CUA PHUONG TRINH BAC HAI
Bai 3.25: Giai cac phuong trinh sau
q
a)
13 1
2x^ + x - 2 1 2x + 7 x ^ - 9
a;^0
b ;^ 0 hoac a = b . a ^b
x + 1 x - 2 x - 3 x + 4 ,
b) - + r + r + - = 4
x - 1 x + 2 x + 3 x - 4 Hwmg dan gidi
a) DKXD: x ^ ± 3 ; x ^ - - 2
13 1 6 j
(x - 3)(2x + 7 ) 2 x + 7 " (x - 3)(x + 3) * H n i t i s n o u r l q ^ / o x^ + X - 1 2 = 0 o (x - 3)(x + 4) = 0 <=>
Vay phuong trinh c6 nghiem x = -4 .
^) Dieu kien: x ô { - 3 ; - 2 ; l ; 4 }
x = 3 x = -4
4 x;!"- * x '.ti')'..-' '
P T ô i + 2 1 4 ^ 6 ^ 8 ,
• + 1 + 1 + 1 + = 4 X- 1
5 x - 8
x + 2 x + 3 x - 4 5x + 12
( x - l ) ( x - 4 ) (x + 2)(x + 3) = 0
<=>x2 + x - ^ = 0 ô x = l
chieu voi dieu ki^n phuong trinh c6 nghiem la x = i
•<<i i * n "
- i ± , —
b) 3x^+1 X'' + - X Bai 3.26: Giiii phuang trinh
2x 13x a) —5 + — ^ = 6
3x ~5x + 2 3 x ^ + x + 2 c) + ^ = 15
' > / rV'-^- 1 •' y"*^ Huang dan gidi
6',} (
= 3
a) Dieu kien: x i 3
Vai X = 0 khong la nghiem cua phuong trinh
Voi X ^ 0 ta CO PT o ^—j + = 6 ' (*) rf""' ^iim'ruiU 3x - 5 + - 3x + 1 + >';;:'•; •;ô'' . •/
X. i \ , X
2 2 13 Dat t = 3x + - phirong trinh tro thanh PT o - — - + - — - = 6
1 4 f ;
Tir do ta tim dugc nghiem cua phuong trinh la ^ ~ 2' ~ 3^' I A T - 1 ± Vs^
b) Dieu kien: xi<0;
, X^ + 3 X^ + 1
P T ằ
I r
X"' + X^ - X
x2+ \ + 3
= 3 o ^ = 3 X - +1
X
i Dat t = X - i phuong trinh tro thanh ^ = 3
! X t + 1
T u do p h u o n g trinh c6 nghiem la x := ^ ~ ; x = 1 ± \ .
4 2
c) Dieu kien: x ? ! : - l ; x * 0
P T ô 1 1
X X +1 x(x + l )
1
x(x + l) x(x + l ) I 15 = 0 Dat ~ — - = t ta duoc phuong trinh t^ + 2t - 1 5 = 0 o t = 3; t = - 5
• x(x + l )
+) t = 3 o — L - = 3 o 3 x ^ + 3 x - l = 0 c ằ x = ~ ^ - ^
x(x + l) 6
+) t - -5 o = -5 o 5x2 + 5x + 1 = 0 o X = -5±S
x(x + l ) 10 Doi chie'u v o l dieu kien (*) thi phuong trinh c6 bo'n nghiem
-3 ± x/2T -5 + N/5 X = ; x = •
10 . , 27- Giai p h u o n g trinh
x + 3 i x - 3 a)
x2 b ) ^ l l . 1 3 ( x . l ) 3x + X 3x +7x + 6 Huong dan giai
^, pieukien: x ; ^ 2 ; x ^ - 3 x+1 x - 2 ^ . Dat u = ; ^ ; v = - - ^ t a duoc
^2 ^. uv = 12v <=> (u - 3v)(u + 4v) = 0 o u = 3v; u = - 4 v
^ 3v ^ ^LLL = 3 ^ L ^. x - 2 <=> x^ + 4x + 3 = 3x2 - -[2^ +12 x - 2 x + 3
ci.2x2-16x + 9 = 0 o x = 8±>/46
0'> f i r ; . ? ,
a u = -4v <=> ^ ^ ^ = -4-^^—^ <=> x^ + 4x + 3 = -4x2 ^ ' x - 2 x + 3
o 5 x 2- 1 2 x + 19 = 0c=>3x
Vay phuong trinh da cho c6 hai nghiem la x = 8 ± V46 b) DKXD: x ^ O , x ^ - 1
Dat u = X +1, V = 3x2 + X, u ^ 0, V 0 Khi do p h u o n g trinh tro thanh
—+ - ^ ^ = 6 ằ 4 u 2 - 7 u v - 2 v 2 = 0 ằ ( 4 u + v ) ( u- 2 v ) = 0 c >
V V + 6u
4u u = Tir do ta t i m dugc nghiem cua pt la x ^
^ai 3.28: Giai va bien luan phuong trinh sau l l - i l
2 ' 3 j ax - 1
X - 1 x + 1 Huang dan giai
f^KXD: x ^ ± l ' •
^ T c : > ( a x - l ) ( x + l ) + 2 ( x - l ) = a(x2 +1)
^ a x 2 + a x - x - l + 2x-2==ax2 + a <=>(a + l ) x = a + 3
a( x 2+ l )
X - £
6b r r : „ ,
' NP'H ^ a + 3 ^ , a + 3 . , , a + 3 . ^ ^eu a * - 1 t h i X = . Ta co ^ 1 , xet - 1 <:> a - 2 a + l a + 1 a + l
a = - 1 thi p h u o n g trinh vo nghiem.
