Giải chi tiết phương trình vo tỉ

            ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 1 S d các phng pháp sau y:  Phng pháp nghiên c lý lu  Phng pháp kh sát th ti  Phng pháp phân tích  Phng pháp t h  Phng pháp khái quát hóa  Phng pháp quan sát  Phng pháp ki tra  Phng pháp t k kinh nghi www.nguoithay.org & www.cunghoctot.edu.vn            I.     2008  2009,          &                   .                     .                  ,       .                  ,     .          ,                33      9B.     : : 10 em : 12 em   : 11 em  7 14 ,        , c.        8     9    9 11 em II.             1.        a) 1: f(x) g(x)  2 g(x) 0 f(x) [g(x)]      . : x 1 x 1   (1) : (1)  2 x1 x1 x1 x3 x 3x 0 x 1 x 1                       :           = 3 b) 2: f(x) g(x) h(x) . : x 3 5 x 2    (2) .     2.   : (2)  x 3 x 2 5                ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 2  2x 1 2 (x 3)(x 2) 25      (x 3)(x 2) 12 x     22 2 x 12 2 x 12 x6 25x 150 x x 6 144 x 24x                   :           = 6 c) 3: f(x) g(x) h(x) . : x 1 x 7 12 x     (3) :     7 12.   : (3)  x 1 12 x x 7      x 1 5 2 (12 x)(x 7)      2 2 19x x 84 x 4     4(19x  x 2  84) = x 2  8x + 16  76x  4x 2  336  x 2 + 8x  16 = 0  5x 2  84x + 352 = 0     22 2 84 352 42 1764 1764 352 5 x x 5 x 2 x 5 5 5 25 25 5 42 4 44 5 x 5 5 x 8 x (x 8) 5x 44 5 25 5                                            x 1 = 44 5 ; x 2 = 8   :          1 = 44 5 ; x 2 = 8 d) 4: f(x) g(x) h(x) k(x)   . : x x 1 x 4 x 9 0       (4) :     4.   : (4)  x 9 x x 1 x 4       2x 9 2 x(x 9) 2x 5 2 (x 4)(x 1)         7 x(x 9) (x 1)(x 4)      22 49 x 9x 14 x(x 9) x 5x 4        45 + 14x + 14 x(x 9) = 0   4                  2)           1. Gi phng trình: 2 x 4x 4 x 8    (1) : (1)  2 (x 2) 8 x       8.   : (1)  |x  2| = 8  x  u x < 2: (1)  2  x = 8  x (  )   x  2 = 8  x  x = 5 HD:  s: x = 5.             ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 3  2. Gi phng trình x 2 2 x 1 x 10 6 x 1 2 x 2 2 x 1           (2) : (2)  x 1 2 x 1 1 x 1 2.3 x 1 9 2 x 1 2 x 1 1               x 1 1 | x 1 3| 2.| x 1 1|        = x1          : y 1 | y 3| 2| y 1|       y = 2  2y  y = 1 ()   y = 2y  2  y = 3  > 3: y + 1 + y  3 = 2y  2 (  m)   = 3  x + 1 = 9  x = 8   :             = 8 3)            a)                 ,        1.  x 1 5x 1 3x 2     1.   1   1 :   : x 1 5x 1         : 3x 2       :        2.    1,   : x 1 5x 1 3x 2      x 1 8x 3 2 (5x 1)(3x 2)       2 7x 2 (5x 1)(3x 2)            1,     1       b)            2. Gi phng trình: 2 2 2 3x 6x 7 5x 10x 14 4 2x x        (1) :   (1)  2 2 2 49 3 x 2x 1 5 x 2x 1 (x 2x 1) 5 35                          2 2 2 3(x 1) 4 5(x 1) 9 5 (x 1)          :    4 9 2 3 5    . =   x = 1   5. =   x = 1   :           = 1 c)             (,         ) 1. Gi phng trình: 2 x7 8 2x 2x 1 x1       :    1 2   = 2    1 x2 2  : VT = 6 1 8 8 3 x1      . Mà: VP > 83   x > 2: VP = 2x 2 + 2x 1 > 2.2 2 + 3 = 83 . VT < 83             ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 4 x 2 x 1 2 1 66 1 1 3 x 1 2 1             :             = 2 2. Gi phng trình: 2 2 2 2 3x 7x 3 x 2 3x 5x 1 x 3x 4          :     = 2.   : 2 2 2 3.4 7.2 3 2 2 3.2 5.2 1 2 3.2 4 1 2 3 6               (1)  2 2 2 2 (3x 5x 1) 2(x 2) (x 2) 3(x 2) 3x 5x 1 x 2             N x > 2: VT < VP N x < 2: VT > VP V: x = 2 là nghi duy nh c phng trình 3. Gi phng trình: 68 6 3 x 2 x   : K: x < 2. B cách th, ta th x = 3 2 là nghi c phng trình. Ta c ch minh  là nghi duy nh. Th v: V x < 3 2 : 6 2 3x   và 8 4 2x    68 6 3 x 2 x   . Tng t v 3 2 < x < 2: 68 6 3 x 2 x   4. Gi phng trình: 22 3x(2 9x 3) (4x 2)(1 1 x x ) 0        (1) : (1)     22 3x 2 (3x) 3 (2x 1) 2 (2x 1) 3 0             22 3x 2 (3x) 3 (2x 1) 2 (2x 1) 3         N 3x = (2x + 1)  x = 1 5  thì các bi th trong cn  hai v b  1 5  là m nghi c phng trình. Hn n nghi c (1) n trong kho 1 ; 0 2     . Ta ch minh  là nghi duy nh. V 11 x 25     : 3x < 2x  1 < 0  (3x) 2 > (2x + 1) 2  22 2 (3x) 3 2 (2x 1) 3      Suy ra:     22 3x 2 (3x) 3 (2x 1) 2 (2x 1) 3 0         (1) không có nghi trong kho này. Ch minh tng t, ta c i  k ông có nghi khi 11 x 25     d)          .  x 4x 1 2 x 4x 1    :    1 x 4              ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 5  ab 2 ba    > 0      1 x x 4x 1 0 4     . Nên: x 4x 1 2 x 4x 1    . =   2 x 4x 1 x 4x 1 0       22 x 4x 4 3 0 (x 2) 3 x 2 3 x 2 3              4.        1. Gi phng trình: 2x 1 x 2 x 3     .  ý th: (2x + 1)  (x  2) = x + 3. Do , nhân l liên h vào hai v c phng trình: (x 3)( 2x 1 x 2 1) 0       x 3 0 2x 1 x 2 1          PT vô nghi 2. Gi phng trình: 2 x 1 2(x 1) x 1 1 x 3 1 x         (1) .     x 1 1 x 2 x 1 1 x 1 0         x 1 = 0; x 2 = 24 25  3. Gi phng trình: 3 2 4 x 1 x x x 1 1 x 1        (1) . Chú ý: x 4  1 = (x  1)(x 3 + x 2 + x + 1). (1)      32 x 1 1 1 x x x 1 0        x = 2 5)           1.  2 x x 1 1   (1) .  x1  y 2 = x + 1  x = y 2  1  x 2 = (y 2  1) 2  (2)  (y 2  1) 2 + y  1 = 0  y(y  1)(y 2 + y  1) = 0. : 15 0; 1; 2        2.    3 x 1 1 2 x 1 2 x      (1) HD:  x 1 1 = y (1)      32 x 1 1 x 1 1 2 0        y 3 + y 2  2 = 0  (y  1)(y 2 + 2y + 2) = 0  y = 1  x = 1   1.  2 + 2) = 5 3 x1 (3) .  x1 , v = 2 x x 1  u 2 = x + 1, v 2 = x 2  x + 1, u 2 v 2 = x 3 + 1.  (3)  2(u 2 + v 2 ) = 5uv  (2u  v)(u  2v) = 0             ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 6  5 37 5 37 ; 22       2.      2 x 5 x 2 1 x 7x 10 3       (1) . 2. (1)     x 5 x 2 1 (x 5)(x 2) 3        x5 = u, x2  u 2  v 2 = 3. (1)  (a  b)(1 + ab) = a 2  b 2  (a  b)(1  a + ab  b) = 0  (a  b)(1  a)(1  b) = 0 Gi ra: x = 1 là nghi duy nh 3. Gi phng trình: x 1 3x 2x 1    (1) .  x1 = u, 3x  b  a = a 2  b 2  (a  b)(a + b + 1) = 0 Mà a + b + 1 > 0  a = b  x = 1 2 là nghi duy nh c phng trình. 4. Gii phng trình: 4 1 5 x x 2x x x x      (1) .  1 x x  = u, 5 2x x   (1)  1 5 1 5 x 2x x 2x 0 x x x x                         u  (v 2  u 2 )  v = 0  (u  v)(1 + u + v) = 0. Vì 1 + u + b > 0 nên: u = v. Gi ra ta : x = 2  1 Gii phng trình: 22 x 3x 2 x 3 x 2 x 2x 3         (1) .  (x 1)(x 2) x 3 x 2 (x x)(x 3)         : x1 = a, x2 = b, x3 = c (a, b,  ab + c = b + ac  (a  1)(b  c) = 0  a = 1 ho b = c. Thay ng tr l ta  x = 2 là nghi duy nh c phng trình 2.  x 2 x. 3 x 3 x. 5 x 2 x. 5 x         .  u 2 x ; v 3 x ; t 5 x    2  2  2 = uv + vt + tu  (u v)(u t) 2 (1) (v u)(v t) 3 (2) (t u)(t v) 5 (3)                2 = 30  (u v)(v t)(t u) 30    (4) : 30 v t (5) 2 30 u t (6) 3 30 u v (7) 5                            ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 7 31 30 31 30 2(u v t) u v t 30 60        (8)  2 30 u 60 11 30 30 239 v x 2 60 60 120 19 30 t 60                       d) S 1.  x 1 2x 1 5    Cách 1:  Cách 2:  x 1 u 0   và 2x 1 v  22 u v 5 v 2u 1       u2 u 12       x = 5. 2  8 x 5 x 5    .  8x = u , 5 x v   22 u v 5 u v 13      u 2 u=3 v v 3 v=2        3.  22 25 x 9 x 2    .  2 25 x = u, 2 9x   22 u v 2 u v 16       u v 2 u 5 u v 8 v 3            4.  1 x 4 x 3    .   1 x u ; 4 x v      22 u v 3 u v 5       x0 x3      5.  2 2 x 2 x 4 x 2      .  