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NGUYEN TAI
CHUNG
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9
PHO
fI DG TRiHH
HE PHIfOnC TRn
iH
BAT PHIDnG TR
P H l J d N G PHAP XAY DL/NG B E
CAC
TDAN
DANG TCAN, CAC PHLfdNG PHAP
GIAI
C A C O E T H I H O C S I N H G I C I GJUCC E I A , O L Y M P I C
3 0 / 4
T A I LIEU B D I DL/QNG H O C S I N H K H A GICI
fx T A I L I E U O N L U Y E N T H I B A I
HOC
T A I LIEU T H A M K H A C C H D GIAO VIEN
N H A X U A T e X N T r a N G H d P T H A N H P H D HO CHI M I N H
LMnoidau
SANG T A O V A GIAI P H U O N G T R I N H ,
HE
PHLfdNG T R I N H , B A T PHaONG T R I N H
HQC sinh hoc toan xong roi lam cac bai tap. Vay cac bai tap do 6 dau ma ra?
Ai la nguai dau tien nghi ra cac bai tap do? Nghl nhu the nao? Ngay ca nhieu
N G U Y I N TAI C H U N G
Chiu trach nhiem xuS't ban
giao vien cung chi biet suti tarn cac bai tap c6 trong sach giao khoa, sach tham
4
;
N G U Y E N THI THANH HlJdNG
Bien tap
: QUOC NHAN
Si^abanin
. : HOANG NHlTX
Trinh bay
: C6ng ty K H A N G V I E T
Bia
: C6ng ty K H A N G V I E T
khao khac nhau, chua biet sang tac ra cac de bai tap. Mpt trong nhimg each do
la tim nhirng hinh thiic khac nhau de dien ta ciing mpt npi dung roi lay mpt
hinh thiic nao do phii hop vai trinh dp hpc sinh va yeu cau hp chiing minh
tinh diing dan ciia no.
' •
Nhu chiing ta da biet phuong trinh, h$ phuong trinh c6 rat nhieu dang va
phuong phap giai khac nhau va rat thuong gap trong cac ky thi gioi toan ciing
nhu cac ky thi tuyen sinh Dai hpc. Nguoi giao vien ngoai nam dupe cac dang
phuong trinh va each giai chiing de huong dan hpc sinh can phai biet each xay
NHA XUAT BAN TONG H0P TP. HO CHf MINH
NHA SACH TONG HOP
62 Nguyen ThI Minh Khai, Q . l
D T : 38225340 - 38296764 - 38247225
Fax: 84.8.38222726
Email:
[email protected]
Website: www.nxbhcm.com.vn/ www.fiditour.com
Tong phdt hanh
dung nen cac de toan de lam tai li^u cho vi|c giang day. Tai lifu nay dua ra
mpt so phuong phap sang tac, quy trinh xay dimg nen cac phuong trinh, he
phuong trinh. Qua cac phuong phap sang tac nay ta ciing rut ra dupe cac
phuong phap giai tu nhien cho cac dang phuong trinh, hf phuong trinh tuong
ling. Cac quy trinh xay dyng de toan dupe tnnh bay thong qua nhiing vi du,
cac bai toan dupe xay dung len dupe dat ngay sau cac vi du do. Da so cac bai
toan dupe xay dung deu c6 loi giai hoac huong dan. Quan trpng hon niia la
mpt so luu y sau loi giai se giiip chiing ta giai thich dupe "vi sao lai nghl ra loi
giai nay".
Nhu vay cuon sach nay se trinh bay song song hai van de: Phuong phap
CONG T Y TNHH MTV
DjCH V g VAN HOA KHANG V I E T
( ^ D i a chi- 71 Oinh Tien Hoang - P.Da Kao - Q.1 - T P . H C M
Dien thoai:'08. 39115694 - 39105797 - 39111969 - 39111968
Fax: 08. 3911 0880
Email: khangvietbookstore ©yahoo.com.vn
1
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So DKKHXB: 1 55-1 3/CXB/45-24ArHTPHCM ngay 31/01/201 3.
Quyet dinh xuat ban so: 296/QD-THTPHCM-2013 do NXB Tong Hop
Thanh Pho H 6 Chi Minh cap ngay 19/03/2013
In xong va nop luU chieu Quy II nam 201 3
sang tac eae de toan va Cac phuong phap giai ciing nhu phan loai cac dang
toan ve phuong trinh, hf phuong trinh. Diem moi la va khac bi?t ciia cuon
sach nay la quy trinh sang tac mpt de toan moi (dupe trinh bay thong qua cac
vi du) va each thiie chiing ta suy nghl, tim ra loi giai mpt bai toan (dupe trinh
bay thong qua eae luu y, chii y, nhan xet ngay sau loi giai cac bai toan). Ngoai
ra cuon sach nay con danh ra mpt ehuong (ehuong 5) de trinh bai cac bai toan
phuong trinh, he phuong trinh, bat phuong trinh trong cac de thi Dai hpc
trong nhiing nam gan day.
Tot nhat, doe gia tu minh giai cac bai toan eo trong sach nay. Tuy nhien, de
thay va lam chii eae ky xao tinh vi khac, cac bai toan deu dupe giai san (tham
chi la nhieu each giai) voi nhiing miie dp chi tiet khac nhau. Npi dung sach da
c6' gang tuan theo y chii dao xuyen suo't: Biet dupe loi giai ciia bai toan chi la
yeu cau dau tien - ma hon the - lam the nao de giai dupe no, each ta xir ly no,
nhiing suy lu^n nao to ra "c6 ly", cac ket lu^n, nhan xet va luu y tir bai toan
dua ra...
Hy vong cuon sach nay la tai li§u tham khao c6 ich cho cac em hpc sinh kha
gioi, hoc sinh cac lop chuyen toan Trung hpc pho thong, cac em hpc sinh dang
luypn thi Dai hpc, giao vien toan, sinh vien toan cua cac tmong DHSP,
D H K H T N cung nhu la tai phyc v\ cho cac ky thi tuyen sinh D ^ i hpc, thi hpc
sinh gioi toan THPT, thi Olympic 30/04.
Cac ban hpc sinh, sinh vien, giao vien va nhirng nguoi quan tarn khac se c6
the a m tha'y thieu sot a cuon sach nay trong qua trinh su dung. Do vay, su gop
y va chi trich tren tinh than khoa hpc va huang thien t u phia cac ban la dieu
chiing toi luon mong dpi. H y vpng rang tren buoc duang tim toi , sang tao
toan hpc, ban dpc se tim dupe nhiing y tuong tot hon, mai hon, nham bo sung
cho cac y tuong sang tao va loi giai dupe trinh bay trong quyen sach nay.
ChiMng 1. PhUcfng phdp sang tdc va giai phUcfng trinh, hf phUOng trinh, bd
1.1
PhiTdng phdp he so bat djnh
3
1.2
Phifdng phdp duTa ve h$
5
1.3
Phufdng phap diTa phiTdng trinh ve phifdng trinh ham
15
1.4
Mot so phep dSt an phu cd ban khi giai h$ phufdng trinh
26
1.5
PhiTdng phdp cpng, phufdng phdp the
35
1.6
PhiTdng phdp dao an. Phifdng phap hiing so bien thien
54
1.7
PhiTdng phdp sijf dung dinh l i Lagrange
63
Nha sach Khang Viet xin trdn trgng gi&i thi?u tai Quy dgc gia va xin
1.8
Phu'dng phdp hinh hpc
69
idng nghe moi y kieh dong gop, de cuon sach ngdy cang hay hem, bo ich hon.
1.9
PhiTdng phap ba't dang thtfc
82
1.10
PhiTdng phap tham bien
95
Tac gia
Thac sy: NGUYEN T A I CHUNG
Thuxingici ve:
Cty T N H H M p t Thanh Vien - Dich V u Van Hoa Khang Vi?t.
71, D i n h Tien Hoang, P. Dakao. Quan 1, TP. H C M
ChUcfng 2. PhUcfng phdp da thiic va phUcfng trinh phdn thitc hOu ti.
11
Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880
2.1
Cdc dong nha't thiJc bo sung
116
Hoac Email:
2.2
PhiTdng trinh bac ba
117
2.3
Phu'dng trinh bac bon
127
2.4
PhiTdng phdp sdng tdc cdc phiTdng trinh da thiJc bac cac
137
2.5
PhiTdng trinh phan thiJc hi?u ti
149
[email protected]
ChUcfng 3. PhUcfng trinh, bdtphUcfng trinh chiia can thiic
158
3.1
Phep the trong doi vdi phiTdng trinh 3/A(X) ± }JB{X) = 3^C(x) .... 158
3.2
PhiTdng trinh (ax + b)" = pJ^a'x + b' + qx + r
160
3.3
PhiTdng trinh [ f ( x ) ] " + b ( x ) = a(x)!i/a(x).f ( x ) - b ( x )
168
3.4
PhiTdng trinh d i n g cap d6'i vdi ^ P ( x ) v^ ^ Q ( x )
174
3.5
Phu'dng trinh doi xiJng d6'i vdi ^ P ( x ) vd ^ Q ( x )
179
3.6
Mpt so hiTdng sdng tac phiTdng trinh v6 ti
184
ChiMng 4. //# phUcmg trinh, h? bat phUOng trinh
231
4.1
He phiTdng trinh doi xuTng
231
4.2
He c6 yeu to d^ng cap
253
4.3
H$ bac hai tdng qu^t
266
4.4
Phi/dng phdp dilng tinh ddn dieu cua ham so'
271
4.5
He lap ba an (hodn vi vong quanh)
277
4.6
SuT dung can bac n cua so phuTc de sang tac va giai he phiTdng trinh ...