Cty TNini M l\' nWH Khang Viet Vay: - V 6 i a - 1 va a ?^ - 2 thi p h u o n g trinh c6 nghiem duy nhat x = 5.'
- V o i a = - 1 hoac a = - 2 thi p h u o n g trinh v6 nghiem.
Bai 3.29: T i m dieu kien a , b de p h u o n g trinh phan biet.
Huong dan giai
x - b X - a = 2 C O hai nghj
Dieu kien: X ;t a,x ?t b :
Ta c6: PT o 2(x - a)(x - b) = a(x - a) + b(x - b) ^ 1 , i - y : •-—
ằ 2 x ^ - 3 ( a + b)x + a^ +b^ + 2ab = 0 o 2x^ - 3(a + b)x + (a + b)^ =Q P h u o n g trinh c6 h a i nghiem la xj = a + b v a X 2 = ^ ^ ^ ' v
T a c o X j 5 t a < = > b ^ 0 , X i 9 t b o a 7 t 0 , X 2^ a o x 2? t b < = > a ; t b u a + t" u
X i ^ X o <=> a + b 7t <=> a / - b
1 ^ 2
Vay v o i a ^ ±b; a 0, b ;^ 0 thi pt c6 hai nghiem phan biet. -'"
j , ) D ] < X D : 2 x - 5 > 0 ô x > - . ^" '
• j H l : V o i x - 4 < 0 c 5 > x < 4 t a c o VT(*) > 0, VP(*) < 0 suy ra p h u o n g trinh v6 nghi?m
T H 2 : V o i x - 4 > 0 <=> X > 4 ta c6 hai ve khong am nen p h u o n g trinh (*) tuong d u o n g voi • • • - r T ' - T w I t M r riôi'ft:>'0'rf:i K t i ' ^ ' v i •„.
X = 3 i r
2 x - 5 = ( x - 4 ) ^ <=>x''-10x + 21 = 0 < ằ x = 7
DANG TOAN 3: PHUONG TRINH CHUA AN TRONG CANBAC HAI.
'Phuang phdp gidi.
De giai p h u o n g t r i n h chiia an d u d i da'u can ta t i m each de khir dau caa bang each: ^ ^ 0 * v , f )
- Nang luy thua hai ve. i .
- Phan tich thanh tich. ô i - Dat an p h u . ;-:>
Q L C A C V f DUMINH HOA
J^ai / . B i n h phuong hai ve ciia phuong trinh.
V i d u 1: Giai cae p h u o n g trinh sau
a) Vx2+2x + 4 = V 2 ^ b) x - V2x-5 - 4 LOT gtfli
a) D K X D : x'^ + 2 x + 4 > 0
2 - x > 0 <=> X < 2
V d i dieu k i ^ n do p h u o n g trinh t u o n g d u o n g voi
r x - - i
Doi chieu v d i dieu kien ta duoc nghiem ciia phuong trinh la x = - 1 va x = x ^ + 2 x + 4 = 2 - x<=>x^+3x + 2 = 0<::>
t - XB + ^xr. <
144
Doi chieu vdi dieu ki?n x > 4 va dieu kien xac djnh suy ra chi cd x = 7 la nghiem.
Vay p h u o n g trinh cd nghiem la x = 7.
Nhatt xet: T u cac Idi giai cac bai toan tren ta suy ra doi v d i cae dang p h u o n g trinh sau ta cd the giai bang each thuc hi^n phep bien doi t u o n g d u o n g :
f(x) = g(x) ^ d f(x) > 0 (hay g(x) >0) i^X itSm B O * gn ằ-!>;
f(x) = [g(x)f : i < ffl• ct> — < L' \ /
g ( x ) > 0 V i d u 2: Giai cac p h u o n g trinh sau
a) X = \ / N/ 3X2+ 1 - 1 b) 7 2 ^ + X2 - 3X + 1 = 0 Lcfi gidi
a) Phuong trinh t u o n g d u o n g vdi x > 0
= 7 3 x ^ + 1 - 1 f x > 0
x > 0
\/3x2 + 1 =x^ + 1 f x > 0
1 3 x 2 + I = ( x 2 + 1 ) 2 l x 4 _ x 2 = o x > 0
2 (x=-T) = 0
x > 0 x = 0 <=>
x = ±l
x = 0 x = l Vay p h u o n g trinh ed n g h i f m la x = 0 va x = 1
^) Taco V 2 x ^ + x 2 - 3 x + l = 0 o V2x-1 = - x 2 + 3 x- l - x ^ + 3 x - l > 0 - x ^ + 3 x - l > 0 2 x - l = { - x 2+ 3 x - l ) [ ( x - l) 2 ( x 2- 4 x + 2) = 0 - x ^ + 3 x- l > 0
X = 1 o
x 2 - 4 x + 2 = 0
- x ^ + 3 x- l> 0
x = l C5
X = 2±N/2
x = l x = 2-V2
Vay p h u o n g trinh cd nghiem la x^^l va x = l - - J l .
Vi dv 3: T i m m de phucmg trinh Vx^ + mx + 2 = 2x +1 c6 hai nghifm phan bi§t.
