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PHIfOnC
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PHIIDnG
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PHlJdNG PHAP XAY DL/NG BE TDAN
CAC DANG TCAN, CAC PHLfdNG PHAP GIAI
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TAO
VA
GIAI
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TRINH,
HE
PHLfdNG TRINH, BAT PHaONG TRINH
NGUYIN
TAI

CHUNG
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trach nhiem xuS't ban
4 ;
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THI
THANH
HlJdNG
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tap :
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NHAN
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. :
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NHlTX
Trinh
bay
:
C6ng
ty
KHANG
VIET
Bia
:
C6ng
ty
KHANG
VIET
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NHA
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phdt
hanh
CONG
TY
TNHH
MTV
DjCH
Vg VAN HOA
KHANG VIET
(^Dia
chi-
71

Oinh Tien Hoang
-
P.Da Kao
-
Q.1
-
TP.HCM
Dien
thoai:'08.
39115694
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39105797
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39111969
-
39111968
Fax:
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Tan Phu, Tp.
Ho
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So DKKHXB:
1
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3.
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do NXB
Tong
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Thanh
Pho H6 Chi

Minh
cap
ngay 19/03/2013
In xong
va nop
luU
chieu
Quy II
nam 201
3
LMnoidau
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
giao

vien cung chi biet suti tarn
cac
bai
tap c6
trong sach
giao
khoa, sach tham
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
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
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,
DHKHTN
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 am 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 tu phia cac ban la dieu
chiing
toi luon mong dpi. Hy 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.
Tac gia
Thac
sy: NGUYEN
TAI
CHUNG
Nha
sach
Khang
Viet
xin
trdn
trgng
gi&i
thi?u
tai Quy dgc gia va xin
idng
nghe
moi y
kieh
dong
gop, de
cuon
sach
ngdy
cang
hay hem, bo ich hon.
Thuxingici
ve:

Cty
TNHH
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71,
Dinh
Tien Hoang, P. Dakao. Quan 1, TP. HCM
Tel: (08)
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-
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Hoac
Email:
ChiMng 1.
PhUcfng
phdp
sang
tdc va
giai
phUcfng
trinh,
hf
phUOng
trinh,
bd'i

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
li

Lagrange
63
1.8 Phu'dng
phdp
hinh
hpc 69
1.9 PhiTdng phap ba't dang thtfc 82
1.10 PhiTdng phap tham bien 95
ChUcfng
2.
PhUcfng
phdp
da
thiic
va
phUcfng
trinh
phdn
thitc
hOu ti. 116
2.1 Cdc dong nha't
thiJc
bo sung 116
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
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
ding
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
307
4.7 Phi/dng
phap
bien
doi
ding
thiJc 314
4.8
MotsohekhongmaumiTc
317
ChUctng
5. Cdc bai todn

phUcmg
trinh,
h^
phUcfng
trinh, bat
phUcfng
trinh trong
dethidt^ihQc
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 368
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
Chi:fc?ng
1
Phi:fdng
phap sang tac va giai
phifcfng
trinh,
he phi:fcfng
trinh,
bat
phi:^dng
trinh
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
lai
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^^ - llx + 21 -
3^4.x-
-4 = 0.
Giai.
Tap xac dinh D = E. Plntdug trinh da cho tu'diig ditdng vdi
•^ikf
•'A/nh
6"
^(4x-4)2-I(4x-4)
+ 12-3x/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. wid ;:,v»fb im V >.
3
Do do (2) <=> i = 2 => X = 3.
LULU
y. De c6 (1) ta can
tiiii
a, (3,7 sao cho
- llx + 21 = a(4x - 4)^
+/3(4a;
- 4) + 7
<!=>2x2
- llx + 21 = IGrtx^ + (4/^ - 32rv);r + (16^ -4/^ + 7)
16a = 2 , , fl 7 \
f 16a = 2
^{
4/3-32a
= -11
^{a;(i\-i)
=
I
16a-4/3
+ 7 = 21
dung
phuong phap he so bat
dinh
cho ta 15i giai bai toan mot
each

rat
tijt
nhien va ro rang.
Bai
toan
2.
Giai
phiCcfng
trinh
-x+3
=
2\/r^-/m^+3\/r^.
(i)
Giai.
Tap xac
dinh
D = [-1; 1]. PhUdng
trinh
(1) viet lai nhu sau :
(1
+ x) + 2(1 - x) -
2v^r^
+
\/l
+ .T -
3\/l
- x2 = 0. (2)
Dat u =
^/^+x,V
= Vl - X {u

>0,v>0),
ta duoc
+
2^2 - 2u + It - 3uv = 0 {u^ - 2uv) + [u - 2v) - {uv -
2^^)
= 0
^(•a
- 2v){u - + 1) = 0 ^ [ « Z 1 ^ 0
• ^/3^
Lu^i
y. Dl CO (2), ta tim a, f3 sao cho
-x + 3 = a(l + x) + /3(l-x)o{
^;^^3^
^{ g = i
Doi
v6i bai toan tong quat : Giai phvfdng
trinh
p{x) = as/1 - X + bVl + X + cVl - x^,
Ta bieu dien p{x)
theo
1 - x, 1 + x va dat
u
=
-/I
+ X,
i;
=
\/l
- X (u > 0,
i;

> 0).
Khi
do dirdc phitdng
trinh
doi vdi u, v c6 thg phan tich ditdc.
Vi
du 1. Ta sc.
sang
tdc mM
phUdng
trinh
duclc
gidi
hhng
phiMng
phdp
he
so bat
dinh
nhu sau : Ta c6
{a-b+ l)(2a - 6 + 3) = 0 ^ 20^ + 6^ - 3a6 + 5a - 46 + 3 = 0.
Tii
day lay a — ^Jl + x vd b = \/l - x ta
diidc
2x + 2 + 1 - X - 3\/l - x2 + 5VI+X -
4s/l-x
+ 3 = 0.
Rut gon ta
duac
bai

toan
sau.
4
Bai
toan
3. Gidi
phuang
trinh 4\/l - x = x + 6 - 3\/l - x^ + 5s/TTx.
X V3
Dap so. X = — J
Bai
toan
4. Gidi
phuang
trinh 4 + 2\/l - x = -3x + SVxTT + Vl - x^.
Dap so. PhUdng
trinh
c6 tap nghiem S — I 0; —; — I. tah
sv,
;: i ;
25 2 j
Bai
toan
5. Gidi
phuang
trinh lOx^ + 3x + 1 = (6x +
l)Vx2
+ 3. (*)
Giai.
Dat u = 6x + 1, ?; = \/.x2 + 3. Ta c6

10x2 + 3x + 1
=
i(6x
+ 1)2 + (x^ + 3) - - = — + ^2 _ 9
4
4 4 4 5,
Thay vac (*) : +
1,2
_ 5 = ^iK:^ (u - 2t;)2 = 9 <^ u - 2u = ±3.
• Vdi
w
- 2D = 3, ta CO v
1
+ 6x - 2v/x2 + 3 =
3<^3x-l
= \/x2 + 3
-'^'^i'l'S
3x
- 1 > 0 ^ ,
x2
+ 3=(3x-l)2 ^x^l.
• Vdi u - 2t; = -3, ta c6
Vay phiMng
trinh
c6 tap nghiem 5 = < 1; ——^ I. ,
Lvfu
y. Phudng phap he so bat
dinh
de
giai

he phudng
trinh
se ditdc de cap
trong
phan phan tich tim Idi
giai
cac bai toan
ciia
bai 1.5 : PhUdng phap
cong, phiTdng phap the (d trang 35).
1.2 Phifcfng phap difa ve he.
Dg
giai phUdng
trinh
bang each
dua ve he
phUdng
trinh
ta
thutdng
dat an
phu,
phep
dat an phu nay
cimg
vdi
phUdng
trinh
trong
gia

thiet
cho ta mpt h$
phitdng
trinh.
Sau day ta se
trinh
bay
phuldng phap sang
tac
(thong
qua cac
VI
du),
phUdng phap giai (thong
qua Idi
giai
cac bai
toan
va
quan trong
hdn
niia
la cac hru y sau Idi
giai).
Cac
phudng phap sang
tac
ciing
nhiT
phifdng