4.7
Phi/dng phap bien doi ding thiJc
314
4.8
MotsohekhongmaumiTc
317
307
ChUctng 5. Cdc bai todn phUcmg trinh, h^ phUcfng trinh, bat phUcfng trinh trong
dethidt^ihQc
Chi:fc?ng 1
Phi:fdng phap sang tac va giai
phifcfng trinh, he phi:fcfng
trinh, bat phi:^dng trinh
328
5.1
Phtfdng trinh, bat phi/dng trinh chiJa can
328
5.2
He phiTdng trinh dai so
332
5.3
PhiTdng trinh liTdng gidc
337
5.4
Phi/dng trinh, bat phi/dng trmh c6 chlJa cdc so n!,Pn, A^, C\5
5.5
PhiTdng trinh, ba't phiTdng trmh mu
5.6
PhuTdng trinh, ba't phifdng trinh logarit
373
5.7
H? mu va logarit
387
5.8
Phtfdng phap dilng dap ham
392
368
Trong chitcJng nay ta se trinh bay nipt so phUdiig phap cO ban va mot so
phUdng phap dac biet di giai va sang tac phitdug tiinh, lie phUdng trinh,
bat phUdng trinh. Co mot so vi du, bai toan c6 sii dung den kien thi'tc cua
plntdug trinh da thi'tc bac ba, ban doc c6 the xcni bai i)liitdiig tiiiih bac ba d
chUdng 2 (chi can c6 kign thv'tc ve lUdng giac la co the hieu bai phUdng trinh
bac ba) trudc khi xem cac bai toan, vi du nay.
- .
, •
1.1
PhifcTng phap he so bat dinh
PhUdng ])hap he so bat djnh la chia khoa giup ta i)han tich, tim dudc l a i giai
cho nhieu locii phUdiig trinh. Chung ta se Ian lUdt tini hieu phUdng phap nay
thong qua cac bai toan va cac km y ngay sau do.
Bai toan 1. Giai phiCdng trinh
2^^ - l l x + 21 - 3^4.x- - 4 = 0.
Giai. Tap xac dinh D = E. Plntdug trinh da cho tu'diig ditdng vdi
•^ikf
•'A/nh 6"
^ ( 4 x - 4 ) 2 - I ( 4 x - 4 ) + 1 2 - 3 x / 4 : r : ^ = 0.
•
^,
(1)
Dat t = ^4x - 4, thay vao (1) ta dUdc f - Ut^ - 2-it + 96 = 0, hay
(f - 2)2(t'' + 4i^ + 12/2 ^
^ 24) = 0.
(2)
Neu t < 0 thi f' - Ut^ - 24f. + 9G > 0, neu t > 0 t hi
+ 4r* + 12*2 + 18( + 24 > 0.
3
wid ;:,v»fb
im V >.
D o do (2) <=> i = 2 => X = 3.
L U L U y . De c6 (1) ta can t i i i i a, (3,7 sao cho
B a i t o a n 3 . Gidi phuang trinh
2x2 - l l x + 21 = IGrtx^ + (4/^ - 32rv);r + (16^ - 4 / ^ + 7)
f 16a = 2
^ {
4/3-32a = -11
I 1 6 a - 4 / 3 + 7 = 21
,
,
fl
^{a;(i\-i)
=
7
Dap
so.
X = - - — - .
J
\
B a i t o a n 4.
dung phuong phap he so bat dinh cho t a 15i giai bai toan m o t each rat tijt
nhien va ro rang.
Gidi phuang trinh
4 + 2\/l - x = - 3 x
D a p so. PhUdng t r i n h c6 tap nghiem S — I 0; — ; —
25
B a i t o a n 2. Giai phiCcfng trinh
- x + 3 = 2\/r^-/m^+3\/r^.
(i)
Dat u = ^/^+x,V
2v^r^ + \/l +
= Vl - X {u >0,v>0),
-
+ 1) = 0 ^
3\/l-
x2 = 0.
10x2 + 3 x + 1
(2)
Thay vac (*) :
• Vdi w -
1 ^ 0
•
^/3^
[ « Z
I . tah s v , ;:
i;
j
lOx^ + 3 x + 1 = ( 6 x + l ) V x 2 + 3.
= i(6x
(*)
+ 1)2 + (x^ + 3) - - = — + ^2 _ 9
4
t a duoc
{u^ - 2uv) + [u - 2v) - {uv - 2^^) = 0
+ 2^2 - 2u + It - 3 u v = 0
^ ( • a - 2v){u
.T -
B a i t o a n 5. Gidi phuang trinh
2
+ SVxTT + V l - x^.
G i a i . Dat u = 6 x + 1, ?; = \/.x2 + 3. Ta c6
G i a i . Tap xac d i n h D = [ - 1 ; 1]. PhUdng t r i n h (1) viet lai n h u sau :
(1 + x ) + 2(1 - x) -
5s/TTx.
V3
X
- l l x + 21 = a(4x - 4)^ +/3(4a; - 4) + 7
4 \ / l - x = x + 6 - 3 \ / l - x^ +
2D =
4
4
4
5,
+ 1,2 _ 5 = ^ i K : ^ (u - 2t;)2 = 9 <^ u - 2u = ± 3 .
3, 1ta+ C6Ox - 2v/x2 + 3 = 3 < ^ 3 x - l = \ / x 2 + 3 v
3x - 1 > 0
^
,
x2 + 3 = ( 3 x - l ) 2
^x^l.
-'^'^i'l'S
• Vdi u - 2t; = - 3 , t a c6
L u ^ i y. D l
CO
( 2 ) , t a t i m a, f3 sao cho
- x + 3 = a ( l + x) + / 3 ( l - x ) o {
^ ; ^ ^ 3 ^
^ {
g =
i
Vay p h i M n g t r i n h c6 tap nghiem 5 = < 1; ——^
Doi v 6 i bai toan t o n g quat : Giai phvfdng t r i n h
p{x) = as/1
Ta bieu dien p{x)
-
X +
bVl +
X
+ cVl
-
x^,
X
(u > 0, i; > 0 ) .
K h i do dirdc phitdng t r i n h doi v d i u, v c6 thg phan t i c h ditdc.
V i d u 1.
,
Lvfu y. Phudng phap he so b a t dinh de giai he phudng t r i n h se ditdc de cap
trong phan phan t i c h t i m Idi giai cac bai toan c i i a bai 1.5 : PhUdng phap
cong, phiTdng phap the (d trang 35).
theo 1 - x , 1 + x va dat
u = - / I + X, i; = \ / l -
I.
1.2
Phifcfng phap difa ve he.
Ta sc. sang tdc mM phUdng trinh duclc gidi hhng phiMng phdp he
Dg giai phUdng t r i n h b a n g each dua ve he phUdng t r i n h t a thutdng dat an
so bat dinh nhu sau : Ta c6
{a-b+
l ) ( 2 a - 6 + 3) = 0 ^
20^ + 6^ - 3a6 + 5 a - 46 + 3 = 0.
Tii day lay a — ^Jl + x vd b = \/l - x ta diidc
2x + 2 + 1 - X - 3 \ / l - x2 + 5VI+X
Rut gon ta duac bai toan
sau.
p h u , p h e p dat an p h u n a y c i m g vdi phUdng t r i n h trong gia thiet c h o t a m p t h$
phitdng t r i n h . Sau day t a se t r i n h bay phuldng p h a p s a n g t a c (thong q u a cac
V I d u ) , phUdng p h a p giai (thong q u a Idi giai c a c b a i toan va q u a n trong h d n
- 4s/l-x
+ 3 = 0.
n i i a la cac hru y s a u Idi giai). Cac p h u d n g p h a p s a n g t a c c i i n g nhiT phifdng
p h a p giai cac phUdng t r i n h b a n g each dUa ve he con dildc de c a p r a t n h i i u
6 s a u b a i n a y ( c h a n g h a n b a i 3.2 d t r a n g
4
5
160).
Lay (1) tru' (2) tlico ve ta du'dc
V i d u 1. Xet I
y ~ 2 ^ 3^^
^ x = 2 - 3 (2 - 3x•'^)^ Ta c6 hdi todn sau.
2(y - x )
B a i t o a n 6. Gidi phUdng trlnh
Giai. D a t , = 2 - 3 x ^ . 1 ^
CO
5(.T2 - y2) ^
x + 3 (2 - 3x^)^ = 2.
^[^ZlZ^.
he
Vdi y = X , thay vao (1) ta dUdc 5x2 _2x
y
=
X
5x + 2
^
-\=i)
1 ± \/6
^ x =
— . thay vao (1) ta du'dc
• Vdi y =
(1) t n r (2) t a fhrac
y - X = 0
2 = - 5 ( x + y)
>;V;!;
o
.•
y = X
x - y = 3{x'^ - y^)
^
X -
y = Q
3(x + y) = l
^
y =
x G |~^'
Vc'ri y = x, thay vao (1) ta diWc Sx^ + x - 2 = 0
Vdi y =
1 - 3x
3
1 - 3x
—^—•
, thay vao (2) ta dudc
1 - 3x
= 2 - 3x^ <^ 9x2 - 3x - 5 = 0
=
-1,
2
X =
1-V21
- ,
3
X =
— — ,
6
1 + V21
()
.
Lxiu y . T i r Idi giai t r c n ta thay iftng neu kliai Irien (2 - 3x'^)'^ t l i i sc dira
phitdng t r i n h da cho vc phiWng t r i n h da thiitc bac bon, sau do Ijien doi thanh
^
= 5x2-l.=.25x2 +
ay + 6 = 5.r2 - 1
8x - 5 («y + bf = - 4
72
1 ± v/O - 1 ±
^
{
+ b + i = hx'\.T + 4 - 5^2 = 5a2y2 +
C fl _
I
_5_ _
,
i{)aby.
^+1
r , _ .
Dg he tren la he doi xftng loai TT t h i < g ~ 5„2 ~ 4 _ 5/^2 => |
Z 9 Vfw
10a/; = 0
'
I « - ta C O phep dat 2y — 5x'^ - 1.
f\
. -jj.,.,
B a i t o a n 8. GidirphtMng trinh 5{5x~ - 17)2 - 343x - 833 = 0.
Y tvtdng. D a t ay + b^5x^-
(x + l ) ( 3 x - 2)(9x2 - 3x - 5) = 0.