2 x u, 2 x v     2 (u v) 2uv 4 (u v) uv 2           6.  4 4 97 x x 5   (1) .  4 97 x = u, 4 x   (1)  44 u v 5 u 2 u 3 x 81 v 3 v 2 x 16 u v 97                        7.  3 3 3 x 2x 3 12(x 1)    .  3 3 x u, 2x 3 v   (1)  3 3 3 3 3 3 3 u v 4(u v ) u v 3uv(u v) 4(u v )                     ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 8 2 2 2 uv 3.(u v).(u 2uv v ) 0 3.(u v).(u v) 0 uv                  6)  1. : 2 x 4 x m   .   : 2 x 4 x m    2 2 2 2 x m x m x 4 x 4xm m 2mx (m 4) 0                0: 2 m4 x 2m   .          2 m4 2m   + > 0: m 2  2  m 2  0 m 2 + < 0: m 2  2  m 2  2 :  2   0 < 2:          2 m4 x 2m    2 < 0   > 2:      2. Gi và bi lu phng trình v m là tham s: mxx  3 2 (        1999  2000) .   : 2 2 2 2 2 x m x m x 3 x m x 3 x m 2mx 2mx (m 3) 0                  = 0:       0: 2 m3 x 2m   .          2 m3 m 2m   + > 0: m 2  2  m 2  0 m 3 + < 0: m 2  2  m 2   3 :   0 m 3    m3 .         : 2 m3 x 2m     3 m 0      m3 :      3. : x x m m   .     < 0:        = 0:        x( x 1) 0  : x 1 = 0, x 2 = 1  > 0:        ( x m)( x m 1) 0    x m 0 x 1 m        + 0 < 1:       : x 1 = m; x 2 = 2 (1 m) + > 1:         : x = m II.                   :  .       .       :             ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 9 1.            a)   :             : 11 em       : 9 em (1 , 2 , 3 3 ) b)   :               : 9 em       : 9 em (3 , 3 3 ) c) H : 1 em (     ) 2.    a)   :               : 8 em       : 5 em (2 , 1 , 1 , 1 gi)                 : 5 em b)   :               : 5 em       : III.                           .  : 1.                           .      qua,              ,                     .                          .         ,                     2.                           .           ,                  ,   .  ,                  Tuy nhiên,                 ,                       .          100%  .    ,     , quan tâm, ,                       .              9,                                  .      ,                        ,      viên,                               .  ,         ,  .  ,                  3.                                                    .                      ,                       .         ,    ,                         ,  .  ,          ,  1,               9              ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn  Trang 10 .  ,                           ,  ,   ,          .      ,        ,      . ,                   ,                .         ,  . Song      ,  , .  : M,        ,          ,  ,               ,               .      ng,                  ,                ,     .                           ,                 ,          ,                                       THCS,  9.  ,              ,                         .  ,          ,                               ,                ******************************************* [...]...́ ̉ ́ MỘT SÔ PHƢƠNG PHAP GIAI PHƢƠNG TRÌNH VÔ TỈ ****************************** ******************************* www.nguoithay.org & www.cunghoctot.edu.vn Trang có nhiều tài liệu giải chi tiết nhất VIÊT NAM Trang 11 . là nghi duy nh c phng trình 3. Gi phng trình: 68 6 3 x 2 x   : K: x < 2. B cách th, ta th x = 3 2 là nghi c phng trình. Ta c ch minh  là.        1. Gi phng trình: 2x 1 x 2 x 3     .  ý th: (2x + 1)  (x  2) = x + 3. Do , nhân l liên h vào hai v c phng trình: (x 3)( 2x 1 x 2 1). phng trình: 2 x 1 2(x 1) x 1 1 x 3 1 x         (1) .     x 1 1 x 2 x 1 1 x 1 0         x 1 = 0; x 2 = 24 25  3. Gi phng trình: