: x [ 1; 2] PT x + 3x + x + = (1) x + x = x + (2) T (2) ta cú: x +1 x +1 x +1 x (3) T (1) v (3) Ta cú x = th vo (2) thoa man.Võy :x = Bi 6:Giai phng trinh : HD: iờu kiờn x> x 4x + =2 x 4x 1 p dng bt ng thc cụ si ta cú: x 4x + 4x x x 4x ì 4x = x x Theo gia thit du bng xay v ch khi: 4x = 4x x x 4x + = (x 2) = x = Du = xay x = 4x x 4x + = x 4x + = (x 2) = x = Võy : x = x = (Thoa man) Bi 7:Giai phng trinh : x 5x = 3x HD: Cỏch iờu kiờn x Vi x thi: V trỏi: x < 5x v trỏi luụn õm V phai: 3x v phai luụn dng Võy: phng trinh a cho vụ nghiờm Cỏch Vi x 1, ta cú: x = 5x + 3x x = 8x + 7x = (5x 1)(3x 2) (5x 1)(3x 2) V trỏi luụn l mụt s õm vi x 1, v phai dng vi x phng trinh vụ nghiờm Bi 8:Giai phng trinh : HD: Ta cú (1) 3x + 6x + + 5x + 10x + 14 = 2x x x + 2x + + ữ + x + 2x + + ữ = (x + 2x + 1) + 5 3(x + 1) + + 5(x + 1) + = (x + 1) Ta cú: V trỏi (1) + = + = Du = xay x = V phai Du = xay x = Võy: phng trinh a cho cú mụt nghiờm x = Bi 9:Giai phng trinh : x+7 + = 2x + 2x x +1 HD: iờu kiờn x D thy x = l mụt nghiờm ca phng trinh Nu x < : VT = Nu x > 2: VP = 2x2 + 1+ 2x +8 < 8+ x +1 > 2.22 + = M: VP > 8+ + VT < + x > x +1 > +1 6 1+ < 1+ =3 x +1 +1 Võy: phng trinh a cho cú mụt nghiờm nht l x = Bi 10:Giai phng trinh : + =6 x 2x HD: K: x < Bng cỏch th, ta thy x = l nghiờm ca phng trinh Ta cn chng minh ú l nghiờm nht Thõt võy:Vi x < : 6 x 2x Bi 11:Tim nghiờm nguyờn dng ca phng trinh: 1 1 + + + ììì+ = 1.2 2.3 3.4 x ( x + 1) 4x +4 x +5 HD:K: x (1) Ta cú: x + = x +5 x = x4 Ta cú: VP(*) = (*) x40 x (2) T (1) v (2) ta cú:x = l nghiờm nht v x0 f ( x) > f ( x0 ) = k ú phng trinh vụ nghiờm Vi x < x0 f ( x) < f ( x0 ) = k ú phng trinh vụ nghiờm Võy x0 l nghiờm nht ca phng trinh Hng 2: Thc hiờn theo cỏc bc Bc 1: Chuyờn phng trinh vờ dang: f ( x) = g ( x) Bc 2: Dựng lõp luõn khng nh rng f ( x) v g(x) cú nhng tớnh cht trỏi ngc v xỏc nh x0 cho f ( x0 ) = g ( x0 ) Bc 3: Võy x0 l nghiờm nht ca phng trinh Hng 3: Thc hiờn theo cỏc bc: f (u ) = f (v) Bc 1: Chuyờn phng trinh vờ dang Bc 2: Xột hm s y = f ( x) , dựng lõp luõn khng nh hm s n iờu Bc 3: Khi ú f (u ) = f (v) u = v ( ) ) ( 2 Vớ d: Giai phng trinh : ( x + 1) + x + x + + x + x + = ( ) ( ) HD:pt ( x + 1) + ( x + 1) + = ( x ) + ( x ) + f ( x + 1) = f ( x ) 2 ) ( Xột hm s f ( t ) = t + t + , l hm ng bin trờn R, ta cú x = Vớ D 2: Giai phng trinh: x+6 + x+2 + x+3 = HD: nhõn thy x = -2 l mụt nghiờm ca phng trinh t Vi f ( x) = x + + x + + x + x1 < x2 f ( x1 ) < f ( x2 ) võy hm s f(x) ng bin trờn R Võy x = -2 l nghiờm nht ca phng trinh Bi ỏp dng: Giai phng