Vay ncu k h i sang tac de toan, t a c6 y lam cho plnMng t r i n h khong v6 nghiem
h i i u t i t h i phildng phap khai trien dua ve phUdng t r i n l i bac cao, sau do phan
tich dua ve phu:dng t r i n h tich se gap nhieu klio khan.
l ( , x - l = 0 ^ x = - ^ = ^ ^
25
5
5
L u t i y. Phep dat 2y = 5 x 2 - 1 chrdc t i n i ra n h u sau: Ta dat n:y-\-b = 5x2 _ ^
vdi a, h t h n sau. K h i do t i i u dUdc he
1 ± v/21
X =
l
PhUdng t r i n h da cho c6 bon nghic'm
X
PhUdiig t r i n h da cho co bon nghiem
X
-
17 (a ^ 0). Klii do
.
jay + b = 5.7:2 _ ^7
\ 5 ( a y + 6)2 - 343x - 833 - 0. (*)
Tir (*) ta
,
, ,,
•
' '
CO
5(ay)2 + lOa^y + ^2 - 343x - 833 = 0 ^ x = 5(ay)2 + 10a6y + ^2 - 833
V i d u 2. Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti
5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _
5x2- 1\
2x = 5
B a i t o a n 7. Gidi phUdng trinh
8x - 5 (5x2 _
- 1. Ta CO bdi todn sau.
_ _^
G i a i . Dat 2y = 5x2 _ ^ YAn do
2y = 5 x 2 - 1
. ^ r 2y = 5 x 2 - 1 (1)
8x - 5.42/2 = - 4 ^ \x = 5y2 - 1. (2)
6
„
* ^
5a-^y2 + i0ft2.;;.y-|.^2^j_y33^
Suy ra ax + b =
+ b.
(**)
Ta hy vong c6 ax + h = by^ - 17, ket hdp vdi (**) suy ra
Hi v,
. 2
5 a ' ^ y 2 + l ( ) o 2 . 6 y + 6 2 . a - 833a
,
o/y-l7=
46
•M! I
<^343.5y2
5831 = 5a''.y2 + l()a2.6y + 62,„
f 343 = a'*
Dong nhat he so ta ditcJc I al^h = 0
t--833a+ 3 4 3 6 = - 5 8 3 1
Idi giai sau.
7
-
g 3 3 „ _|. 3435.
r - 7
1 / = (1
"^''^^
'
''"^
COS
7y = 5x2 _ 17
= 5x2 _ 17
1^5,y2 - 343a; - 833 = 0
J7y
7x = 5 y 2 _ 1 7 .
(1)
(2)
x = y
5x + 5y = - 7 .
7±\/389
x =
10
.
Lay (1) t r i t (2) t a c6 7{y - x) = 5(x + y ) ( x - y) ^
* Neu X = •(/, thay vao (1) : Sx^ - 7x - 17 = 0
* N i u 5x + 5j/ = - 7 , ket hdp (1) t a c6
llTT
1
llTT
llTT
= 4 cos-* 18 - 3 cos 18 '
6
137r
137r
137r
COS — —
= 4 cos'' 18 — 3 cos
6
18 •
llTT
137r
TT
la tat ca cac nghiem ciia phuong
r = cos ——, X = C O S
18' •
18 '
18
t r i n h (4) va cung la t a t ca cac nghiem cua phUdng t r i n h da cho.
L t f t i y. Phep dat 6y = 8x3 _ ^ dUdc t u n ra n h u sau : Ta dat
G i a i . D a t 7y = 5x^ + 17, t a c6 h? phitdng t r i n h
ay + 6 = 8x3 - V3
^^^^
^jj^
^
Ket hdp v6i phudng t r i n h da cho c6 he
Ket luan: Phudng t r i n h c6 tap nghiem 5 =
V i d u 3. Ta
CO
Ax^ - 3x = ~
<^ Qx = 8x^
fSx^
{
>1296x + 216v/3 = 8 (Sx^ - ^/fj
Ta
CO hai todn
' 7 ± \/389 - 3 5
50
10
- V^.
- v/3\
ay + 6 = 8x3 - v/3
162x + 27V3 = a3y3 + 3a26y2 + Safety + ^3
± 5 v ^ l
J •
73
8
Can chgn a va 6 sao cho :
162
o?
27\/5 - 63
3a26-3a62 = 0
Vay t a c6 phep dat 6y = 8x3 _
^
Vdy xet
V i d u 4. Xet tarn thiic bdc hai luon nhdn gid tri duang : x'^ + 2. Khi do
-V3
^ 162x + 2 7 ^ = (Sx^ - Vs^
/
.
x^ + 2) dx = — + 2x + C.
^
3
x3
Chi cdn chon C = 0 ta diicfc mot da thiic bdc ba dSng bien la h{x) = — + 2x.
sau.
B a i t o a n 9. Gidi phUOng tnnh
162x + 27\/3 = (8x^ - s/zf
Ta CO /i(3) = 15. Vdy ta thu duoc mot ham so da thiic bdc ba dong bien g{x)
x3
vd thod man g{2>) = Q la g{x) — — + 2x - lb. Ta se tim mot da thiic bdc ba
.
G i a i . Dat 6y = 8x^ - \/3. Ta c6 h ^
r 6?y =
6y - 8x3 - v/3
162x + 27\/3 = 216?y3
dong bien k{x)
8x3 _ ^ 3
\x = 8?y3 - v/3 (2)
o
sao cho k{x) = x <^ g{x) — 0, muSn vdy ta xet
^
+ a x - 15 =
X
0 nen 8 (x^ + xy + y^) + 6 > 0. Do do t i t (3) t a dUdc x =
x3
^
A;(x) = y tuang dudny U(H .• — + 3x - 15 = y <^ x^ + 9x - 45 = 3y. TH phiiang
Thay vao (1) t a dUdc
tnnh cuoi ndy thay x bdi y ta thu duac he doi xilng loai hai
6 i = 8x3 - \/3 <^ 4x3 - 3x = ^
a
x3 + 9x - 45 = 3y
y3 + 9y - 45 = 3x.
4^3 _ 3^, = cos ^ .
Til: he tren, sU dung phep the ta thu duoc phuong
a
Sii di^ng cong thiic cosa = 4cos3 - - 3cos - , t a c6
x3 + 9x - 45
cos-=4cos
1 8 - 3 C O S - ,
8
I
+ 9
trinh
/x3 + 9 x - 4 5 \
9
- 45 = 3x
0fiOJ i.'
<^ ( x ' ' + 9x - 4 5 ) ^ + 81 {x^ + 9x - 45) = 1215 +
Vay ta thu dUdc hai todn sau.
Bai
Gidi
t o a n 10.
phuong
81x.
G i a i . D i c u kiCm
,
hdp v d i phUdng t r i n h d a cho, t a c6 he |
; V
trmh
(x^ + 9 x - 4 5 ) V 8 1
D a t y = l o g n ( l O x + 1), Ivhi d o I P = 1 0 x + 1. K e t
>
JJy ^
L a y (1) t n r (2) tlieo ve t a d i t d c
(x^ + 9 x - 4 5 ) = 1215 + 81X.
(1)
I F
-
ir^ =
^
^ |
"» " • - - w.,
• f-: , A.
l O y - l O x <^
ir^ +
•
lOx = I P + lOy,
(3)
..ji
G i a i . T a p xac d i n h M. D a t
:^i^:,^.^7j
•
•
.
-
•
/
x^ + 9 x - 45 = 3y
(2)
\ + 9 7 / - 45 = 3x. (3)
J
X e t h a m so / ( / ) = 1 1 ' + 10/.. T a c6 f'{f.)
+ 9 x - 45 = 3y. K e t h d p v 6 i (1) t a c6 he
^
= 1 1 ' h i 11 + 1 0 >
(1)
t a d i f d c I L ' ' = 1 0 x + 1 <^ I F -
(4)
X e t h;\ so f / ( x ) = I F -
lOx -
l O x - 1 =0.
.•
1 tren khoang
(
; +00
V
Lay (2) t i i f (3) t h e o v e , t a dUdc
x^ -
g\x)
+ 9 x ^ 9)/ = 3?/ - 3 x ^
. ^ ( x - y)(x^ + xy +
-
x =
Vi
6.
Ta se su dung
trinh
= 0, y ( 0 ) = 0 n e n x = 0 v a x = 1 l a t a t ca
phuang
phdp
doi xvtng loai hai. Xuat
the ta diMc phuang
c6
cac
lap di
phdt
sang
tit \^^Z
tdc phuang
tit. he
•^^ ^^'^'"'(1 P^^'^P
^^^4^30
trmh Ax = ^ 3 0 + | v / x + 30. Til phuong
trinh
nay ta lai,
trinh
thu dtWc he doi xtCng loai liai
do
x^ + 9 x - 45 = ay
a'Uj^ + Slay = 1215 + S i x
^
;
| xj^ + 9 x — 45 = ay_
\ + Slay - 1215 = 8 1 x .
, . ,
«3
81«
1215
81
„
„
Vay can chon a thoa m a n dieu kien — = —
= — 7 - = — => a = J . U o cto
••'
•
1
9
45
a
d a t x^ + 6 x ~ 45 - 3 y , t a so t h u ducJc m o t ho d o i x i ' m g l o a i h a i .
V i d u 5. Chon
mot phMng
phiCOng trinh
trmh
nay
trmh chi c6 hai nghiem
/ d 0 v d 1 IdlV
=
^
-
10
quay
4x
= l o g , , ( l O r + 1)
logiiuux+ij.
Ta CO bdi todn
Bai
ra I F = l O l o g n ( 1 0 x + 1) + 1 ^
- v / a M ^
4x=
-v/^r+30.
^/30+
.Zi
the ta thu diMc phuang
trmh
=
\
sau.
t o a n 12 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 0 ) .
Gidi
phuang
I F = 2 1 o g i i ( 1 0 x + 1)^ + 1. Ta c6 bdi
sau.
t o a n 11.
Gidi
phuang
trinh
IV
= 2 1 o g i i ( l O x + 1)^ + 1.
10
nst,-
lOx+1.
ta thiel lap mot he doi xvCng loai hai, sau do lai
^ I y = l o g n ( l O x + 1)
^ \ l F = 1 0 y + l
4M = ^ / 3 0 +
Tir he niiy, ticp tuc s'li dung phcp
nhu sau :
f l l ' - = lOy + 1
\ i r y = 10x + l
Bai
du
phuang
x^ + 9 x - 45 = ay ( v d i a t i n i s a u ) .
todn
1
^ - j ^ ; + 0 0 j , s u y r a d o t h i cvia
n g h i e m c i i a ( 4 ) . N g h i e m c i i a p h U d n g t r i n h d a cho l a x = 0 v a x = 1.
3.
LuTu y . P h c p d a t x^ + 9 x - 45 = 3y dittfc t u n r a u h u s a u : T a d a t
Suy
J
h a m g v a t r u e h o a n h co v d i n h a u k h o n g q u a h a i d i e m c h u n g , s u y r a (4)
P h i l d n g t i i i i h d a c h o c6 n g h i e m d u y n h a t x = 3.
Tii
. T a c6
10
= l F ( l n l l ) 2 > 0.
/
k h o n g q u a 2 n g h i e m . M a g{l)
x^ + 9 x - 45 = 3 x <=> ( x - 3) (x^ + 3 x + 15) = 0 ^
ve phuang
10, g'\x)
V a y h a m so g c6 d o t h i l u o n 16m t r e n k h o a n g
T h a y vao (2) t a d i W c
Khi
= ll-^nll -
+ 1 2 ( x - ?y) = 0
+ 12) = 0 <^ X = y.
0, V/. G K . V a y h a m
so / d o n g b i e n t r e n E . M a (3) c h i n h l a / ( x ) = / ( y ) n e n x = y. T h a y vao
4x
=
30 + - W 30 + - ^ 3 0 + - \/^T30.
11
trinh
4
G i a i . Do x la n g h i f n i t h i x > 0. Dat u =
30 + --^x + 30, t i t phitdng t n n h
da cho t a c6 ho
4u = J 3 0 +
-y/xT3Q
(1)
4x = A / 3 0 + - v / w + SO.
Gia s\t X
>
u.
(2)
Dat . = \V^FT30, t i t (2) ta c6 he | ^J I
(3)
V.
Nhit vay
dang nay la j)hn'dng t r i n h vo t i , infi san k h i dat an phu dita ve he, r o i dimg
phep the dan t d i phudng t r i n h da thitc, do do k h i sang tac de toan t a phai
dac biet chii y cac chi so can. Chang han d v i d n 7 t h i m = n = 4 nen t a yen
tam rang se dan tdi phitdng t r i n h da thite bac 4 co i t nhat m o t nghiem dep.
B a i t o a n 14. Gtdi phiMng trinh
+ 30 > ^ 3 0 + - \ / u + 30 = 4x =^ u > x =^ x = ?i.
Vay t i t he (1) t a c6 a; = u va 4x = ^ 3 0 + - y x T S O .
Gia sii x>
- JJx) + "\/b + / ( x ) = c, t a co each giai :
Dat u = 'ija - fix), v = '^h + f(x), dan den he { ^ H ^+,n"s'^['^
V i d u 8. Vd'i. x = - 2 thi 2
0. K h i do
5
I"2 —
Z t'0, 7 ^ox
+ Sv'^ = 5(3x - 2) + 3(6 - 5x) = 8.
.
M a t khac t a lai co 2u + 3r - 8 = 0. Vay t a co he
K h i do
4v = Vx + 30 > VtTTSO = 4a:
4u > 4a: =^ V > a; =^ u = X .
,
f T> 0
Vay r = .x va 4.T =
^ | J g p ^ ^ ^, ^ 30
1 + 71921
t r i n h da cho co nghiem d n y nhat x =
32
1 + \/l921
—
. PhUdng
{^t +'fv= 8^ =^
+ 3 ( ^ ^ ) ' = 8 ^ 15..^ + 4^2 - 32z. + 40 = 0
Phu'dng t r i n h nay c6 nghiem d u y nhat u = - 2 nen
v'Sx - 2 = - 2
B a i t o a n 1 5 . Giai phiMng
X-
= -2.
trinh
V i d u 7. Vdi X = 8 thi ^/x-\-8-\- \ J x - l — 3, ia c6 bai todn {ch&c chan co
mot nghiem
1 + \ / l - x 2 [ V ( l + :r)-* - ^ ( 1 - x)'A^ =2+
dep x = 8) sau.
B a i t o a n 1 3 . Giai phUdng trinh
y/x + 8 + \/x - 7 = 3.
G i a i . Dieu kien - 1 < x < 1. D a t ^l + x = a, \ / r ^
G i a i . Dieu kien x > 7. D a t u = ^x + 8 > 0 va u = v ' x - 7 > 0. T a c6 he
u
+ r = 3
{V =
— u
^ i 0/2--,,2)(„2
[u,(;>0
U.,V>{)
{
2)
u4-t;'*-15
u = 3- u
< 3
u2
+ (3 - uf
3
^ \ 4 u ^ - 18u2 + 36u - 32 = 0
T i t do t a t h u dudc 1
=
2
^ ro <
^
\
<=>{^
u< 3
=
2 -
+ f=p
^ x
= 8 (thoa m a n
dieu kien). Vay phitdng t r i n h da cho co nghiem d u y nhat x = 8.
12
s/lT^=-^{a
vdi a > 0, 6 > 0.
S ""
'
(2) ,
+ b) [do
a,b>0).
V2
Ket-hop (2) t a co
ft; = 3 - u
=
K h i do a' + l? = 2. T a co he sau (
\
.
.
\l + ab{a-^ -b^) = 2 + ab.
(1) =^ {a + bf = 2+ 2ab^
,,2)^15
0 < i< < 3
^ ' ( 2 u - 3 ) ( 2 u 2 _ 6 u + 9) = 5
= 5
^ ro < 1/- <
+
ffif, t uM..
yjl - xK
' .
,|
1
1
' (
-7= (a + b){a - b){a^ + b^ + ab) = 2 + ah => ^ ( a ^ ~ h'-) = I.
v2
v2
T i t do t a c6 he |
~ ^2 ! l 2 ^
'
Cong hai phitdng t r i n h ve theo ve t a co
2 a 2 - - = 2 + y 2 ^ a 2 = l + 4 = ^ l + a ; = l + ^ ^ x = 4=V2
s/2
V2 ,
Vay phitdng t r i n h co nghiem d u y nhat x = — .
13
vj / j
B a i t o a n 16. Gidi phuang
irinh
1 -x +
\/\/2
=
n + J' =
v2
G i a i . D i n i kien 0 < x < \/2 - 1. Dfit \/\/2 - 1 - x = u va ^
0 < u < \/s/2-l
va
- 1. N h i t vay t a c6
u = —^
(
.u:^ + v^ = v/2 - 1
1
-
<
8 - vfei
8+
h"^^
18
he
8 Vay w, f la nghiem ciia
V
8 +
1
Ttr phudng t r i i i h thi'i: hai, ta co
-
vfei
18
•,
yi94
18_
3'
18
nen nghiem duy nhat ciia phudng t r i n h la
+ 7-4 = v / 2 - 1 .
V
^
= v. K h i do
71 + 7) =
-
!i i
= 0
(1)
= 0.
(2)
Do (2) v6 nghiem
/
-2 + ^ 2 ( 7 1 9 4 - 6 ) + ^ ^
/
1
2v
\
v/2
v/2
+ i ; ' = \/2 - 1
+
1.3
1.3.1
Phifcfng phap difa phifdng trinh ve phifdng t r i n h
ham
Phu'dng phap giai.
Dita vao ket ciua : Neu ham so y = f{x)
- 3
1 ±
,72
B a i t o a n 17. G'jdv phifdng trmh
G i a i . Dieii kien |
2
« > 0, v < - .
v / l - -x^ = Q
r 1 - x^ = 1 - « 4
Do do 1(1
\2
2u..v
.
Ta CO he
I [(i7,+ r ) 2 - 2 n . r
2
3
W+ U=
- 2u2.,;2 ^ 1
9
14
-
t a CO the sang tac va giai dUdc nhien phitdng t r i n h hay va kho, thudng gap
trong cac k}' t h i hoc sinh gioi. D6 van dung dildc phitdng phap nay, t a thirdng
bien ddi phiWng t i i n h da cho thanh phitdng t r i n h ham f {{x) = tpix). De giai dUdc cac bai toan bang phitdng phap nay t h i nhftng
kien thifc ve ham so nlut dao ham, xet sit bien thien va k l nang doan nghiem
la cite k i ciuan trong, c6 nhitng bai doan dUdc dap so la da hoan t h a n h den
hdn 90% Idi giai. Phitdng phap nay ditdc si't dung nhien, chang han d muc
3.6.3 d trang 191. M o t so tritdng hdp dac biet thitdng gap :
• Neu / la ham ddn dieu tren khoang (a; 6) t h i plutdng t r i n h / ( x ) = k {k la
hang so) CO khong qua 1 nghiem tren khoang {n; h).
B a i t o a n 18 ( H S G Q u a n g N i n h 2 0 1 1 ) . Gidi phieang tnnh
u.v
2u^.i)-^
vdi
a; = 7^
• Neu f yk g \h hai ham ddn dieu ngitdc chieu tren khoang (a; b) t h i phitdng
t r i n h / ( x ) = g{x) c6 khong qua 1 nghiem tren khoang (o;6).
• Neu t a thay cum t i t " / la ham ddn dieu tren khoang (a; 6)" bdi cum t i t
" / la ham ddn dieu tren m5i khoang (a; 6), {c]d)"
t h i hai ket qua d tren se
khong dung, ti'tc la plutdng t r i n h co thc^ sc c6 nhien hdn mot nghiem. Ban
doc hay xein bai toan 20 d trang 16.
-
(«2+t.^)^-2»^.7>2^1
- -
-
ddn dieu tron khoang (a; b) va
(a; b) t h i
/(^) = /(y)
- ^' <^ 0 < x < 1. Dat u = sji: va v; = ^ -
U + V=
I
x,ye
1
=0
81
v^5x 15
=7 +
1
v / ^ ^
= 0.