Lcri gidi Phuong trinh <=> 2
3 x 2+ ( 4 - m ) x - l = 0
Phuong trinh da cho c6 hai nghi^m <=> (*) c6 hai nghi^m phan biet Ion hon hoac bang -^<=> thi ham so y = Sx^ + ( 4 - m ) x - l tren
friic hoanh tai hai diem phan bi^t.
Xet ham so y = Sx^ + (4 - m)x -1 tren
fat
• 1 ^ b m - 4
. Ta CO =
2a 6 + THI: Ne'i. —^<-—<=>m<l thi ham so dong bien tren
6 2
m < 1 khong thoa man yeu cau bai toan.
+ TH2:Neu - ! ^ ^ > - - ô m > l : 6 2 Ta CO bang bien thien
- - ; . = c nen
' 1
m - 4
+00
+00
^m-4^
. 6
Suy ra do thj ham so y = Sx^ + (4 - m)x -1 tren hai diem phan bi|t
c3t true hoanh tai
ô y >0>y >0>y
. 1) J I 6 J <=> 2 m - 9 12^ m^ +8m-28) (1) Vi -m^ +8m-28 = - ( m - 4) 2 -12<0, Vm nen
(1) o 2m - 9 > 0 <=> m > - (thoa man m > 1) 9 ' ' "•^
Vay m > - la gia tri can tim. 05; f- xC+ "'x-- l = x
11
Cty TNHIl \f! \ Vi?t
jCgai 2; Phan tich thanh tich bAng each nhan lien hqip.
i' true can thue ta nhan voi cac dai lugng lien hgp: \
( ^ - V B ) ( V A + V B ) A - B V , V . - ,
A - B , V A - V B =
Voi A , B khong dong thoi bSng khong.
Vi dy 4: Giai eae phuong trinh sau ^ ^
>2 a) 2 ( x - l ) ^
(3-V7T2^f = x + 20
b) N/SX - 2 + ^ = 2 Ldi gidi
a) DKXD: 7+2x>0 7 x > —
x ^ l 2
2 ( x - l) 2 ( 3 + j7T25^f
Phuong trinh <=> \ = + 20 , ^
( B - T T T Y X ) (3 + V7 + 2x) ^
2 ( x - l) 2( l 0 + 2x + 6V7 + 2x)
- = X + 20 (2-2x)^
o 10 + 2x + 6 V 7 + 2x = 2(x + 20)
O Jj + lx = 5 ox = 9 (thoa man dieu ki^n) Vay phuong trinh c6 nghi^m x = 9
b)DKXD: x > - ' . S c • - J . Nham ta thay x -1 la nghi^m ciia phuong trinh nen ta taeh nhu sau Phuongtrinh ằ ( V 3 ^ - l ) + ( ^ - l ) = 0 ••
^ ( V 3 ^ - l ) ( V 3 ^ + l) P M F ^ ^ _ 5 ^ - O •
^ 7 3 ^ + 1 ^ ^ + ^ + 1
<:>_iiz3_^ = O o ( x - l )
V 3 ^ + l 3/7 + ^ + 1 ,. >;
1
3 1 y s ^ ^ + i 3/7+3/^+1,
=o(*:
Phdn loai vii phutnig phiip gini D i i i sd'lO
D o ^ + ^ + l = f ^ + l l + - > 0 nen ^_ + - ^ P h u o n g t r i n h (*) o x = 1 (thoa man dieu kien) • M <i /. \
V a y p h u o n g t r i n h c6 n g h i e m d u y nha't x = 1.
> 0
V i d u 5: Giai cac p h u o n g t r i n h sau ... ..^
a) (x + 3 ) 7 2 x ^ + 1 = x^ + X + 3 b) (3x + 1 ) 7 x ^ + 3 = Sx^ + 2x + 3
a) Ta thay x = -3 k h o n g la n g h i e m ciia p h u o n g t r i n h Xet x*-3, p h u o n g t r i n h <=> V2x^ + 1 = ^ ^
x + 3
1 1 X 2x
+ 1 - 1 = <=> X +
3 V2x2 + 1 + 1 x + 3
<=>
x = 0
2(X + 3) = N/2X^ +1 +1 (*
P h u o n g t r i n h ( * ) ô V2x^+1 = 2x + 5
V t : : U - x >
2
2 x ^ + 1 = 4x2+ 2 5 + 20X
2 x^ + 1 0 x + 12 = 0
•"Tt^^^y•^~•.;,r,,:'A•':^
x > -
2 <=> X = 5 + x/ l 3 (thoa man)
X = - 5 ± VT3 I >
V a y p h u o n g t r i n h da cho c6 n g h i e m x = 0 va x = - 5 + y/lS b) Ta thay x = - ^ k h o n g la n g h i e m ciia p h u o n g t r i n h
Xet x ^ - - , p h u o n g t r i n h da cho <=> Vx^ t- 3 = + 2 x + 3 3 3x + l
1 8 Den day, chii y Sx^ + 2x + 3 = 3(x + -)2 + - > 0
3 3
X :a>'
N e n p h u o n g t r i n h c6 n g h i f m phai thoa m a n x > - - Vx^ + 3 + 2x > 0 3
3x + l D o d o p h u o n g t r i n h da cho o 7 x^+3 - 2x = + ^ _
x ^+ 3 - 4 x 2 3x2+2x + 3 - 6 x 2- 2 x
<::> = —
V x 2 + 3+ 2 x 3x + l
2x,
_ 3( 1- x 2 ) ^ 3( l- x 2 ) ^ V x 2+3 + 2x 3x + l
x 2= l
fc.-x£;
'5 •r + s - x £ \ Vx2 + 3 + 2x = 3x + l
, T H l : x2 = l ô x = ± l
islhung X = - 1 k h o n g thoa m a n ^ > - - nen p h u o n g t r i n h c6 n g h i e m x = 1
T:H2: N/ X^ + 3 + 2x = 3x + 1 <=> Vx^ + 3 = x + 1 x > - l
< ^ L 2 + 3 = x 2+ l + 2x
1": ••; f.'
<=> X = 1 (thoa man)
VSy p h u o n g t r i n h c6 n g h i e m d u y nha't x = 1.
Xgai 3 ; D a t a n p h u
V i d u 6: Giai cac p h u o n g t r i n h sau ^ ^•^x,'^,:^m<4S'^'^io E
a) x2 + 7 x^ + 1 1 = 31 b) (x + 5)(2 - x) = 3Vx2 + 3 x Laigidi
a) Dat t = V x ^ f H , t > 0 . K h i d o p h u o n g t r i n h da cho t r o thanh: ,
" t = 6 t = - 7
t 2+ t - 4 2 = 0 ô /;J r!,nri!
-, - 111"!',
V i t > 0 => t = 6 , thay vao ta c6 Vx^ +11 = 6 x 2+ n = 36<:=>x = ±5 ' r^J' Vay p h u o n g t r i n h c6 n g h i e m la x = ±5
b) P h u o n g t r i n h ô x^ + 3x + sVx^ + 3 x - 1 0 = 0
Dat t = Vx^ + 3 x , t > 0 . P h u o n g t r i n h da cho t r o t h a n h
s +• xs;
r + 3 t- i o = 0 o t = 2
t = - 5 ' iiiJgnr:.,;:.-
<^x^ + 3 x - 4 = 0<=>
V i t > 0 =:> t = 2, thay vao ta c6 Vx^ +3x = 2 " 7j;:^v^*''
" x = l •
. X = - 4 ^ : l
, _ V | y p h u o n g t r i n h c6 n g h i e m la x = 1 va x = -4 .
^1 7: Giai cac p h u o n g t r i n h sau
a) 7 4 ^ + 4x2 - 6 x + l = 0 b) V3X2- 2X + 9 + N/3X2-2X + 2 = 7 Lot gifli
^) D K X D : x > i :>o..^/ *
4
Dat t = V 4 x- 1 , t > 0 => x = t 2+ l
149,
r .2 Phuang trinh tra thanh t + 4 - 6
c:>4t + t * + 2 t ^ + 1 - 6 ( t ^ + l ) + 4 = 0
< = > t''-4t2+4t- l = 0 o ( t - l ) ( t 3 + t2-3t + l ) = 0 t = l
+ 1 = 0
: ,< •gnu.
> ( t - l f ( t 2+ 2 t- l ) = 0 o
Vol t = 1 ta CO 1 = V 4 x - 1 o X = -
t = -l±x/ 2 1
(loai t = - 1 - V2 )
Vol t = - l + V2 taco - I + V2 =V4x^<=>4x-1 = 3- 2 7 2 o x = ^ - ^ 2
A 1 2 — / ?
Vay phirang trinh c6 hai nghiem x = - va x = ^ .
b) Dat t = V3x2 -2x + 2, dieu kien t > 0. Khi do VSx^ -2x + 9 = V t V z . Phuang trinh tro thanh 7t^ +7 +1 = 7
o Vt^ + 7 = 7-t<=> t<7 ft<7
<r><^, _ O t = 3 t^+7 = t ^ - 1 4 t + 49 It = 3
Vai t = 3 ta c6 V3x^ - 2 x + 2 = 3
<ằ3x2- 2 x + 2 = 9 o 3 x^ - 2 x- 7 = 0 o
X =
X =:
1 + 722 3 I- V 2 2 Vay phuong trinh c6 hai nghiem x = 1±V22 V i du 8: Giai phuang trinh
a) (x + l)2-2V2x(x2+ l)=0 b) 10Vx^ + l=3(x2+2) Lai giai
a) DKXD: 2 x( x 2+ l) > 0 o x > 0 i Dat V2x =a,\/x^ +1 = b; a > 0 , b > 0
Suyra a^ + b^ =2x + x 2 + l = (x + l)2
Phuang trinh tra thanh a^ + - 2ab = 0 o (a - b)^ = 0 o a = b Suyra N/2^ = Vx^ +1 <=> 2x = x^ +1 o (x -1 ) ^ - O o x = l (thoa man) Vay phuang trinh c6 nghiem la x = 1 .
pKXD: + 1 > 0 <=> x> - l .
phuong t r i n h ô loV(x + l)(x2 - x +1) = 3(x2 + 2) , | / pat Vx + l = a ' V x 2 - x + l = b , a > 0 , b > 0 , A • n , . , Suy ra a^ + b^ = x2 + 2 khi do ?'"^V ^^^-H ,„ ^ •;
phuang trinh tro thanh f r x V
I0ab = 3(a2+b2)o3a2 -10ab + 3 b 2= 0 o( 3 a- b ) ( a- 3 b ) = 0 o Vai 3a = b taco 3N/X + 1 = V X ^ - X + 1<=>9(X + 1) = X ^ - X + 1
^ x ^ - 1 0 x - 8 = 0<=>x = 5 ± >/33 (thoa man dieu ki^n) . ; ' Vai a = 3b ta c6 Vx+T = 3\/x^ - x + 1 <=> x +1 = 9(x^ - x +1) ^^^.j,
c>9x - l O x + 8 = 0 (phuang trinh v6 nghiem) ^ Vay phuang trinh c6 nghif m la x = 5 ± N/33 .