phap giai
cac
phUdng
trinh
bang each
dUa ve he con dildc de cap rat
nhiiu
6 sau bai nay
(chang
han bai 3.2 d
trang
160).
5
Vi
du 1. Xet I y ~ 2 ^ 3^^ ^ x = 2 - 3 (2 -
3x•'^)^
Ta c6 hdi todn sau.
Bai
toan 6.
Gidi
phUdng trlnh x + 3 (2 - 3x^)^ = 2.
Giai.
Dat, =
2-3x^.1^
CO he
^[^ZlZ^.
(1)
tnr (2) ta fhrac
x-y
= 3{x'^ - y^) ^

X
- y = Q
3(x
+ y) = l ^
y = X
1
- 3x
y = —^—•
Vc'ri
y = x, thay vao (1) ta diWc Sx^ + x - 2 = 0 x G |~^'
1
- 3x
Vdi
y =
3
1
- 3x
,
thay vao (2) ta dudc
=
2 - 3x^ <^ 9x2 - 3x - 5 = 0 X
1
± v/21
PhUdiig
trinh
da cho co bon nghiem
2
1-V21 1 + V21
X
=

-1,
X = -, X = ——, X = .
3 6 ()
Lxiu
y. Tir Idi
giai
trcn
ta thay
iftng
neu
kliai
Irien
(2 -
3x'^)'^
tlii
sc dira
phitdng
trinh
da cho vc phiWng
trinh
da
thiitc
bac bon, sau do
Ijien
doi thanh
(x
+ l)(3x -
2)(9x2
- 3x - 5) = 0.
Vay

ncu khi
sang
tac de toan, ta c6 y lam cho
plnMng
trinh
khong v6 nghiem
hiiu
ti thi phildng phap khai
trien
dua ve phUdng
trinli
bac cao, sau do phan
tich
dua ve phu:dng
trinh
tich
se gap nhieu
klio
khan.
Vi
du 2. Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti
5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _
2x
= 5
5x2-
1\
-
1. Ta
CO
bdi todn sau.

Bai
toan 7.
Gidi
phUdng trinh 8x - 5 (5x2 _ _ _^
Giai.
Dat 2y = 5x2 _ ^ YAn do
2y
= 5x2-1 . ^ r 2y = 5x2-1 (1)
8x
-
5.42/2
= -4 ^ \x = 5y2 - 1. (2)
6
Lay
(1) tru' (2)
tlico
ve ta du'dc
y = X
y
- X = 0
2
= -5(x + y) ^
2(y
- x) 5(.T2 - y2) ^
Vdi
y =
X,
thay vao (1) ta dUdc 5x2 _2x -\=i) ^ x =
5x
+ 2

1
± \/6
•
Vdi y = —. thay vao (1) ta du'dc >;V;!; .•
o
-l^
= 5x2-l.=.25x2 +
l(,x-l=0^x=-^=^^
PhUdng
trinh
da cho c6 bon nghic'm
25
1
± v/O -1 ± 72
5 5
Luti
y.
Phep
dat 2y = 5x2-1 chrdc
tini
ra nhu sau: Ta dat n:y-\-b = 5x2 _ ^
vdi
a, h thn sau. Khi do
tiiu
dUdc he
ay + 6 = 5.r2 - 1 { + b + i = hx'- ,
8x
- 5 («y + bf = -4 ^ \.T + 4 - 5^2 = 5a2y2 +
i{)aby.
C

fl _ _5_ _ ^+1 r , _ .
Dg
he tren la he doi
xftng
loai
TT
thi < g ~ 5„2 ~ 4 _ 5/^2 => | Z 9 Vfw
I
10a/; = 0 ' I « - -
ta
CO
phep dat 2y
—
5x'^ - 1. f\ .
-jj.,.,
Bai
toan 8.
GidirphtMng
trinh 5{5x~ - 17)2 - 343x - 833 = 0.
Y
tvtdng.
Dat ay + b^5x^- 17 (a ^ 0).
Klii
do . ,
jay
+ b = 5.7:2 _ ^7 , , ,
\5(ay + 6)2 - 343x - 833 - 0. (*) • ''
Tir
(*) ta CO
5(ay)2 + lOa^y + ^2 - 343x - 833 = 0 ^ x = 5(ay)2 + 10a6y + ^2 - 833