(1)
Vay p h U d n g t r i n h da cho c6 t a p n g h i e m l a 5 = {3, - logs 2}-
G i a i . D i c u k i c u ^ - > T- K l i i d o
o
(1) ^ (5.T
^
6)2 -
-
f{5x-6)
• ,; ^ , ,
hiiu y. X e t h a m so /(x) = 5 ^ . 8 " ^ , Vx ^ 0. K h i do / ( 3 ) = 500 v a ,.
'
1
^
= 5 ^ 8 ^ . In 5 + ^ . 5 ^ 8 ^ . In 8 > 0, Vx ^ 0.
fix)
v d i f{t) =
= f{x),
(2)
t^-
' "
Suy r a h a m s6 / d o n g b i e n t r e n m o i k h o a n g (-CXD; 0 ) , (0; + 0 0 ) . T u y n h i e n
n l u k e t l u a n 3 l a n g h i e m d u y n h a t c i i a p h U d n g t r i n h t h i se m i c p h a i sai l a m .
T a c o f'{f) = 2t + _
1
,
> 0,V/, > 1. Vay / doug bieu t r c u (1; +oo),
t h i p h U d n g t r i n h / ( x ) = k {k \h h a n g so) c6 k h o n g q u a 1 n g h i e m t r e n {a; b)".
2v/rn:(t-i)
tit (2) CO 5x - 6 = X
Vay t a c a n n h d c h i n h x a c k e t q u a " N e u / l a h a m d d n d i e u t r e n k h o a n g (a; b)
b) T u d i i g t i t c a u a ) .
X = 1,5. Phifdng t r i n h c6 nghiem duy nhat x = 1,5.
B a i t o a n 19 ( H S G L a m D o n g , n a m h o c 2 0 1 0 - 2 0 1 1 ) . Gidi phuang
trinh
G i a i . Dieu kien x > 1. Dg thay x = 1 khong l a nghigm ciia phirong t r i n h nen
B a i t o a n 21 ( C h o n d o i t u y e n N i n h B i n h n a m h o c 2 0 1 0 - 2 0 1 1 ) .
phuang trinh
Xet ham so / ( t ) = s/T+G +
= -
+
+ \ / x - 1 = 7.
(2x^ -
, + 2t +
= 7 nen phitdng t r i n h d a cho c6 nghiem duy nhat la x = 2.
-.fu.:':
h)
3 ^ 8 ^
X
( l + ilog52)
2^
^
X
. X
logs 2
[(x^ + 2x) - (2x^ -
hix) = k[g{x)
= 3 ,
log3
X
sao cho
+ 2)] =^k = l.
^ fix)].
,
=0
2x- 1
, = 3 x 2 - 8 x + 5.
(x - 1)^
G i a i . Dieu kien 0,5 < x 7^ 1. K h i do (1) titdng ditdng
„
X = - logs 2.
logs (2x - 1) - log3 (x - 1)2 = - (2x - 1) + 3 (x - 1)2 + 1.
trinh
(1)
=•
(2)
^ l o g g (2x - 1) + (2x - 1) = log3 (3(x - 1)2) + 3 (x - 1)^
4»/(2x-l) = /(3(x-l)2),
16
-2
1.
Phitdng t r i n h da cho c6 hai nghiem x = - 2 , x = 1.
= 500 ^ 5 ^ 2 ^ = 5^2^ ^ 5 - ^ 2 ^ - 2 ^ 1 ^ 5 ^ ^ - ^ ^ logs ( 5 ^ - ^ 2 " ? ' ) = logs 1 ^ logs 5 " " ' + log5 2 ^ = 0
l + -logs2 = 0
X =
X =
B a i t o a n 22 ( H S G T h a i B i n h n a m h o c 2 0 1 0 - 2 0 1 1 ) . Gidi phUdng
= 1
x-3 = 0
= 3 * h i 3 + 1 > 0 , G E . Vay
+ 2 = x^ + 2x <^ x^ - 3x + 2 = 0 <^
,^
—log52 = 0^(x-3)
, , ( 3 )
t h i t a dung plntdng phap he so bat (Hnh u h u tren de difa vc
Giai.
+
h)\
Con phitdng t r i n h t6ng quat a-^(^) - a^^^) = h{x) dUdc giai tUdng t u . T h u d n g
= 36.
a) Dieu k i c u x 7^ 0. K h i do
(2)
+ 2) = / (x^ + 2x) , v d i /(<) = 3* + i .
- (x^ - 3x + 2)=k
5 ^ 8 ^ = 500 ;
^ x - 3
X
L i r t i y . Phep phan tich (2) difdc t i m r a n h u sau : T a can t i m
' B a i t o a n 2 0 . Gidi cdc phiCcfng trinh
5 - 8 ^
(1)
tiino
H a m so / d o n g b i e n t r e n E v i f'{t)
> 0 , V i > 1.
Do do ham so n a y d o n g bien. Suy ra (*) c6 khong qua mot nghiem, mat khac
a)
Gidi
' '
^ 3 2 x 3 - x + 2 ^ (2x^ - X + 2) = 3 ^ ' + 2 . ^ (^3 ^ 2x)
(*)
+ yTH:,Vi > 1. K h i do
1
I
s/^Te
(3) ^ 2x^ -
f{2)
+ x^ - 3x + 2 = 0.
3^'+^^
3 2 x ^ - . + 2 _ 3 x 3 + 2 x ^ _ (23,3 _ ^ + 2) + (x^ + 2x)
+ x^ = 7 - v / ^ ^ ^
f'(t)
32^'-^+2 -
G i a i . PhUdng t r i n h (1) viet lai
ta chi xet x > 1. T a c6
^/^T6
''
vdi fit)
- log3 t + t.
THLT VIEN TiNHBINHTHUAN
r-\^
A
1
AQ
,
,
A A
(3)
Vi /'(/:) = — ^ + 1 > 0 ,
/.In 3
\2 ^
Dong nhat he so vdi ve trai cua (1) ta dildc
> 0 neii / (long bicii ticii (0; +oo), tit (3) c6
2x - 1 = 3(a; - 1)' <^ 3x' - 8x + 4 <^ x e | 2 , ? |
o
2
-12u = -36
6w2 _ 1 = 53
-u^-u
+ 5 = -25
(thoa di^u kien).
Tap nghiem ciia phudng trinh (1) la 5 = | 2 ,
Lifti y. Pliep phan t i d i (2) dUcJc tini ra nhit sau : Ta can tini n, ft, 7 sao cho
3x-2 - 8x + 5 = a (2x - 1) +
( x - - 1)2 + 7
(3x - 5)2 + 3x " 5 =
9x2 _ 28^; + 21 = sjx - 1.
3
2
Giai. Dicu kien x > 1. Neu
fix)
X
- 1 + v/x - 1
': :
^ / ( 3 x - 5) = / ( ^ F ^ ) , v6i J{t) = e + t
phap he so bat dinh nhir tren dfi dua ve mot trong cac tnrdng hdp
<^3x - 5 = \/x - 1 ^do ham / dong bien tren ( - - ; + 0 0 ) j
+ a'=g(x) + k, hix) = fix) - a'gix)
Bai toan 23. Giai phucing trinh
+ k, /i(x) = k[gix) -
Sx^ - SGx^ + 53x - 25 = \/3x - 5.
fix)].
(1)
Giai. Ta CO
(1) ^ 8x^ - 36x2 + 54x - 27 + 2x - 3 = 3x - 5 +
^
+
X -
^
^
1
^ fii - 3x) = fi^/^i^)
X G {2; ^ - ^ ^ } .
r 4 - 3x > 0
1 (4-3xf = x - l
L i f u y• Bai toan nay con c6 each giai khac, dirdc de cap 6 bai toan 8 d trang 163.
• Do ve trai c6 bac 3 con v6 phai co bac - nen ta can difa 2 ve ve bieu thiic
o
dang fit) = mt'^ +nt. De y rang hang tit v'3x - 5 6 ve phai c6 bac thg,p nhat
nen no tiidng ling vdi nt trong / ( i ) , vay n= 1. Lifu y r^ng 8x^ = 8(x^) = (2x)^
nen d day ta phai xet 2 trUdng hdp, m = 8 hoSc m = 1. Ngu m = 1 thi
fit) = t^ + t. Do do can dita (1) ve dang
(2x -
M)3
+ (2x - u) = 3x - 5 + ^3x - 5
<^8x^ + x2(-12u) + x(6i/2 -l)-u^
18
-u + 5= \/3x - 5.
= 2 (^thoa a: >
(vdi fit) =t^ + t)
4 - 3x = i / x - 1 ^do f { t ) dong bien tren ( - ^ ; +c)o)^ •
^ (2x - 3)^ = 3x - 5
<^8x^ - 36x2 + 51x - 22 = 0
<^ X
(1) <^ (4 - 3x)2 + 4 - 3 a ; = x - l + \/x - 1
(2)
Vi fit) = 3/2 + 1 > 0, Vi e R nen / dong bien tren R, vay tir (2) ta c6
2x - 3 =
r 3x-5>0
5p =
^ 1 (3x -
3
1
Neu l < x < - = > 4 - 3 x > - - thi
N/3X^
(2x - 3)^ + 2x - 3 = 3 x . - 5 + s / 3 ^ ^
^ / ( 2 x - 3 ) = / ( s y 3 ^ ^ ) , vdi/(«) =
^
(1)
1
3x - 5 > - - . Ta co
Dang tong quat log„ —T-T- = h(x) dUdc giai titdng tu. Ta thitcing dung phudng
h{x) = -fix)
*'
s
Vay trifcJng hdp rn = 1 da cho ket qua, do do khong can xet m = 8.
• Nhiing budc phan ti'ch tren nhhi tuy dai nhung khi da quen roi tlii ta c6
t h i tinh rat nhanh. Tuy nhien, trong mot so bai toan, ham f { t ) khong dong
bien tren R nhitng ta co th^ chi can xct ddn dicu tren mien xac dinh D.
Bai toan 24. Giai phuang trinh
( ft = 3
(
a=-l
=> ^ 2Q - 2/3 = - 8
^\ S
[ -a + ft + 'y = 5
I 7 = 1-
:
u = 3.
