Chun đề 1 PHƯƠNG TRI ̀ NH VƠ TY ̉ * Dạng 1 : A 0 (hoặc B 0 ) A B A B ≥ ≥  = ⇔  =  T      n n A B= * Dạng 2 : 2 B 0 A B A B ≥   = ⇔  =   T     n A B= Da ̣ ng 3   A B= T        n n A B + + = Da ̣ ng 4   A B A B= ⇔ =        n A B + = + − = − − = − + − + − − = − + + = + + − − + = 2 2 2 3 3 1. : ) 4 2 2 (1) ) 4 2 8 12 6 (4) ) 3 1 4 1 (2) ) 12 14 2 (5) ) 11 11 4 (3) Ví dụ Giảicác phươngtrình a x x x d x x x x b x x e x x c x x x x    ( )  − ≥  ≥  ≥   ⇔ ⇔ ⇔ ⇔ =    = ∨ = − = + − = −      = 2 2 2 ) : 2 0 2 2 (1) 3 0 3 3 0 4 2 2 : 3 a Tacó x x x x x x x x x x x Vậy x ( ) ≥ − ⇔ + = + + ⇔ + = + + + + ⇔ + = −  − ≥  ≥  ≥   ⇔ ⇔ ⇔ ⇔ =    = ∨ = − = + = −      = 2 2 1 ) : 3 (2) 3 1 1 4 3 1 1 4 2 4 4 2 2 0 2 2 5 0 5 5 0 4 2 : 5 b Tacóđiềukiện x x x x x x x x x x x x x x x x x x Vậy x ( ) ± + ≥ ⇔ + + + − + + − − = ⇔ − − = −  − ≥  ⇔ ⇔ =  − − = −   = 2 2 2 ) : 11 0 ( ) (3) 11 11 2 11 16 11 8 8 0 5 ( ( )) 2 11 8 : 5 c Tacóđiềukiện x x a x x x x x x x x x x x thỏa a x x x Vậy x = − + ≥ = − +   − + = − + = −  ⇔ = − ⇔ − = ⇔ = ∨ = ⇔ ⇔ =  − + =   − + =   = 2 2 2 2 2 2 2 2 2 ) 2 8 12 0, 2 8 12 2 8 12 0 2 8 12 0 12 (4) 6 2 0 0 2 2 2 2 8 8 0 2 8 12 2 : 2 d Đặt t x x ta có t t x x x x x x t t t t t t x x x x x Vậy x  ( ) + = + + + ⇔ − + + + − + − + + =  − + = −   − + + =   ⇔ − + = − ⇔ + − = ⇔ = − ∨ = = − ∨ = = − = + 3 3 3 3 3 3 3 3 3 3 3 3 3 ) 1: : ( ) 3 ( ) (5) 12 14 3 12 . 14 12 14 8 12 . 14 .2 6 ( ) 12 14 2 ( ) ( ) 12 14 27 2 2 195 0 15 13 ( ( )) : 15 13 2 : 12 ; 14 . e C Tacó a b a b ab a b x x x x x x x x a x x b a x x x x x x thỏa b Vậy x x C Đặt u x v x Ta   + = + =    + = = − =   ⇔ ⇔ ⇔ ∨      = − = = − + = + − + =        − = − ⇔ = − = ⇔ = 3 3 3 3 3 : 2 2 2 1 3 3 3 1 26 ( ) 3 ( ) 26 * 12 1 13 * 12 3 15 có u v u v u v u u uv v v u v u v uv u v x x x x            * Phương pháp 1 : Biến đổi về dạng cơ bản Ví dụ : Giải phương trình sau : 1)  −=− xx (x=6) 4) !  =−++− xxx 1 (x ) 2 = − 2) "  −=−+− xxx ( "  = x ) 5)   +=+− xxx ( #  " ±− = x 3) $ =−− xx ( " = x ) 6)     xx =− (  ±= x ) * Phương pháp 2 : Đặt điều kiện (nếu có) và nâng luỹ thừa để khử căn thức 1) ! ++−=+ xxx ( 11 x 0 x ) 3 = ∨ = 4) " =−−−−− xxx (x=2) 2) $ =+−− xx ( ! = x ) 5) % +=−+ xxx (  = x ) 3)  +=++ xxx (   +− = x ) 6)  +−=+ xx ( = x ) * Phương pháp 3 : Đặt ẩn phụ chuyển về phương trình hoặc hệ pt đại số 1) xxxx ##&"&  +=−+ (x 1 x 4)= ∨ = − 5) "##&& =−++−++ xxxx (x 0 x 3)= ∨ = 2)   =+−+− xxx (x 1 x 2 2)= ∨ = − 6)   −−=− xx (x 1 x 2 x 10)= ∨ = ∨ = 3) #"#&&" =−++−++ xxxx (  " ± = x ) 7) '  −+−=−++ xxxx (x=5) 4) '  =+−++− xxxx (x=1; x=2) 8) "!  +−+−=−+− xxxxx (x=2) Lu ̣ n tâp:&  ( ) *(    +,-  ./ ) /- ) 0  10.  2 )    ) -*&2 )   +   *(    #.33# 145( )   26(   #  "  & #x x x− = − = #   !   & #x x x x− + + = = #    $  & 7 #x x x x x+ + = + = = #   & %#  $ x x x x − = − = − 3#     & #x x x x− − = − = − 8#  ' "  & #x x x x− = − − − − = −  #       $ 7  x x x x x   − = − − = =  ÷   #   "    x x x x   − − = − =  ÷   245( )   26(   #      & #x x x− + + = = #         7 7  x x x x x x   − + − = − = = =  ÷   345 )   26(   # ( ) ( )     "  ' & $7 #x x x x x x+ + − + + = = − = #      %   " $ 7   x x x x x x   − + + − + = = − =  ÷   #    $    ! & 7 #x x x x x x x x+ + + + + = + + = − = # %  $   $ & #x x x x x+ − + = − + − + = 445 )   26(   #       & #x x x x x− − + + − = = #    '  & #x x x x x− + − = − + = #    "     7  x x x x   − ± + = − = =  ÷  ÷   #     $  7     x x x x   + + − = = ± = −  ÷   3#    %  %  &   '#x x x x x− − + = + = ± 8#      ' " % & #x x x− + − − = = − 545 )   26(   #     & #   x x x x x + − + = = − #     & %#x x x x x− + − − = = #    "  "     x x x x x   ± + + − + + − = =  ÷  ÷   # "   ' & #x x x+ + + = = 3#  !   & #x x x− + + = = 8#      " & #x x x+ + − = = #      & #x x x x− + − + = = #      & 7 "#  x x x x x x x + + − + − − = = = 645( )   26(   #      x x x x+ + + = + + &9:# #        x x x x x x + + + = − + + + + &  .  x x= − = + # #        x x x x+ + + = + + + &9: 79:;# #      x x x x x+ + = + + &9:# 3#     x x x x + + = + &9:# I. C ơ bản : 4    ! x x x+ = − − 4      x x x x+ + + = + + 4 & #  'x x x+ − = + + 4        x x x x+ + + = + + + "4      x x x x x+ + = + +  '4       x x x x x x+ + + = + + + $4    & #x x x x x x− − − − + − %4    % '   x x x x+ + + − = + !4 "     x x x− − − − − =  4 ( )     x x x x+ − = − − 4      x x x x− − + + − − = 4 "        x x x x x + + + + + + − + = II. È n phu : 4     x x x x− − + + − = 4         =++ + xx x "4 "  'x x+ + − = '4      x x x x+ − = + − $4      x x x x+ − = + − %4     x x x x x + − = + !4     x x x x+ − = +  4 + − + + − = − − 1 ( 3)( 1) 4( 3) 3 3 x x x x x 4      !   " x x x x x − + − = − + − + 4    x x+ + = 4 ( )     " x x+ = + 4 2 3 2 5 1 7 1x x x+ − = − "4 − + = + + 2 2 (4 1) 1 2 2 1x x x x '4 2 2 2(1 ) 2 1 2 1x x x x x− + − = − − $4 ( )       ' x x x x− + + − = %4        x x x x x+ + − = + + !4 ( )        x x x x+ − + = + +  4        x x x x+ − = + − + − 4   & #    x x x x− + = + + 4     & # x x x x+ + = + + 4  $ $x x+ + = 4 + = − 3 3 1 2 2 1x x "4      x x+ = − '4   '  %  x x x+ = − − $4      x x− + − = %4 2 2 3 2 1( 99)x x x x NT− + − + − = − !4    x x − = − −  4    !   x x x x x+ + + − + = + 4     x x x− = − 4   '   "x x x− − = + 4 ( )       x x x+ − = + − 4 ( ) ( )       x x x x+ − = − "4 ( ) ( )          x x x x   + − − − + = + −     '4  '  x x+ =  . 15 có u v u v u v u u uv v v u v u v uv u v x x x x            * Phương pháp 1 : Biến đổi về dạng cơ bản Ví dụ : Giải phương trình. (1) ) 4 2 8 12 6 (4) ) 3 1 4 1 (2) ) 12 14 2 (5) ) 11 11 4 (3) Ví dụ Giảicác phươngtrình a x x x d x x x x b x x e x x c x x x x    ( )  −