BDHSG-PT VÔ TỈ

BDHSG Toán 9 Ngày 22 tháng 09 năm 2011 Chuyên đề: PHƯƠNG TRÌNH VÔ TỈ. I. PHƯƠNG PHÁP 1: NÂNG LUỸ THỪA                 f x f x g x g x f x g x ≥   = ⇔ ≥   =               g x f x g x f x g x ≥  = ⇔  =                          f x f x g x h x g x f x g x f x g x h x  ≥  + = ⇔ ≥   + + =                      n n f x f x g x g x n N f x g x ≥   = ⇔ ≥ ∈   =                   n n g x f x g x n N f x g x ≥  = ⇔ ∈  =                  n n f x g x f x g x n N + + = ⇔ = ∈                 n n f x g x f x g x n N + + = ⇔ = ∈  Bài 1: !"#$%&'($! )  ) + = −  *⇔   )   )  )  )  )  )  ) )  − ≥ ≥ ≥    ⇔ ⇔    = + = − − =    x⇔ = Bài 2: !"#$%&'($!   x x− + = *+,-   x x− + =   x x⇔ + =          =⇔         = −= ≥ ⇔    =−− ≥ ⇔    =+ ≥ x x x x xx x xx x Bài 3: !"#$%&'($!    x x x+ − − = − *+,-    x x x+ − − = −     x x x⇔ + = − + −               x x x x x x x  − ≥  ⇔ − ≥   + = − + − + − −   GV: Trần Công Tiến Trang 1 BDHSG Toán 9         x x x x  ≤  ⇔   + = − +               x x x x x  ≤   ⇔ + ≥   + = − +                 x x x x x x x −  ≤ ≤ −   ≤ ≤   ⇔ ⇔ ⇔ =   =    + =    = −   Bài 4: !"#$%&'($!     x x− − − = *.       x x x − ≥  ⇔ ≥  − ≥    ( ) ( )                     / x x x x x x x x x ⇔ − − − + = ⇔ − − + = =   − =   ⇔ ⇔ −  =  − + =   0&!1 23&+4"1,)5 Bài 5. !"#$%&'($!  x x x− = + HD:.6  x≤ ≤ 6!4- &47,!8&"#$%4"#$%     x x x+ + − =           x x −   ⇔ + = ⇔ =  ÷   Bài 6. !"#$%&'($!9+:    / x x x+ = − − HD:.6 x ≥ − !"#$%&'($!&"#$%4"#$% ( )         /  /    ; x x x x x x x x =   + + =  + + = ⇔ ⇔  − −  = + + = −     Bài 7. !"#$%&'($!9+: ( ) ( )       /     x x x x x+ + = + + HD: & ( )       x x x⇔ + − = ⇔ = Bài 8.23<=$>:?$ !"#$%&'($!  )  ) @− = − *+,-  )  ) @− = − ⇔     ) @ ) @ )  ) )@ @ @) @   ≥ ≥   ⇔   − = − + − + =   A0:@5 !"#$%&'($!2B$%!=@ A0:@C  @  ) @ + = .D:6=$4E,-$%!=@)F@⇔  @  @ + F@ G0:@H@  GF@  ⇔@  I⇔  @ < ≤ G0:@J@  GI@  ⇔@  F⇔@IA -@>KA0:@IA!8L,J@I !"#$%&'($!,-@M&$%!=@  @  ) @ + = GV: Trần Công Tiến Trang 2 BDHSG Toán 9 A0:AJ@I!8L,@H !"#$%&'($!2B$%!=@ Bài 9.23<=$>:?$ !"#$%&'($!2N@>3&!+@9O mxx −=−   Bài 10.23<=$>:?$&!P8&!+@9O@ !"#$%&'($! ) ) @ @− = − *.D:6=$)F A0:@J !"#$%&'($!2B$%!=@ A0:@5 !"#$%&'($!&'Q&!3$! ) )  − = ⇒,-!+$%!=@)  5R)  5 A0:@H !"#$%&'($!47,!8&"#$%4"#$%2N  ) @ ) @  − + − = ) @  )  @  − = ⇔  = −   G0:J@I !"#$%&'($!,-!+$%!=@)  5@S)  5   @− G0:@H !"#$%&'($!,-@M&$%!=@)5@ III-Bài tập áp dụng: Bài 1:,T, !"#$%&'($!9+:   x x+ − =      x x+ − − =      x x+ − − =      x x x+ + = +   )   ) + = − −   )  )   )+ − − = −   ) )  )  ) / − − − − + + = ;   x − − = /5  x x−        x − + =  /     x− + =   ;     x− − =    ; x x+ = −     x x+ + − =      x x− − = − Bài 2 !"#$%&'($! +   x x− = − <   x x− + = ,   x x+ + = U   x x+ + − = P    x x− + − = V   x x+ − − = % /   x x+ = − + !     x x x+ − + = +    x x− − = Bài 3(@@4E !"#$%&'($!9+:,-$%!=@     x x m x x− + − = + − Bài 4!8 !"#$%&'($!  x x m− − = +  !"#$%&'($!6!@5 < (@@4E !"#$%&'($!,-$%!=@ Bài 5!8 !"#$%&'($!   x mx x m+ − = − +  !"#$%&'($!6!@5 < WN%T&'X$38,Y+@&!( !"#$%&'($!,-$%!=@ Bài 6: ,T, !"#$%&'($!9+: +   / x x− − − = U  /        x x x− − − + − = − %     x x x x − − = − − <   x − = P    /       x x x− − − + − = − !    x x+ − − = ,    x x− + = V      x x x x+ − = − −     x x− + + = GV: Trần Công Tiến Trang 3 BDHSG Toán 9 II. PHƯƠNG PHÁP 2:  ĐƯA VỀ PHƯƠNG TRÌNH TRỊ TUYỆT ĐỐI I-KIẾN THỨC: Z[U\$%!]$%4^$%&!_,9+:                          f x g x f x f x g x f x g x f x g x f x = ≥  = ⇔ = ⇔  = − <   II-BÀI TẬP: Bài 1:  !"#$%&'($!  ) )  ) ;− + + =  *⇔  )  ; )− = − ⇔`)A`5;A) A0:)J⇒A)5;A)2B$%!=@ A0:) ≥ ⇒)A5;A)⇔)5&!8@7$W?a)5 Bài 2:  !"#$%&'($! )   )  )   )   )   ) + + + + + − + = + − +  * ⇔ )   )   )   )   )  /  )   )   + ≥    + + + + + + − + + = + − + +   ⇔ )  )   ` )   ` ` )  ` ≥ −    + + + + − = + −    .L&a5 ) + aF⇒ !"#$%&'($!47,!8&'Q&!3$! a  ` a `  ` a `+ + − = − A0:IaJaGGAa5Aa⇔a5A>8K A0:IaIaGGAa5aA⇔a5 A0:aHaGGaA5aA2B$%!=@ WNa5⇔)G5/⇔)5;&!8@7$W?a)5; Bài 3: !"#$%&'($!         x x x x− + − + + + − = *.   x ≥             / x x x x⇔ − + − + + − + − + =        x x⇔ − + + − + =   x⇔ − = x⇔ = !8@7$W?a)5 Bài 4: !"#$%&'($!     x x x x+ − + − − = *. x ≥ &         x x x x⇔ − + − + + − − − + =     x x⇔ − + + − − = 0: x >  &     x x⇔ − + + − − =  x⇔ = b8K 0: x ≤  &     x x⇔ − + + − − =  x⇔ = b:B$4c$%2N x∀  W?a&? $%!=@,Y+ !"#$%&'($!>3 { } ` S x R x= ∈ ≤ ≤ III-Bài tập áp dụng: ,T, !"#$%&'($!9+:     x x+ + =    x x− + =    /  x x x− + = −     x x x+ + = +        x x x x− + + + + =      x x x x− + − − + = GV: Trần Công Tiến Trang 4 BDHSG Toán 9      /  ; ;  x x x x x x− + + + + = − + ;      / x x x x− + + − + =  /     x x x x+ − + − − =        x x x x− − − + − − =        x x x x+ − + + + − + =          x x x x− + − + + + − =        x x x x+ − + + − =   =−−−+−++ xxxx      x x x− + + =      ;x x x− + + =       x x x+ + + + = ;     =−−++ xx /       x x x x x + + − + − − =     x x x− + = −          / x x x x− + − − + − − − + =  ;   x x+ − − = III. PHƯƠNG PHÁP 3:  ĐẶT ẨN PHỤ 1. Phương pháp đặt ẩn phụ thông thường  .O2N$!D: !"#$%&'($!2B2B&dR4E%,!c$%&+,-&!E4L& ( ) t f x= 23,!ce4D: 6=$,Y+ t $0: !"#$%&'($!<+$4f:&'Q&!3$! !"#$%&'($!,!_+@M&<0$ t g:+$&'h$%!#$ &+,-&!E%4"1, !"#$%&'($!4-&!P8 t &!(2=,4L& !\)P@$!"i!83$&83$j Bài 1.  !"#$%&'($!     x x x x− − + + − = HD:Điều kiện x ≥ !?$)k&     x x x x− − + − = .L&  t x x= − − &!( !"#$%&'($!,-UK$%   t t t + = ⇔ = !+a238&(@4"1, x = Bài 2.  !"#$%&'($!      x x x− − = + HD:.D:6=$   x ≥ − .L&   t x t= + ≥ &!(    t x − = !+a238&+,- !"#$%&'($!9+:              ;     t t t t t t t − + − − − = ⇔ − − + =        t t t t⇔ + − − − = +&(@4"1,<O$$%!=@>3 R R   S   t t= − ± = ± *8 t ≥ $l$,!d$!?$,T,%T&'X     R   t t= − + = + m4-&(@4"1,,T,$%!=@,Y+ !"#$%&'($!>     vaø x x= − = + Cách khác: +,-&!E<($! !"#$%!+20,Y+ !"#$%&'($!2N4D:6=$     x x− − ≥ +4"1,        x x x− − − = R&m4-&+&(@4"1,$%!=@&"#$%_$% GV: Trần Công Tiến Trang 5 BDHSG Toán 9 .#$%$$!n&>3&+4L&    y x− = + 234"+2D!=4O)_$%(Xem phần đặt ẩn phụ đưa về hệ) Bài 3.  !"#$%&'($!9+:   x x+ + − = *.D:6=$  x ≤ ≤ .L&  y x y= − ≥ &!( !"#$%&'($!&'Q&!3$!        y y y y y+ + = ⇔ − − + = 2N y ≤      y y y y⇔ + − − − =     R   (loaïi)y y + − + ⇔ = = m4-&+&(@4"1,,T,%T&'X,Y+    x − = Bài 4  !"#$%&'($!9+: ( ) ( )    x x x= + − − HD:.  x ≤ ≤ .L& y x= − &!( !"#$%&'($!&'Q&!3$! ( ) ( )        y y y y x − + − = ⇔ = ⇔ = Bài 5. !"#$%&'($!9+:     x x x x x + − = + HD:.D:6=$  x− ≤ < !+,!+20,!8)&+$!?$4"1,    x x x x + − = + .L&  t x x = − R&+%4"1, Bài 6. !"#$%&'($!     x x x x+ − = + * x = 6!B$% !>3$%!=@R!+,!+20,!8)&+4"1,    x x x x   − + − =  ÷   .L&&5   x x − R+,-   t t+ − = ⇔     t x ± = ⇔ = Bài 7. !"#$%&'($!     ;    x x x x+ + + + + = *.L&a5   x x+ + S y ≥ !"#$%&'($!,-UK$%a  Ga5    y y −  =  ⇔  =   y⇔ = WNa5    x x⇔ + + =    x x = −  ⇔  = −  b3$%!=@,Y+ !"#$%&'($!47,!8 Nhận xét.O2N,T,!4L&o$ !\$!"&'l$,!c$%&+,!d%g:a0&4"1,@M&>N <34#$ %$R4B6! !"#$%&'($!4O2N t >Kg:T6!-% 2. Đặt ẩn phụ đưa về phương trình thuần nhất bậc 2 đối với 2 biến :  !c$%&+47<0&,T,!% !"#$%&'($!   u uv v α β + + = <]$%,T,! pk& v ≠  !"#$%&'($!&'Q&!3$!   u u v v α β     + + =  ÷  ÷      v = &![&'q,&0  GV: Trần Công Tiến Trang 6 BDHSG Toán 9 T,&'"r$%!1 9+:,s$%4"+2D4"1,  ( ) ( ) ( ) ( )  a A x bB x c A x B x+ =    u v mu nv α β + = + !c$%&+!7a&!+a,T,<E:&!_,t)R)<Q,T,<E:&!_,2B&d&!(9u$!?$4"1, !"#$%&'($!2B&d&!P8UK$%$3a a) . Phương trình dạng : ( ) ( ) ( ) ( )   a A x b B x c A x B x+ = !"2?a !"#$%&'($! ( ) ( ) Q x P x α = ,-&!E%<]$% !"#$% !T &'l$$0: ( ) ( ) ( ) ( ) ( ) ( ) P x A x B x Q x aA x bB x  =   = +   p:n& !T&&m4^$%&!_,  ( ) ( )     x x x x+ = + − + ( ) ( ) ( )            x x x x x x x x x+ + = + + − = + + − + ( ) ( )        x x x x x+ = − + + + ( ) ( )           x x x x x+ = − + + + 7a&K8'+$!v$% !"#$%&'($!2B&dUK$%&'l$2wU\$!"       x x x− + = + .E,-@M& !"#$%&'($!4x R,!c$%&+ !,!h$!=9O+R<R,9+8,!8 !"#$%&'($!<?,!+  at bt c+ − = %i$%!=@4x j Bài 1.  !"#$%&'($! ( )      x x+ = + HD:.L&      S     u x u v x x v= + ≥ = − + ≥  !"#$%&'($!&'Q&!3$! ( )        u v u v uv u v =   + = ⇔  =  (@4"1,    x ± = Bài 2. !"#$%&'($!         x x x x− + = − + +  **y&!na ( ) ( ) ( )            x x x x x x x x x+ + = + + − = + + − + +20& ( ) ( ) ( ) ( )         x x x x x x x x α β + + + − + = − + + − + .z$%$!n&20&'T2N&+4"1, ( ) ( ) ( ) ( )           x x x x x x x x− + + + − + = − + + − + .L&      S    u x x u v x x v     = + + ≥ = − + ≥  ÷  ÷     !"#$%&'($!&'Q&!3$!:G25  uv u v⇒ = m4{a&+9u&(@4"1,) GV: Trần Công Tiến Trang 7 BDHSG Toán 9 Bài 3: !"#$%&'($!9+:       x x x+ − = −  *.6 x ≥ !?$)k&+20& ( ) ( ) ( ) ( )       x x x x x x α β − + + + = − + + .z$%$!n&20&'T2N&+4"1, ( ) ( ) ( ) ( )        x x x x x x− + + + = − + + .L&   R  u x v x x= − ≥ = + + > R&+4"1, /      v u u v uv v u =   + = ⇔  =  +4"1,  x = ± Bài 4. !"#$%&'($! ( )        x x x x− + + − = *!?$)k&.L& y x= + &+<0$ &&'l$2D &&!:f$$!n&<?,4O2N)23a               x y x x y x x xy y x y =  − + − = ⇔ − + = ⇔  = −  &,-$%!=@ R   x x= = − Bài 5: !"#$%&'($! ( )      x x+ = + *. x ≥ − &       x x x x⇔ + − + = + .L&    R   u x u v v x x  = +  ≥  = − +   !"#$%&'($!&'Q&!3$!:25:  G2   ⇔ ( ) ( )   u v u v− − =   u v v u =  ⇔  =  0::52      /  ; x x x x x⇔ + = − + ⇔ − + = 2B$%!=@ 0:25:         ;    x x x x x x x  = − ⇔ − + = + ⇔ − − = ⇔  = +   >3$%!=@ b).Phương trình dạng :   u v mu nv α β + = + !"#$%&'($!,!8QUK$%$3a&!"r$%6!-i !T&!=$i!#$UK$%&'l$R$!"%$0:&+<($! !"#$%!+20&!(4"+2D4"1,UK$%&'l$ Bài 1.  !"#$%&'($!       x x x x+ − = − + *+4L& ( )   R S  u x u v u v v x  =  ≥ ≥  = −   6!4- !"#$%&'($!&'Q&!3$!   u v u v+ = − !+a:G2:25 ( ) ( ) u v u v+ − Bài 2. !"#$%&'($!9+:        x x x x x+ + − = + + *.6   x ≥ ($! !"#$%20&+,- ( ) ( ) ( ) ( ) ( ) ( )              x x x x x x x x x x+ − = + ⇔ + − = + − − GV: Trần Công Tiến Trang 8 BDHSG Toán 9 +,-&!E4L&     u x x v x  = +  = −  6!4-&+,-!=       u v uv u v u v  − =   = − ⇔  + =   *8 R u v ≥  ( )           u v x x x + + = ⇔ + = − Bài 3. !"#$%&'($!     /   x x x x x− + − − − = + *.6 x ≥ !:aE$20<($! !"#$%&+4"1, ( ) ( )        x x x x x− + = − − + Nhận xét : !B$%&z$&K9O R α β 4E ( ) ( )       x x x x x α β − + = − − + +  2?a&+6!B$%&!E4L&    u x x v x  = − −  = +   !"$%@+a@|$&+,- ( ) ( ) ( ) ( ) ( ) ( ) ( )          x x x x x x x x x− − + = + − + = + − − +20&>K !"#$%&'($! ( ) ( )            x x x x x x− − + + = − − + .0$4{a<3&8T$ 4"1,%g:a0& 3. Phương pháp đặt ẩn phụ không hoàn toàn  m$!v$% !"#$%&'($!&w,! ( ) ( )     x x x+ − + − + = R ( ) ( )      x x x x+ − + − + = !+&'E$23'c&%h$&+9u4"1,$!v$% !"#$%&'($!2B&d6!B$%&f@&!"r$%,!c&$38R4M 6!-,Y+ !"#$%&'($!UK$%$3a !\&!:M,238 !"#$%&'($!&w,!@3&+):n& !T& m4-,!c$%&+@N4&(@,T,!% !"#$%&'($!UK$%$3a!"#$% !T %4"1,&!E !=$g:+,T,2wU\9+: Bài 1. !"#$%&'($! ( )        x x x x+ − + = + + *.L&  t x= + S t ≥ R&+,- ( )        t t x t x t x =  − + − + = ⇔  = −  Bài 2 !"#$%&'($! ( )      x x x x+ − + = + *.L&   R t x x t= − + ≥ !4- !"#$%&'($!&'Q&!$! ( )   x t x+ = + ( )    x x t⇔ + − + = {a%r&+&!l@<N&R4E4"1, !"#$%&'($!<?,&!P8&,- ∆ ,!}$ ( ) ( ) ( ) ( )               t x x x t x t x t x t x =  − + − + + − = ⇔ − + + − = ⇔  = −  Bài 3: !"#$%&'($! ( )      x x x x+ + = + + *.L&  S t x t= + ≥ GV: Trần Công Tiến Trang 9 BDHSG Toán 9 !"#$%&'($!&'Q&!3$!&  )G&G)5 ⇔ &)&5  t x t =  ⇔  =  0:&5)  x x⇔ + = WB>e0:&5     x x⇔ + = ⇔ = ± W?a  x = ± 4. Đặt nhiều ẩn phụ đưa về tích  p:n& !T&&m@M&9O!=i4K9Oi4x ,!c$%&+,-&!E&K8'+4"1,$!v$% !"#$%&'($!2B &d@36!%$-,!c$%&+>K4L&$!D:o$ !\23&(@@Og:+$!=%v+,T,o$ !\4E4"+ 2D!= p:n& !T&&m4^$%&!_, ( ) ( ) ( ) ( )     a b c a b c a b b c c a+ + = + + + + + + R+,- ( ) ( ) ( ) ( )     a b c a b c a b a c b c+ + = + + ⇔ + + + = m$!?$)k&$3a&+,-&!E&K8'+$!v$% !"#$%&'($!2B&d,-,!_+,~$<?,<+       ; ;  x x x x x+ − − − + − + =         /   x x x x+ + − + − − − = Bài 1.  !"#$%&'($!         x x x x x x x= − − + − − + − − *. x ≤ .L&  S   S   S  u x u v x v w x w  = − ≥   = − ≥   = − ≥   R&+,- ( ) ( ) ( ) ( ) ( ) ( )          u v u w u uv vw wu v uv vw wu u v v w w uv vw wu v w u w  + + =  − = + +   − = + + ⇔ + + =     − = + + + + =   R%!=&+ 4"1,  /   u x= ⇔ = Bài 2. !"#$%&'($!9+:            x x x x x x x− + − − = + + + − + *+4L&             a x b x x c x x d x x  = −   = − −   = + +   = − +   R6!4-&+,-      a b c d x a b c d + = +  ⇔ = −  − = −  Bài 3. ,T, !"#$%&'($!9+:        / x x x x x+ + − − + = − *.L& ( )      S   a x x a b b x x  = + +  ≥  = − +   +4"1,!= !"#$%&'($!    /   /  a b x a b x  − = −  − = −  m4-&+,-+  <  5+< ⇔ +<+G<5    a b a b =  ⇔  = −  0:+5<          x x x x x⇔ + + = − + ⇔ = &!8@7$ GV: Trần Công Tiến Trang 10 . BDHSG Toán 9 Ngày 22 tháng 09 năm 2011 Chuyên đề: PHƯƠNG TRÌNH VÔ TỈ. I. PHƯƠNG PHÁP 1: NÂNG LUỸ THỪA                 f x f x g x. !"#$%&'($!47,!8,-$%!=@>3       x x  = −    =   5.2 Giải phương trình vô tỉ bằng cách đưa về hệ đối xứng loại II  +!7a4&(@$%:z$%O,,Y+$!v$%<3&8T$%