trinh: a) 4x + 4x2 = c) x = + x x2 e) x + x + = b) x = x3 x + d) x = x + x x3 f) 2x + x2 + = x VI PHNG PHP 6: S DNG BIU THC LIấN HP - TRC CN THC Mụt s phng trinh vụ t ta cú thờ nhm c nghiờm x0 nh võy phng trinh luụn a vờ c dang tớch ( x x0 ) A ( x ) = ta cú thờ giai phng trinh A ( x ) = hoc chng minh A ( x ) = vụ nghiờm , chỳ ý iu kin ca nghim ca phng trỡnh ta cú th ỏnh gớa A ( x ) = vụ nghim Bi 1:Giai phng trinh: HD: C1: K ( 1) x ( x + ) + x ( x 1) = x (1) x 2; x x2 x x2 2x x ( x 1) x ( x + ) x x ( x 1) x ( x + ) Nu x ta cú =2x ( 2) =2x x ( x 1) x ( x + ) = x ( x 1) = x + x ( x 1) + x ( x + ) = x ( 3) Giai (3) ta tim c x Nu x -2 ta cú x ( x 1) x ( x + ) = x ( x 1) = x + x ( x 1) + x ( x + ) = x ( 4) Giai (4) ta tim c x C2: K: x 2; x Nu x ta chia ca hai v cho x ta c: ( x + ) + ( x 1) =2 x Binh phng hai v sau ú giai phng trinh ta tim c x Nu x -2 t t = -x t Thay t ( t + ) + t ( t 1) = t ( t ) + t ( t + 1) = Chia ca hai v cho t ( t) ( t ) vo phng trinh ta c 2 ta c ( t ) + ( t + 1) =2 t Binh phng hai v tim c t Sau ú tim x Trong C1 ta a s dng kin thc liờn hp Cũn C2 ta võn dng kin thc miờn xỏc nh vờ n ca phng trinh.nhin chung thi viờc võn dng theo C2 n gian hn x x + x = ( x x 1) x x + Bi Giai phng trinh sau : HD: 2 2 Ta nhõn thy : ( 3x x + 1) ( 3x 3x 3) = ( x ) v ( x ) ( x 3x + ) = ( x ) Ta cú thờ trc cn thc v : x + x x + + ( x x + 1) 3x = x + x 3x + Dờ dng nhõn thy x = l nghiờm nht ca phng trinh Bi Giai phng trinh sau: x + 12 + = x + x + HD: phng trinh cú nghiờm thi : x + 12 x + = 3x x Ta nhõn thy : x = l nghiờm ca phng trinh , nh võy phng trinh cú thờ phõn tớch vờ dang ( x ) A ( x ) = , thc hiờn c iờu ú ta phai nhúm , tỏch nh sau : x + 12 = x + x + 2 x2 x + 12 + = 3( x 2) + x2 x2 + + x+2 x +1 ( x 2) 3ữ= x = 2 x2 + + x + 12 + x+2 D dng chng minh c : x + 12 + x+2 x2 + + < 0, x > Bi Giai phng trinh : x + x = x3 HD :k x Nhõn thy x = l nghiờm ca phng trinh , nờn ta bin i phng trinh x + x = x ( x 3) + Ta chng minh : ( x 3) ( x + x + ) = 2 3 x x3 + ( ) + x + x+3 1+ (x 1) + x + Võy pt cú nghiờm nht x = x+3 = 1+ ( x+3 < < x + 3x + x2 + + x3 + ) Bi 5:Gii phng trỡnh sau: HD:K: x x2 + x + x2 + + x2 x x2 =x Nhõn vi lng liờn hp ca tng mu s ca phng trinh a cho ta c: (x ) )( ) )( ( x + x x + x x = 3.x (x x > x ( ) (x + ) +( x + + ) ) = 3.x +2 (x 3) = 27 x x > x > ; x ( 2x ) 4 4 2 ( x 3) = x x ( ) 4( x 3) = x ( x ) Giai hờ trờn ta tim c x= Bi 6:Gii phng trỡnh: (3 x2 + 2x ) = x+9 x HD:K: x Pt ( 2x2 + + x ( )( + 2x ) + + 2x ( x 18 + x + + x 4x2 ) = x+9 ) = x+9 + 2x = x= l nghim Bi