3a = b a = 3b
Vi du 9: T i m m de phuong trinh sau c6 nghiem
a) ( 2 x - l ) ^ + m = Vx^-x + l (1) b) 3 ^ / ^ + m V ^ ^ = 2^x^- 1 (2) Led giai
a) Dat t = V x 2 - x + l o t^ = x^ - x + 1 ^ (2x - i f = 4x2 - 4x +1 = 41^ - 3
,2 . . -t^Yoemertp.
Vi x'^ - X +1 = l Y 3^3 . ,^>/3
X — + - > - nen t > — , r Phuong trinh (1) tro thanh 4t2 - 3 + m = t o -4^^ +1 + 3 = m (1') Xet ham so y = -41^ +1 + 3 voi t > ^ . Ta c6 = 7 < "V"
2 2a o 2 Bang bien thien ^ [j
73 +00
-12 + 73
-00
Phuong trinh (1) c6 nghiem <=> phuang trinh (1') c6 nghi?m t > — 73
<^ do thi ham so y = -41^ + t - 3 tren -12 + 73
73 -;+oo cat duong t h i n g
y = m o m < •
Vay phirang trinh (1) c6 nghiem khi va chi khi m < -12 + x/3
D a t t = 4 ^ = 4 / l - 2 b) D K X D : x > 1 .
Chia ca hai ve'cho Vx + 1 , ta c6 '1 .0 £, B - d = I + x - " / V ,f. •
Vx + 1 N/X + 1 V X + 1 V X + 1 r l n i w j . ; ( i
d = £;
1 3 r / x + 1 'V x + 1
Phuong trinh (2) tro thanh - 3 t ^ + 2t = m (2')
Xet ham so y = - 3 t ^ + 2t tren [ 0 ; l ) , t a c 6 - — = - , y''^'^
2a 3
I
3 Bang bien thien
Phuong trinh (2) c6 nghiem <=> p h u o n g trinh (2') c6 nghiem t e [0;1)
<=> do thj ham so y = - 3 t ^ + 2t tren [0;1) cat d u o n g th^ng y = m < = > - l < m < i
ft) r n -
... ' i .
Vay p h u o n g trinh (2) c6 nghi|m khi va chi k h i - 1 < m < -1 3
hm y: K h i giai bai toan bSng each dat an phu, doi v o i loai toan khong chiia tham so t h i c6 the khong neu dieu ki§n(hoac dieu kien "long") ciia an phu \ sau k h i t i m duqyc nghiem an phu roi chiing ta phai thay lai de giai. Nhung voi bai toan chua tham so thi chiing ta can phdi neu dieu kien "chat" doi voi an phu.
Xgai 4; D a t a n p h u k h o n g h o a n toan V i 10: Giai p h u o n g trinh 3^fx + 3 = 3x^ + 4x - 1
Lai giai
D K X D : x > - 3 ' ^ ' > — . f e l i d l p b Phuong trinh <=> - 2 7 ( x + 3) - 3Vx + 3 + 3x^ + 31x + 80 = 0
Dat t = Vx + 3 ( t > 0 ) p h u o n g trinh tro thanh: -27t^-3t + 3x^+31x+80 = 0
C6 A . = (18x + 93)^ suy ra t , = = ^ ^ , t , = ^
_ -^x^ 16 nghiem vi vol x > - 3 thi '^^^ < 0
/ 7 r 3 = l l ^ < = > x ^ + x - 2 = 0 ằ x = l hoac x = - 2 , r • . P
• 3
Vay phu'ong trinh ban dau c6 hai nghiem x = 1 va x = - 2
j^an xet: Trong loi giai tren ta thay kho nha't la bien doi p h u o n g trinh ban j i u thanh - 2 7 ( x + 3 ) - 3 7 x + 3 + 3x^ +31x + 80 = 0de sau khi dat an phu t = thi p h u o n g trinh an t c6 A = (18x + 93)^ (la binh p h u o n g ciia mot nhithuc) - i ;
Neu ta tach khong hop ly thi A khong la binh p h u o n g ciia mot nhj thuc hoac la mot hang so, trong truong hop do viec giai p h u o n g trinh theo huong tren la khong the thuc hien dugc.
Vay lam thenao de tach dugc phuong trinh ma thoa man cac dieu kien tren va vi?c tach ra n h u theco la duy nha't?. De tra loi dugc cau hoi nay ta thuc hi?n theo cac buoc n h u sau: r , , i >-\t .
Bl: Viet (1) o m (x + 3) - 3Vx + 3 + 3x^ + (4 - m ) x - 1 - 3m = 0 ( m ^ O ) B2: Dat t = N/X + 3 (t > 0) pt tro thanh
mt^ - 3 t + 3x^ + ( 4 - m ) x - l - 3 m = 0 ^_ j ^ ^.^^^
Co A, = - 1 2 m x ^ - 4 m ( 4 - m ) x + 12m^ + 4 m + 9 = f ( x ) B3: T i m m sao cho
I < y ;
-12m > 0
, , \m = -27
A' = 4 m ( m + 27)[m^ + m + I j = 0 ho tr.i; ' - 1 2 m > 0
Den day viec giai pt n h u da trinh bay 6 tren.