4
" - 3
,25±\/l3.
18
«>x =
25 - \ / l 3
18
Vay (1) CO tap nghiem S'= {2; ^ ^ - j ^ ^ } .
Lxiu y. Ta xay dvtng ham fit) = mt^ + nt. Dg y rftng ha,ng t i i v/x - 1 d v6
phai CO bac thap nhat nen n = 1. V i 9x2 _ 9 (^.2^ ^ l.(3x)2 nen ta phai xet
2 tritdng hdp m = 9, m = 1.
• Ngu m = 9 t h i / ( / ) = 9/2 + / . Can dua (1) ve dang
9(x - 'u)2
+ X
- u = 9(x - 1) + v/x - 1
-»9x2 + x ( - 1 8 u - 8 ) + t i 2 - u + 9 = v ^ x - 1 .
19
Dong nhat he so t a dUdc:
<=>/(3x - 3) = / ( ^ / 9 ( : : 3 x 2 + 21.7?T5)) vdi i{t)
/ - 1 8 w - 8 = -28
^ j i i
= —
^ 3 x - 3 = > y 9 ( - 3 x 2 + 21x + 5) o
f . . s
=
+ 27t.
3x^ - 6x2 _
^^
- 8 = 0.
Day l a phUdng t r i n h da thUc b a c 3, dUdc do cap d b a i 2.2.1 (d t r a n g 117).
• Neu 771 = 1 t h i fit)
L i i u y. N h a n 9 cho 2 ve c i i a (1) t a dUdc
= t'^ + t. Ta can diia (1) ve dang
(3x - uY + 3x
<^9x^ + x{-6u
Dong nhat he so t a duoc |
— u
= {x — 1)
+
\/ X —
27X''' - 54x2
1
- u + 1 = y/x - 1.
+ 2) +
^^^1'^^'^'^"
= 5. Den day c6 le bai
- 30x + 25 + 3x - 5 =
^/(3x-5) = /(\/^^),
L u u y rang f{t)
X
^27x^
- 1 + \/x - 1
v d i / ( ( ) = f2 + i .
+ x 2 ( - 2 7 7 / + 27)
^f^^
"2*^J "
"f^
la -
+ 3 i . Viec nhan
trinh
(1)
Hu'dtng d a n . N h a n 9 vao hai ve ciia (1) t a d\tdc
( 3 x - 3)'^ + 27(3x - 3) = 9 ( - 3 x 2 + 21x + 5) + 2 7 ^ 9 ( - 3 x 2 + 21x + 5)
20
-2777 -
9
<^ 77 = 3.
7x^-45 =
'
-153
chi ddn gian la k h i i mau so.
1627/2 + 27 ^ 3 =
7i = 4.
3x^ - 6x2 _ 3^. _ 17 = 3 y 9 ( - 3 x 2 + 21x + 5).
45)
wi-iart.R
ciing c6 the
Si'ssi;:
30.7:2011 ^ 4^^,^2010 ^ 30y4022 ^ 4y2012
3
1
K i e m t r a lai : T a c6 x < - <;=^ 4 - 3 x >
do do chon u = 4. Den day bai
z
z
toan mdi thuc sU dudc giai quyet. N h u vay t a can l i n h , h o a t t r o n g viec xay
dung ham so, nhat l a doi vdi ham bac chan. Ta cuiig c6 the giai bai toan
tren bang each dat ^x-\ 3?/ - 5 de dUa ve he doi xi'tag loai 2.
B a i t o a n 25. Giai phuong
2777 -
B a i t o a n 26 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Giai he phxcang
+ 77, + 1 = V x - 1.
^
-
3x'' = Q(3.X)^ (trudng hdp nay t h a t ra hiem gap). N h u vay f{t)
( u - 3x)2 + u - 3 x = x - l + \/x - 1
D6ng nh^t he s6 : {
(2)
Co the ban doc se thac mac t a i sao lai nhan 9 ma khdng phai la so khac.
T h a t r a dieu nay da dUdc de cap den roi. K h i xay dUng ham f{t) = mt^ + 3t,
ta thudng nghi t d i 3x'^ = 3(x^) nen cho m = 3 ma quen r i n g ngoai ra con c6
3
Con 1 < x < - t h i sao ? L a i de y rang ham so
+ x ( ~ 6 7 / - 4) +
+ (-u^
f -2717 + 27 = - 5 4
D5ng nhat he so t a dirdc : { 9u'^ - 108 = - 2 7
I
bac 2 cung c6 cai hay cua no, do l a ( - / ) 2 —
t r e n , dua vao he so bac cao
nhat l a 9, t a chi m d i xet t = 3 x - u nen bay gid t a se xet i = u - 3 x . Can
dUa (1) ve dang
^Six^
+ x ( 9 7 7 2 ~ 108)
= 2 7 ^ 9 ( - 3 x 2 + 21x + 5).
(2)
1
1
( - o o ; - - ) , hon niTa \/x - 1 > 0 > - - . N h u vay t a chi c6
1
3
khi 3 x - 5 > - - < ( ^ x > - .
'
(3x - uY + 27(3x - 77) = 9 ( - 3 x 2 + 21x + 5) + 27 V 9 ( - 3 x 2 + 21x + 5)
= t^ + t chi dong bien tren ( - ^ ; +oo) v a nghich bien tren
(2) 4^ 3 x - 5 =
153 = 2 7 ^ 9 ( - 3 x 2 + 21 + 5 ) .
Do bieu thiJc chita can c6 he so la 27, hang t i i bac cao nhat la 27x^ = (3x)^
nen t a se difa hai ve ciia (2) ve dang f{t) = t^ + 27t. Ta ])ien doi (2) thanh :
toan d a ditdc giai quyet, nhung that r a "chong gai" con ci phia trudc.
(1) <^
273.
,
(8x^-^3) ^
,
,,,
f;,?
trinh
„
(2)
G i a i . T h a y 7/ = 0 vao he thay khong thoii man, vay chi xet y 7^ 0. T a c6
™2011
„
( 1 ) ^ 3 0 . ^ + 4 . - = 30^2011 + 47/.
(3)
y
y
Xet ham so f{t)
= 30<20ii ^ 4^^^^ ^
y/^^^ ^ 3 0 . 2 0 i u 2 0 i o + 4 > Q nen
ham / dong bien tren M . Do do t i f (3) t a c6
' x \, ,
X
f ( - ) = f i y ) ^ = y^x
\y/
y
o
=
y\
Vay (!) <^ x = 7/2. T h a y vao (2) t a dUdc
162x + 2 7 V 3 = (Sx^ - ^ / 3 ) ^
21
• Mki
... (
•
(4)
Then l)ni loan 9, cJ trang 8, cac nghiem cua (4) la cos-;^. cos^-i^. c o s i ^ lo
18
18
117r
137r
TT
Do cos —— < 0, ('OS - — < 0 nen ta chi nhan nghiem x = cos — . Cac nghiem
18
18
18
cua he phuring trinh da cho la
^ ,
Xet ham so dac trung f{t) = t'^ + t, t eR. Ta co / ' ( i ) = Si^ + 1 > 0, vay ham
so dong bien tren R nen (*) <=i> a + 1 =
Ta co he sau :
Sii dung phep the ta co
^3 _
;
_ 1)2 = l<=^ 6'^ - 362 + 66 - 4 = 0 o
;;
.
(6 - ^^(,^2 _ 26 + 4 ) = 0 ^
6 = 1.
Tit do X = - 1 . Vay phUdng trinh co nghiem duy nhat x = - 1 .
Bai toan 27 (De nghi Olympic 30/04/2011). Gim phuang trinh
I'
1.3.2
V i du 1. Xudt phdt tic mot phuong
Hu'dng dan. Xet, /(,;:) = Vx'-^ + 3.7-2 + g^. _ 13 _ ^^332^. Ta c6
6^. + 0
3.7:2 +
2\/jr'^ + 3a;2 + 9.7: -
4
Vay ham / rlong bion trcn
13 +
1
v/3 - 2x
, mat khac /
/ 4 \ \/543 - \/27
Uj ~
cd each gidi rat cd ban, do la :
0(7--^)=0(6x-5)
^ 7 - ^ - ' + 6 log7 7^-^ - (6x - 5) + 6 logy (6x - 5)
nen - la
3
9
trinh
-'..^
- r x - i = 6x - 5. Xet mot ham. so dOn dicu (j){t) = t + 6 logy t. Khi do
04
> 0, V x e
Phu'dng phap sang tac bai toan mdi.
^r~^ + 6 (x - 1) = (6x - 5) + 6logy (6x - 5)
^7^-^
= 1 + 2 logy (6x - 5)^ .
i : '-ih ; /
•[ ',
, , , , , ,
nghiem duy nhat cua phitdng trinh da cho.
x
<
^,
Ta duoc bai toan sau.
Bai toan 28 (De thi hoc sinh gioi cac trtrdng Chuyen khu vu'c Duyen
Hai va Dong Bang B a c B o nam 2010). Gidi phiMng trinh
Bai toan 30. Gidi phuang trinh
7'^''^ = 1 + 2 logy (6x - 5 ) ^
(1)
Giai. Dieu kien .x > | . Ta co
6
2.x'* - x2 + v/2x^ - 3 x + 1 = 3 x + 1 + v/x2 + 2
(1) <^ 7^-^ - 6 logy (6x - 5) = - 6 (x - 1) + (6x - 5 ) .
Hu'dng dSn. Tap xac dinh D = M. Bien d5i phildng trinh ve •
(2)
7-^-^ + 6(x - 1) = (6x - 5) + 61ogy (6x - 5)
2x'* - 3x + \/2x'-^ - 3 x + 1 = x2 + 1 + \/x2 + 2..
<^ 0 (7^-1) = (;6 (6x - 5), vcii (/>(«) = « + 6 logy t.
Xet ham ,s6 / ( / ) = t + 0, ^t > 0. Vay
(4)
7^-1 = 6x - 5 <(=> 7"^-' - 6x + 5 = 0.
Cach 1. De thay r i n g x = 1, x = 2 thoa ( 4 ) . Xet ham f{x) = T'^ - 6x + 5
tren R. Ta co / ' ( x ) = 7 ^ - i l n 7 - 6; / " ( x ) = 7^-^ (In7)^ > 0, Vx G R. Vay
ham / CO do thi luon luon loin nen cat true Ox tai khong qua hai diem, suy
ra (4) CO khong qua 2 nghiem. Vay x = 1, a; = 2 la tat ca cac nghiem cua
(4). Phitdng trinh (1) co tap nghiem la 5 = {1,2}.
f,
^
Bai toan 29. Gidz phuang trinh (x + 5)\/x + 1 + 1 = \/3x + 4.
Giai. Dicu kiOn x > -1. Dtlt a =
Ham so 0 dong bien tren (0; + 0 0 ) vi (i>'{t) = 1 +
(3)
_^
Cach 2. Ta co / ' ( x ) = 0
" ^ Cong ve fheo ve ta co
7^-' = —
x = xo = 1 + logy (6. logy e). V i
v6i moi x G R thi / " ( x ) > 0 nen suy ra / ' la ham dong bien tren R va
a'* + 30.2 _^
2 = 1/ +
( a + I ) ' * + (a + 1) =
22
+ b.
(*)
/ ' ( x ) < 0, Vx G
( - 0 0 ;
xo) ; / ' ( x ) > 0, Vx €
23
(XQ; + 0 0 ) .
,
Vay ham / nghich bien tren {-OO;XQ)
va dong bien tren ( x o ; + o o ) , do do
(4) C O khong qua 2 nghiem. Vay x = 1, a; = 2 la l a t ca cac nghiem ciia (4).
PhUdng t r i n h (1) c6 tap nghiem la S = { 1 , 2 } .
L i f u y. Phep phan tich (2) diidc t i m ra nhil sau : Can chon a, /3, 7 sao cho
l=a(x-l)+/3(6x-5)=.{
- + 6/?j0^
" '
a-''^^) + klog^gix)
= ^(x) - / c / ( V ) ,
(doi vdi (*))
f /^V^
= 3' + t. Tic phuang
1\ l-x
= 2x^-2x-l.
-
2x
1\
2x (v/3) ^ - 2x ( -
B a i t o a n 31. Gidi phudng trinh
= 2x2 _ 2x - 1.
Hu'dng d a n . T i f d n g t i f bai toan 21 d trang 17. PhiTdng t r i n h c6 hai nghiem
l + V^
-
—
,
l-\/3
X
=
—
-
—
.
V i d u 3. Xet ham so nghich bien tren khodng (0; + 0 0 ) la f{t)
Tii phuang trinh ham j (\{x
\
logi
(\{x-\f
= x'^ ~ ISx - 3 1 .
+ in
9x2 + (ix + 3126
^,^3^^^
.
'• '
vdi / ( i ) =
-
'Ux
/
= l o g i t — t.
= f {2x + \) ta cd
- if]
= l o g i (2x + 1) - (2x + 1)
^ 3 + l o g ! ((x - 1)-) - l o g i (2x + 1) = (\{x
2
2
\
24
trmh
, ^ • •
'' "' 6^56^30 = 30
g«l.>lifii|
+ 3125
< 1
nen ham c6 / ' ( < ) > 0, suy ra ham / dong bien tren E.
-i
•
B a i t o a n 34 ( D e n g h i cho k i t h i h o c s i n h gioi c a c tru'dng C h u y e n
k h u v y c D u y e n H a i v a D o n g B a n g B a c B o n a m 2 0 1 0 ) . Gidi phudng
trinh
Ta CO bai toan sau.
—
8 l o g i -~—-j-
•••
trmh
2x2-2x-l
f^V~"
2x(v/3)'
=
1 '
T a c 6 / ' ( i ) = 3i2 + i + - ^ J ^ i _ , ma
,
X
^
x^ - 2x = 1 + ^ / ^
/(x) = /(A/S^TT),
= / ( ^ ~ 1)' ^" ^-'^
or-i
Ta diCcfc bdi toan sau.
i
Htfofng d a n . PhiJdng trinh viet lai
( d 6 i v d i (**)),
+ kf(x),
V i d u 2. Xet ham .so ddiig hiev. tren M Id f{t)
1
= x2 - 18x - 31.
2 ^ ^ - - '
B a i t o a n 33 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi phuang
Dudc giai tudng t i f n h i l tren bang phitdng phap he so bat d i n h , phan tich
ham f
1)2
(v6i 0 < a < 1, A: > o)
= hix)
h{x)=!jix)
-
Hii'ding d a n . T u d n g t i t nhxi bai toan 22 d trang 17. Phitdng t r i n h c6 hai
nghiem x = 9 - 2^22,x = 9 + 2\/22.
/^y. i^...;^. .y)
Q
..V,.
= h{:r.) (vdi a > 1, A; > 0
- klog,jj{x)
X
B a i t o a n 32. Gidi phuang trinh
^ { ? = T.'
Cac phiTdng t r i n h long quat
' ''
^ 8 logi
5
- lf \ (2x + 1)
/
(6^' - 3^) (19-^- - 5^) (10^ - r)
+ (15^ - 8^) (9^ - 4^') (5^ - 2^) = 2 3 F .
(1)
G i a i . Ta c6 cac nhan xet sau :
N h a n xet 1. Vdi a > 6 > c > 1 t h i
> 6^ neu x > 0 va
N h a n xet 2. Vdi a > 6 > 0 cho tritdc t h i ham so / ( x ) =
dong bien va lien tuc tren tap D = fO; + 0 0 ) do
/ ' ( x ) = a"" In a - 6^ in 6 > 0, Vx > 0.
< 6^ neu x < 0.
- 6^ xac djnh
'' '
N h a n xet 3. T i c h hai ham so dong bien, nhan gia t r i ditdng tren tap D la
ham dong bien, tong hai ham dong bien tren D la ham dong bien tren D.
Ta se ap dung ba nhan xet tren de giai bai toan nay.
Ngu X < 0 t h i
's.ji. - •<
'i t
,
(6^ - 3^) (19x - 5^) (10^ - 7^) + (15^ - 8^) (9^ - 4^) (5^ - 2^) < 0,
25
trong k h i 231'^ > 0, auy r a phitdng t r i n h khong c6 nghi^m khong ditdng.
V6i X > 0, chia hai vg phitdng t r i n h cho 2 3 F = (3.7.11)^ dudc
B a i t o a n 3 6 . Gidi he phitang
trinh |
^
J ^2 _ ^ _ ^^^^
Hu'o'ng d i n . D a t u =^ x + y, v = x - y- K h i do
u
+
iv
)
Goi y l a hani so d ve t r a i ciia (2). T i t nhan xet 2 va nhan xet 3, suy ra y
dong bien t r e n D = (0; +oo) va
9 _ 4\5 _ 2\
5(1) = ( 2 - 1 ) ( { ^
_
14 3
+-V
=
2x,
uv = x2 - y2, ^2 + 1-2 = 2(x2 + y^).
(2)
7
7)
[3
Thay vao he ta ditdc |
4v'^ - t;2
{v - 1) (4i;2 _^ 4,^ _^ ^2) = 0
11 7 ^ 1 1 ' 7 3 ~ •
Co r a t nhieu each dat an phu k h i giai he phitdng t r i n h . D a t an p h i i nhit the
nao con t u y thuoc vao tifng he phitdng t r i n h cu the. B a i nay se neu r a mot
so phcp dat an p h u cd ban, thit6ng gap. N a m ditdc cac phep dat nay t a se
CO dinh hudng t o t hdn k h i giai he phitdng t r i n h .
4
G i a i . Dieu kien x 7^ 0 va y
M- + V
X =
K h i do
uv =
+
X
9
u-v
9
= 2x (x" + 10x2y2 + 5y4)
u^-v''
= 2y (3x2 ^ ^2)
u'-v'
= 8xy (x2 + y^)
=»7/7;
2x + 2x2 _2y2 = 7
2 (.r2 + y2) ^ 5
26
—
2
W. -
y =
"
I
2
2x
1
2 {u - -t))
z(, + w
2 (^2 - 7/2)'
u+
01;
[uvY = - 5
{u^ - v^) = u - {uvfv.
(3)
1 +
=0
V {u^ - ( / ) = 1 - uS;" <=> u'^v{l + v'') = 1 + If' ^u'^v
= 1.
Khi 1 +
= x2 - 2/2^^2 ^. ^2 ^ 2{x'^ + xf).
T h a y vao he t a ditdc he doi x i i n g loai 1 doi v d i u va v. { ""2"^^ t
.,. ,
Neu u = 0 t h i y = - . T , the vao phitdng t r i n h thit hai ciia he thay khong thoa
man. Vay xet u 0. T i t (*) t a c6
Hvfdng d i n . D a t u = x + y, v = x - y. K h i do
u + v = 2x,uv
1
0. D a t u = x + y, v = x - y. K h i do
uv (^2 + t'2) ^
- v'^ = 2y (5.7;'* + 5x2y2 ^ ^^4^
B a i t o a n 3 5 . Giai he phiCOng trinh
3
trinh
Thay vao he (*) t a du'dc
= 2y
+ -.2 = 2 (x2 + y2)
- y
4
1
4y
u + v = 2x
=0
= 1.
(*)
P h e p d a t « = x + y, v — x - y.
1.4.1
V
V
B a i t o a n 37 ( D l n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi he phxMng
M o t so phep dat a n p h u cd b a n k h i giai he
phufdng t r i n h .
1.4
3
+ 3^;2 = 4v^ 4v^ + 8t'2 - 12u = 0
{4V^ + 8v -12) =0<^v
Vay (1) ^ g{x) = y{\) o x = 1. Suy ra x = 1 la nghiem d u y nhat ciia (1).'"'I
4r^ - ,-2
T i t (1) suy ra u
thav vao (2) dUdc :
3)
7_b3_
^f+^;^ I
^
= 0 t a CO v; = - 1 , suy ra u = v^S. Vay
X
=
s/5 - 1
y=
27
V5 + 1
Khi
7/.^?;
= 1 t a c6
1'
^ ~ ^ _ 2/- 1 =
u-v=
= —r, thay vao {v.ny = - 5 t a ditdc
x+1
1
5
V-5
,
X = —-—z
) y =
2
' "
(x + l ) ( y + l )
•i> ( S I ofiv "/,fii!