dng: 1) x ( x 3) + x ( x ) = x 2) ( x + 3) ( x + ) + ( x + 3) ( x 1) =2 ( x + 3) Tng quỏt: 3) 3x x + 10 f ( x ) g ( x ) + f ( x ) h ( x ) = f ( x ) = 3x + BI TP TNG HP Bi 1: Tim tt ca cỏc s thc x1; x2; ; x2005 thoa man: x1 12 + x2 22 + + 2005 x2005 20052 = ( x1 + x2 + + x2005 ) Bi 2: Tim cỏc s thc x, y, z tha man iờu kiờn: x + y + z = ( x + y + z) Bi 3: Gii cỏc phng trỡnh sau: x + 2x = 3( x x + 1) = ( x + x 1) x + x +1 = x + 48 = x + x + 35 2( x + 2) = x + x2 x + = ( 27 x10 x + 864 = ) x x = x x = x x x = ( x + 2)( x + 4) + 5( x + 2) x + 17 x + x 17 x = 10 x = x )( x+4 =6 x+2 x 3x = x2 + x + x2 x + = x2 + x + x x2 + = x2 x + 3+x 3x + x = x + x + x x + 24 + = x + x + Bi 4: Gii cỏc phng trỡnh sau: 25 x 10 x = ( x) x + ( x 5) x x + x5 ( x + 3) 10 x = x x 12 x2 4x + + x2 4x + + x2 x + = + =2 ( x 3) ( x + 1) ( x 3) x + x + 20 = x + 10 x +1 +3= x 2x + = x x2 + x + = 2 x + x + = x 20 3x + 3x x + x = + x + x = x + x x 12 = 48 + x 2x 2x = +1 3 +1 x + ữ x + ữ+ = x x x 20 + 2+ x + 2+ x 4x + + = x4 + x5 x 45 = + x x x x ( x) = x + x2 =4 x + ( x 5) x x + x5 9x + + x + 2005 = 2005 x= 3x 3+x =2 (a , b > 0) a + b x = 1+ a b x 64x6 - 112x4 + 56x2 - = x + x + x + x + 28 = x2 Bi 5: Ký hiờu [x] l phn nguyờn ca x Giai phng trinh sau: + + + Bi 6:Cho phng trinh: x x + x +2 x3 = 855 = x x + 62 x Gi tng cỏc nghiờm ca phng trinh l S,tớnh S15 Bi 7:Giai phng trinh nghiờm nguyờn sau: a/ x + y = 1960 b/ x + y = 1980 c/ x y = 48 Bi 8:Giai phng trinh nghiờm nguyờn sau: + x2 1225 + = 74 x y z 771 y z 771 Bi 9:Gii cỏc phng trỡnh sau : d/ x + y = 2000 x 14 x + x x 20 = x + 2x + ( x + = x3 x 15 30 x x ) = 2004 ( x 1 = + x x x x ( x 1) x3 + = x3 + x + ( x x 10 = x x 10 ) ( x + = x + 3x + x + ) x + x + 12 x + = 36 ( + x ) + 3 x2 + ( x ) = 2008 x x + = 2007 x x = (2004 + x )(1 x ) ( x + x + 2)( x + x + 18) = 168 x 3 ) 30060 x + + x + x3 + x + x + = + x ( x + ) + 16 ( x ) + 16 ( x ) = x + 16 x + + x = + x3 + x x + 3x + = x x + + 2 x x + 16 x + 18 + x = x + 12 x + x = x + x + 3x3 = x 2 x 11x + 21 3 x = ( x) ( x) = x+ ( x ) ( 10 x ) x2 + = x + 2x x + + 3x + = x + + x + x + x + = ( x + 3) x + x + x +1 = x 3 x3 + (1 x ) = x 2x2 x + x + = 2x + x2 + x + = 3x + 3x + 3x + Bi 10: Giai phng trinh: a) x + x + x + = 12 x b) x x + 3x + = 3x c) x x + = x x + 12 d) 3x + 15 x + x + x + = e) ( x + 4)( x + 1) x + x + = f) g) h) x + x + 11 = 31 x + 3x + 2 x + x + = 2 x2 + 5x + 2 x2 + 5x = Bi 11: Giai phng trinh: x3 + x+ (1 x ) x x = + x2 ( x ) = x ( x2 ) 35 12 ( x 3) ( x + 1) + ( x 3) x 2x x2 x2 + = ( 1+ x) 64 x 112 x + 