- j ^ d y . 11; Giai p h u o n g trinh V 6 0 - 2 4 x - 5 x ^ = x ^ + 5 x - 1 0 Lai giai
DKXD: 60 - 24x - Sx^ > 0
^at t = V 6 0 - 2 4 x - 5 x ^ (t > 0) pt tro thanh L2 t ^ + t - i x ^ - x = 0 o t ^ + 6 t - x ^ - 6x = 0 6 6
Phuong trinh an t nay co A' = (x + 3)^ nen ta t i m d u g c t j = x,t2 = - x - 6 f x > 0
• V 6 0 ^ 2 4 x - 5 x ^ =x<=>
+ 4 x - 1 0 = 0 ô x = - 2 + V l 4
153
• V60-24x-5x2 _ x _ 6 <=> - x - 6 > 0
2 ô X = - 3 - TTI x'' + 6 x - 4 = 0
Vay p t ban dau c6 hai nghiem X j = - 2 - \flA, X j = - 3 - Vl3 m 2. BAI TAP LUYEN TAP - -i i > y i: ;x t - Bai 3.30: Giai cac p h u a n g trinh sau
a) V2xTT = 3x + 1 fj !:=ri?r ô:ff b) N/TTSX + I = V x ' ^ - x ^ - l c) ^i2x + ^yi
a) P t o
ex-^ +1 =x + i
3x + 1 > 0 2x + l = (3x + l)2
d) x^ + = 7 - /i,.r.
Huang dan gidi ., i f ) .
<=> <
b) P T o
c) Pt<ằ>
1 x >
3 9x^ +4x = 0
\x > —
X = 0, X = —
9
" x = 0 4 9
<=>
- 1 <=> X = - 2 . x^ + 3x + 2 = 0
x > - ]
\/6x^+l = x ^ + l x > - 1
, <=>x = 0,x = 2 . 4x2 = 0
x' ' - x 2- l > 0
<^+3x + l = x'*-x2 x + 1 > 0
2x + N /6 X^TI = (x + 1
x > - l
d) -(x+7)+(x+v >rr7 )=oo(x+ v>rr7)(x-V )rr7 +i)=o
T u do p h u a n g trinh da cho c6 hai nghiem x = 2; x = - — . ., Bai 3.31: Giai cac p h u a n g trinh sau:
a) slx^ +12 + 5 = 3X + N/X2 +5 b) 3 ^ + sjx^ + 8 - 2 = 7x2+15
• c) V 5 x- 1 + ^ ' 9 ^ = 2 x 2+ 3 x - l d) ^ / ^ + x2 = 7- V x ^ -i -- Huang dan gidi
a) D K X D : x > -
3 ' <
Phuang trinh da cho t u a n g d u a n g v o i : N/X2+12-4-3X-6 + N/X2+5-3
x 2 - 4
7 x^ + 1 2 + 4 = 3 (X- 2 ) + x 2- 4
ô ( x - 2 ) x + 2
7X2+5+3 x + 2
, 7 x 2+ 1 2 + 4 7X2+5 + 3 , = 0
rv = 2
x + 2 x + 2 J^^7u+4 7X2+5 + 3
1 1 x + 2
- 3 = o n . f •• ^--M ;
x + 2
X2TI2T4"^7X2+5 + 3^7X2+12 + 4 7X2+5+3 <0 nen pt (*) v6 nghiem V?y phu'ang trinh da cho c6 nghiem d u y nhat x = 2.
^ Xa du" doan dugc nghiem x = ±1, va ta viet lai p h u a n g t r i n h n h u sau:
pX<r:>3(\'x2 - ] j + |7x2 + 8 - 3 J = (7x2+15-4J H:) fib ftf 1!T• • > '•
— r = = _ I •
x 2- l x 2- l
^ + 7 ? + l 7 x 2+ 8 + 3 7X2+15 + 4 x 2= l
_ ^ + \ / x 2+ l 7 x 2 7 8 + 3 7x2+15 + 4 Mat khac, ta c6: 7 , r 7x2 +15 > 7x2 + 8 =^7x2 +15 + 4 > 7x2 + 8 + 3:
Nen p h u a n g trinh thiic hai v6 nghiem.
7x2+15 + 4 7 x 2+ 8 + 3
Vay pt CO 2 nghiem x = l , x = - l . x + ^icV - £ :m ui S. ; c) DKXD: X > - .
Phuong trinh da cho t u a n g d u a n g v o i : 5 N/5XM" - 2 + ^ / 9 ^ - 2 = 2x2 + 3x - 5
J ( x - l )
75x - 1 + 2 " /3,
1 - x
( ^ ^ ^ ) + 2 ^ / 9 ^ + 4
= ( x - l ) ( 2 x + 5)
^ ( x - 1 )
^ ( x - l )
2x + 5 -
V 5 x - 1 + 2 /3/qr = 0
^ 5V5x - 1 + 5
2x + , — = + 1
*nv;;
0^
7 5 ^ + 2 (^/9^)^ + 2 ^ / 9 ^ + 4
= 0
p h u a n g trinh da cho c6 m o t nghiem d u y nhat x = 1.