X
+ y
he phuang
x - y
| ^2
3-x
3y2^_""i^
Ta t h u dUdc he
Hu'dng d a n .
I
C a c h 1. D a t D a t u = x + y, v = x - y. K h i do
\+
<^u^ +
- UV = 1
^
= .,j3 _ ,^3 _^ 2ur2 - 2u^u <^ 2v^ + 2u^v - 2uv^ = 0
r ^' = 0
<^t; (t;^ - u r + u^) =
0
2
_|. " j ' ^
4. ^ = 0
1.4.2
Phep dat u =
x+ l
u
^ 1
r u= 0
1-4.3
Phep dat u ^ x +
22:y - 2
r +
x+1
y + 1
(x+l)(y+l)
28
=
'.\> g i i b - J i i - i
2 - y • Jiftwrfs ,ijtjt.3,i VB
1 -
2w - 2
2 -
v+l
V - 1
^-v
u + 3'
V+1
kb .QJTi
u + •() ~ 2u + 4
uv — \ — V
^ / tx2 = 2z;
^ I
= 3.
v = ij + - .
X
K h i do
u + u=
X
\. uv + 1
7' + 3
2^2 + 4u - 2uv - 4v = Au + 4(; - 2u'^ - 2uv
I uv^ -v + 3uv - 3 = 3uz; + 3 - uv'^ - v
^
y + 1
K h i do
-
u - V _ 4 - 2u
^ / 4u2 - 8u = 0
^\2ut;2 = 6
{ " - 0
C a c h 2. D u a ve phildng t r i n h d i n g cap bac ba doi vdi x va y.
3
V = - — r - K h i do
t; + 1
X + y
uv — \
u + v' I + xy
uv -\-1
3u - 3
1 u+\ 4 - 2 ^ 1 - 2 y
" - 1
2it + 4 ' 2 - y
3u +1
1 -3x
trinh
1 - xy
u — V
x-y
I - xy
'* '
<
1 + xy
Thay \ao he t a dvtdc { JJ3 | J^3 ^ 35
x,,ff,ij 11
^r:^g''\-^i-iu..W.a
3^ + y ^ 1 - 2y
Hu'dng d a n . Dtit x = ^^—4u+1
Hifdfng d a n . D a t u = x + y, v = x - y. K h i do
B a i t o a n 3 9 . Giai he phUOng
trinh
| ^3 :t | ^ 2 ^ ?! 17^ 5. V i . Ui*o..
trinh
,.
,
, (x;y) =
B a i t o a n 4 0 . Giai he phuong
B a i toan 38.
(x + l ) ( y + l )
^2/ - (x + y) + 1 ^
2 (x + y)
(x+1) (y+1)
(x + 1) ( y + 1 )
" - ^ _ X- y
,
1 — uz)
(x; y) =
2xy + 2
\ - u v ^ \
-. Cac nghiem ciia h§ 1^ ' ^ f "
2
(x + 1) ( y + 1 )
x y - ( x + y) + l
(x+1)(y+1)
_ x y - (x + y) + 1
'^""^
Do do
y + 1
x - l y - 1
uv — x + l ' y + 1
2 (x - y)
+
1- +
" + w = (x + y)
\'
y
2, v^ = r/ + \ 2 vk
1
1+ — )
u'-* +
29
= (x^ + y'
+ 4
n _ y x'^ + 1
V
x' y'^ + 1 '
uv = xy -\
xy
xy
{x + y)
B a i t o a n 4 1 . Giai he phudng trinh •
xy +
1
xy
/
\
1 4.4
1A
1+ — = 4
x2 +
h
xy
B a i t o a n 4 2 . Giaijie
phmng
|
Irinh
^ ^2)
K l i i do
u + t) = (x + y) f l +
— = 4.
Hifcing d a n . D a t u = x + i , u = y + ^, t a t l m dudc he { ^
uv = x y H
= 4
.
4
1^-11 = ( x - y ) H
V
—)
1-2
B a i t o a n 46. Gidi he phiMng
irinh
0.
Ho titdng ditrmg vrJi
f
xyy
1 \
1+ —
=4
\
V
xyy
xy + — = 2
/
^xy
(x + y)
Hifctng d a n . D a t u = x + - i , v = y + - , t a t h u dudc he | " , + 2 ' r
y
X
•
u t — 4t.
1
1
\
= 208.
B a i t o a n 4 7 . G?d? he phuong
B a i t o a n 4 3 . Gidi he phMng
trinh
\ .2 _^ 2^
V
4
xyJ
Hu-dng d a n . Dat u = x + ^, i ; = y + ^, t a t h u dUdc he | 4("2"^_j_^]2y f
= 16.
^^
i
trinh
t a t l m diOTc he | "2"t|.\,2 ^ 2 1 2 .
y (x2 + 1) = 2x (y2 + l )
/
B a i t o a n 48. Gidi he. phudng
Uvtdng
xy
V
xy
_^ 3.2^2^ ^ 208x2y-.
Hvfdng d a n . De thay (x; y) = (0; 0) la nghieni cua h§. T i e p tlieo xet xy
Dat u = X + ^ , v^y^-K
P h e p d a t a — x + i , v = y + -•
y
•
X
•!
•2\
1\
45.
25
trinh
d a n . Dieu kieu x y ^ 0. Dat u = x + - , 7; = y + ^, t a t l m ditcfc he
^
y
Hifdng d a n . Dat u = x + J , ^ = y + ^ , t a t h u dudc he |
+ i;2 = 20.
{
= 1
B a i t o a n 4 4 . Gidi he phmng
trinh
B a i t o a n 4 5 . G i d i he phUdng trinh
/
1
X +
-
V
H i f d n g d a n . He phimng t i i i i h tucing diMiig
2 (x+ x)
30
1-4.5
(
y + ..V/
xy (2x + y - 6) + y + 2x = 0
^^2 _|_ y2^ ^2 _)_ ^
Hifdng d i n . D a t u = x + - , v = y + - , t a t l m dUdc he
y
X
r 2x'^y + y^x + 2y 4- x = Qxy
I
J_ +
4_ E = 4.
I
xy
X
y
= 4
= 6.
+ J J ^ j I ^35
M o t s o p h e p d a t §n p h u k h a c .
_ g
2u + w = 6
u2 + t;2 ^ 8.
\
Cac phep dat fiu p h u r a t da daug va phong phii. Ta can kliai thac cac dac
fiipm rieng, cac t i n h chat dilc biet cua tftng he phUdng t r i n h de dita ra phep
<5§.t phii hdp.
f.
31
Bai toan 50. Gidi
x = 4- y/W va y = 3 + / l O
he phUdng trinh
Giai. Bien d6i he da rho, ta tlni ditrJc
x^
do. ta
+ x/ - xy{x + y) = 3
+ xy{x + y) = 15.
CO
x-'^ +
=9
>^ <
xy{M '+y) = 6
•
y = 21
2
X =
. y. = 3
=2
.X =
y = l-
he CO nghiem (x; y) = (1; 2), (x; y) = (2; 1).
Bai toan 51 (HSG Hai Phong, bang A, nam hoc 2010-2011). Gidt
/x + - + ^ x + y - 3 = 3
he •phuanc) trinh sau
V
2x + y + -1 =
y
Giai. Dieu kieii y 7^ 0, x + - > 0, x + y > 3. Dat
1
a=
^ x + i ,
"J'^^t
He da oho viet lai la {
• V6i a = 2 va 6 = 1, ta c6
xJr-
y
X+
6=
v^x
+ y
- 3, a , 6 > 0.
a = 2 va 6 = 1
a = 1 va 6 = 2.
5
'= 2 va Jx + y - i = l < ! = > x + 1- = 4 v a x + y = 4
V
f x 2 - 8 x + 15 = 0
X = 3 va y — I
= 4
<^ x ^ 4
<^
4 - X
X = 5 va ?y = - 1 .
I y=4-x
[ y= 4
Vdi a = 1 va 6 = 2, ta c6
-
+ v'lO va y = 3 - ^10.
' V'
''•''
Thii lai, thay tat ca deu thoa man. He phirdng trinh da cho c6 4 nghiem la
(3,1). ( 5 , - 1 ) , (4 - v ^ , 3 + \/ro) , (4 + v ^ , 3 -/To) .
LuM y- Dang he ijhitong tiinh giai bang each (hit an phu nay thittJng gap tj
nhieii ky thi, tit DH-CD den thi HSG cap tinh va khu vuc.
Bai toan 52 (HSG tinh Ha Tinh, nam hoc 2010-2011). Gidt he
3
. 2y = 1
+
x2 + y2
_
1
x' + i/V 2x = 4
X =
X
X• + - = 1 va Jx + y - 3 = 2^'
v * / »» •
.^.7^;
""'^'^
^^
V i d u 1. Xuat phdt tit mot bien doi tiMng dudng do ta chon
/ f + y = 3
I z' + y = b
k h i (y; z) = (4; - 1 ) : I
CO
fti:^^i*i,3>
1)' '^"y '•'^ (^; 2/) = (1; 0).
G i a i . Xet x = 0 => y = 0. Vay (0;0) la mot nghiem ciia he. Xet x ^ 0, chiii
hai ve cua (1) cho x, hai ve ciia (2) cho x^, ta dudc
X
4COS(Q
^ - 2 cos(3a - 45") = ^ 3 ^
= x - y, dieu kien u > 2. Thay vao (1), ta ditoc
{ a + 6 =^3""
f ^ f ^ I'^i •;
8sin(rv - 45"). sin(a + 15") cos(a - 15") = \/3
..ill
G i a i . Dieu kien x + y 7^ 0. He viet lai £ /Of v
3(x2+y2) + (x2-y2)
+ s i n 2 a ^ = ^3
+
X
9.1/2
+ 27?/ + 27
- 2= y + 3 ^
X
•
= y + 5.