56 x = x a) Giai phng trinh vi m = b) Tim m phng trinh cú nghiờm c) Tim m phng trinh cú nghiờm nht =m Bi 13: Cho phng trinh: x + x2 a) Giai phng trinh vi m = + b) Tim m phng trinh cú nghiờm Bi 14: Cho phng trinh: ( x x ) + x x m = a) Giai phng trinh vi m = b) Tim m phng trinh cú nghiờm Bi 15:Giai cỏc phng trinh nghiờm nguyờn sau: y= x + x + x x y2 = + x2 4x = + x2 x +1 = x3 Bi 12: Cho phng trinh: + x + x + ( + x ) ( x ) = m x+ x+ x+ x = y y = x + x + + x +1 y = x + 2x + x 2x y = x x + x + x Bi 16: Giai cỏc phng trinh nghiờm nguyờn sau: x + x + x + + x = y nu: a/ V trỏi cú 100 du cn b/ V trỏi cú n du cn Bi 17:Giai cỏc phng trinh nghiờm nguyờn sau: x + x + x + + x + x = x (V trỏi cú 100 du cn) Bi 18:Tim cỏc s hu t a v b thoa man: Bi 19:Cho hai s x , y thoa man: ( a+b x2 + x )( ab ) = 20 y2 + y = Tớnh x + y Bi 20:Giai phng trinh: x + + x = Bi 21:Cho cỏc s thc dng x,y,z thoa man iờu kiờn: x y + y z + z x2 = Chng minh rng: x + y + z Bi 22:Cho cỏc s thc dng a,b,c thoa man iờu kiờn: Bi 24:Tim cỏc s hu t a v b bit: Bi 25:Giai phng y = + 199 x x a b = 11 28 x + x2 trinh: =1 x2 Bi 26:Tim cỏc s nguyờn k thoa man: 1+ 1 1 1 2009 + + + + + + + + = 2009 12 22 22 32 k ( k + 1) Bi 27:Giai phng trinh: 1/ 8+ x3 + x3 = a + b c = a+bc Chng minh rng: 2010 a + 2010 b 2010 c = 2010 a + b c Bi 23:Giai phng trinh nghiờm nguyờn: = 2/ x + x2 + x x2 = x + 3/ x x 30 2007 30 + x 2007 = 30 2007 4/ x + x 3x = x + x + + x x + 5/ x + + x + + x + + + 100 x + 100 = 165 6/ 1 + + =1 x+3 + x+2 x + + x +1 x +1 + x 7/ x + 25 x + 125 x + 45 16 x + 80 + =9 12 16 8/ x + 712671620 52408 x + 26022004 + x + 712619213 56406 x + 26022004 = 9/ 2009 + 2010 x + x + = 20 + 2009 2010 x + x + 10/ ( x + 5)(2 x) = x + 3x Bi 28:Gii cỏc phng trỡnh sau: 15 x x = x 15 x + 11 ( x + 5)(2 x) = x + x (1 + x)(2 x) = + x x x + 17 x + x 17 x = 3x + x = x + 3x x + x + x + 11 = 31 n (1 + x) + n x + n (1 x) = ( x + x + 2)( x + x + 18) = 168 x x = (2004 + x )(1 x ) x2 + x2 = ... Vi n s a1, a2,, an Du = xay a+b ab a=b b) Vi ba s a, b, c Du = xay thi ta cú: a+b+c abc c thi ta cú: a+b+c+d abcd c=d thi ta cú: a1 + a2 + + an n a1.a2 an n a1 = a2 = = an 3.GTLN,GTNN... Bunhiakụpxki: Cho hai bụ s : ( a , b), (x , y) thi ta cú: (ax + by)2 Du = xay a b = x y 2.Bt ng thc cụsi: (a + b )( x + y ) a) Vi hai s a, b Du = xay thi ta cú: a = b= c) Vi bn s a, b, c, d Du... ta cú thờ giai c phng trinh ú theo t thi viờc t ph xem nh hon ton Bi Giai phng trinh: x x2 + x + x2 = HD:iu kin: x Nhõn xột x x x + x = 1 t t = x x thi phng trinh cú dang: t + = t =