' ^ ^ K X D : x . f
^ < ằ ^ / 7 + 6 + V ^ + x2 - 7 = 0
1W
ô ( ^ x + 6 - 2 ) + (N/ ^ - 1 ) + (X2 - 4 ) = 0 (1)
Ta CO Vx > 1 : ^{x + bf + 2^x + 6 + 4 = (^x + 6 + 1)^ + 3 > 0 & ^ ^ ^ ^ " + 1 > o x - 2 x - 2
Do do PT ô
J , . , , . ^(x + 6)^ + 2 ^ x + 6 + 4 + 1
1 1 + ( x - 2 ) ( x + 2) = 0
^(x + 6)2 + 2 ^ / ^ + 4 + 1 Vay p h u a n g trinh da cho c6 nghiem duy nhat x = 2 . Bai 3.32: Giai cac p h u o n g trinh sau , ..
+ X + 2 = Oci> X =2
X + 2 = X + X b) ( 2 x - ] ) ^ = V x 2 - x + l c) 13x + 2(3x + 2)N/X + 3 + 42 = 0
e) x"" + 2x, X - - = 3x + l
d) x^ - 2x - 22 - V-x^ +2X + 24 =
1
0 +^[^^^~^ = 2x + 'l
0
' Huong dau gidi a) Dat t = Vx^ + X + 2, ( t > 0) x^ + x = - 2
X V -r
:0J £j ,06f.
t = - l ( l ) t = 2 Phuong trinh tro thanh: t = t ^ - 2 c : > t ^ - t - 2 = 0<=>
V o l t = 2 taco; 2 = V x ^ + x + 2, ( t > 0 ) o x ^ + x - 2 = 0<=>
b) Dat t = Vx2 - x + 1, (t>0) r:>X^-X = t ^ - l
x = l x = - 2
1 1 , , r , j .
t = l
>;: Phuong trinh tro thanh: 4 ( t ^ - l j + l = t < = > 4 t ^ - t - 3 = 0<;=>
, T u do p h u a n g trinh c6 nghiem la x = 0, x = 1
c) Dieu kien x + 3 > 0 <=> x > - 3 . Dat t Jx + 3, t>0i=>x = t ^ - 3 Luc do p h u a n g trinh da cho tra thanh: •-•' + xS
13(t2 - 3) + 2 [3(1^ - 3) + 2] t + 42 = 0 o 6t'V 13t^ - 1 4 1 + 3 - 0 , = 2
4
<:^(t + 3)(6t^ - 5 t + l ) = 0 o 6 t 2 - 5 t + l = 0 , ( t > 0 ) ô
-p, 11 26
T u d o x = ; x = .
4 9
. 1 2
, = 1 ' 3 .'1 '.ijtovfi'
I < X :^
156
CtifTNHH Ml \ ' } i Khattg Viet
t = / - x ^ + 2 x + 24, ( t > 0) => -x^ + 2x + 24 = t^ => x^ - 2x - 22 = 2 - 1 ^ r t = l
phuong trinh tro thanh: 2 - t ^ - t = 0 c i . t ^ + t - 2 = 0c:> t = -2(l)
e)
t = 1 ta c6: 7-x^ + 2 x + 24 = l ằ x ^ - 2 x - 2 3 = 0<=>x = l ± 2N/6 pi^u kien: - 1 < x < 0
Chia ca hai ve'cho x ta nhan dugc: x + 2Jx - — = 3 + —
Dat t = X - - , ta duac t^ + 2t - 3 = 0 <=>
X X t = l
t = -3 •
x = 0 khong phai la nghiem. Chia ca hai ve'cho x ta dugc: n J 1
Dat t = 3/ x - l , Taco : t^ + t - 2 = 0 c:> t = 1 ^ x = *
X — + 3 x — = 2 X/ V X
X 2 Bai 3.33: Giai cac p h u a n g trinh sau -r^f;
a) 4x^+22 +V 3 x- 2 =21x b) x ( l - SVxTs) = 3(x2 - 4)
c) 51N/X^ = 3X^-58X + 110 d) x^ + xV3x - 1 + 2 = 6x Humig dan gidi , 4 f a) P T o - ( 3 x - 2 ) +V 3 x- 2 + 4 x ^ - 1 8 x + 20 = 0 ' £ - . f / ^ v f i :
Dat t = V 3 x- 2 , t > 0 . — '
Phuong trinh tra thanh - t ^ + t + 4x^ - 18x + 20 = 0, c6 A, = ( 4 x - 9 ) ^
T . Vi i r a o ta co nghiem p h u a n g trinh la x = , x = ^ A f - u - u ur- 19 + 773 23-x/97 ' ' 8 8 ; if,i!..,)
P T o 2 ( x + 3) + 5XN/X+^ + 3x2 - 3 x - 1 8 - 0
D§t t = V^TTs, t > 0. :,
Phirong trinh t r a thanh li^ + 5xt + 3x2 - 3x - 1 8 = 0 ' * i
^ ° A, = (x +12)2 . T u do ta CO nghiem p h u a n g trinh la x = 1, x = - i A i ^ - l ^
^) P T C: > - 2 7 (X- 2 ) - 5 1N / X^ + 3X2-31x + 56 = 0 E>^tt = V ^ , t > 0 .
Phuong trinh tra thanh: -27t2 - 51t + 3x2 - 31x + 56 = 0 A, = ( l 8 x - 9 3) 2
TCrrfA. - u . u v 25 + 3733 41-3793
" ao ta CO nghiem p h u a n g trinh la x = , x =
2 6
Phdn lo^i va phuong phdp gidi Dai so 10
d) P T < ằ - 2 ( 3 x - l ) + x V 3 x - l = 0 ằ (x - 73x - l)(2V3x-1 + x) = 0 . , , , ^ 3 + V5
Tu do ta C O nghiem phuong trinh la X = —-— . v^,. , Bai 3.34: Giai phuong trinh x + Vx^ - 9 = 'I'' . s ' , .
Huang dan gidi J ' 1
< , (a^+b^=2x
• Vol X > 3: Dat a = vx + 3; b = N/x-3;a>0,b>0=><
a 2 - b 2= 6
Phuong trinh tro thanh:
2 1 2 ^ 1 4a , - a a + b + 2ab = — r - o a + b = 2 - ^
b^ b^
o ( a + b ) ( a - b ) = ^ i i ^ ô 6 b 2 = 2 a ( a - b )
ô a 2- a b - 3 b 2 = 0 ô a = l ^ b 2
f ^ " 1 + V T 3 ^ •''
Do a > 0,b > 0 =:> a = b . 2 h
Suy ra Vx + 3 = "* "^^^ Vx - 3 ox = 8- ^/\3 (thoa man).
• Voi X < -3 tuong tu ta c6 phuong trinh v6 nghiem.
• Voi -3 < X < 3 khi do phuong trinh khong xac djnh nen no v6 nghiem.
Vay phuong trinh c6 nghiem la x = 8 - >/T3 .
x^ + 2x2-3x + l Bai 3.35: Giai phuong trinh Vx^ - x + I = •
x2+2 Huong dan gidi Xet phuong trinh Vx^ - x + 1 = -x - 2
x<-2
X — ^
7 2
X - x + l = x + 4 x + 4
3 (v6 nghiem)
X — 5
r - ... ...,.;m --f5^x)^£-
Suy ra \ / x - x + l + x + 2;^0 do do .—- J,;
Phuong trinh o V x ^ - x + 1 - (x + 2) - x^+2x^-3x + l _ + 2)
X + 2 " . - 5 x - 3 - 5 x - 3 r 5x + 3 = 0
• <=> I
^/x^-x + 1+ x + 2 = x^+2 C'ii- V x ^ - x + l + x + 2 X +2
^ ' - ' ' ^ O l2 o X = ± V 3 7 2 ; i
X ' - X +1 = (x^ - xj
Suy phuong trinh c6 nghiem la x € • 3 I- V 3 + 2V5 1 + V3 4- 2V5
•5' 2 ' 2
DANG TOAN 4: PHUONG TRINH BAC CAO
i b a i 7. D u a ve phuong trinh tich. | 1. Phtfcmg phap giai ^ ; f | ,
Degiai phuong trinh f(x) = 0 ta phan ti'ch f (x) = fj (x).f2 (x)...fn (x) khi do fi(x) = 0
f(x) = 0 ô f 2 ( x ) = 0
De dua ve mot phuong trinh tich ta thuong dung cac each sau: j ,|- J-K ,
• Su dung cac hang dang thuc dua ve dang a^ - b^ = 0, a^ - b'' = 0,...
• Nha'm nghiem roi chia da thuc: Ne'u x = a la mot nghiem ciia phuong trinh f(x) = 0 thi ta luon c6 su phan thich: f(x) = (x - a)g(x).
De du doan nghiem ta chu y cac ket qua sau:
Cho :bthuc f(x) = a n x " + a „ _ ] x " ' ' + . . . + a i X + ao ' " " "
^ Ne'u phuong trinh f(x) = 0 c6 nghiem nguyen thi nghiem do phai la uoc cua ao.
' ^eu da thuc C O tong cac h^ so bang khong thi phuong trinh f(x) = 0 c6 mot
"ghiembSngl.
da thuc C O tong cac h§ so bac ch^n bang tong cac hf so bac le thi phuong trinh f(x) = 0 C O mot nghiem bang-1.
phan tich f(x) ta su dung luge do Hooc-ne nhu sau: <• • '^^'y f(x) C O nghiem l a x = XQ thi f(x) chua nhan tu ( x - X Q) tuc la:
^('^) = ( X - xo).g(x), trong do g(x) = b^.jx""' + b„ _ 2x" " ' + ... + biX + bo
Vdi h f so' b| dugc xac djnh nhu sau:
Luge do Hoocne
- 1 ^0
a b„_2 = a . a „ + a n _ i b j =a.a2 + 3 ] 0
V i dy: Giai phuong trinh x'* + x'' - x - 1 = 0
Nhan thay : 84 + 8 3 +a2 +a, + a,, = 1 + 1 + 0 + (-1) + (-1) = 0 'J
Va : 3 4 + 8 2 + a() = 1 + 0 + (-1) = 83 + 3 ] = 1 + (-1)
Suy r3 phuong trinh c6 hai nghiem x^ = 1, X2 = - 1 Luge do Hoocne
1 1 0 - 1 - 1
x = 1 1 2 2 1 0
x = - l 1 1 1 0
Ta CO phuong tinh thuong duong vol (x - l ) ( x + l ) | x ^ + x + l j = 0c5'X = ±l
• Su dung phuong phap h^ so bat djnh ^