„
* ^ 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 5a'^y2+l()o2.6y+62.a- 833a ,
•
M!
I -
o/y-l7=
46
<^343.5y2 5831 = 5a''.y2 +
l()a2.6y
+ 62,„
g33„
_|.
3435.
f
343 = a'* r - 7 '
Dong
nhat he so ta ditcJc I
al^h
= 0
1
/ =
(1
"^''^^
''"^

t 833a+
3436=-5831
Idi
giai
sau.
7
Giai.
Dat 7y = 5x^ + 17, ta c6 h?
phitdng
trinh
J7y
= 5x2 _ 17
1^5,y2
- 343a; - 833 = 0
7y = 5x2 _ 17 (1)
7x =
5y2_17.
(2)
Lay
(1)
trit
(2) ta c6 7{y - x) = 5(x + y)(x - y) ^
* Neu X = •(/,
thay
vao (1) : Sx^ - 7x - 17 = 0 x =
* Niu 5x + 5j/ = -7, ket hdp (1) ta c6 .
x = y
5x + 5y = -7.
7±\/389
10

Ket
luan:
Phudng
trinh
c6 tap nghiem 5 =
'7± \/389 -35 ±5v^l
50 J
•
10
Vi
du 3. Ta CO Ax^ - 3x = ~ <^ Qx = 8x^ -
V^.
Vdy xet
fSx^
- v/3\
-V3
{
>1296x + 216v/3 = 8 (Sx^ - ^/fj ^ 162x + 27^ = (Sx^ - Vs^ .
Ta
CO hai todn sau.
Bai
toan 9.
Gidi
phUOng
tnnh
162x + 27\/3 = (8x^ - s/zf .
Giai.
Dat 6y = 8x^ - \/3. Ta c6 h^
6y - 8x3 - v/3
162x + 27\/3 =

216?y3
r
6?y = 8x3 _ ^3
\x =
8?y3
- v/3 (2)
Lay
(1)
tilt
(2) theo ve ta dudc
6(y - x) = 8(x3 - 2/3) ^ (a; _ [g (x^ + xy + y^) + 6] = 0.
Vi
x^ + xy + y2 > 0 nen 8 (x^ + xy + y^) + 6 > 0. Do do tit (3) ta dUdc x =
Thay
vao (1) ta dUdc
6i
= 8x3 - \/3 <^ 4x3 - 3x = ^ 4^3 _ 3^, = cos ^.
a a
Sii
di^ng
cong
thiic
cosa
=
4cos3
- -
3cos
-
,
ta c6

cos-=4cos
18-3COS-,
8
llTT
1
llTT
COS
=
4
cos-*
18
6
18
137r 137r
COS
——
=
4 cos''
18
6
18
TT
llTT
137r
r
= cos ——,
X
= COS
18'
•

18 ' 18
- 3 cos
—
3 cos
llTT
18 '
137r
18 •
la tat ca cac nghiem ciia phuong
trinh
(4) va cung la tat ca cac nghiem cua phUdng
trinh
da cho.
Ltfti
y. Phep dat 6y = 8x3 _ ^ dUdc tun ra nhu sau : Ta dat
ay + 6 = 8x3 - V3 ^^^^ ^ ^jj^
Ket
hdp v6i phudng
trinh
da cho c6 he
ay + 6 = 8x3 - v/3
162x + 27V3 =
a3y3
+
3a26y2
+ Safety + ^3
Can
chgn a va 6 sao cho :
8 73
162 o? 27\/5 - 63

3a26-3a62
= 0
Vay
ta c6 phep dat 6y = 8x3 _ ^
Vi
du 4. Xet
tarn
thiic
bdc hai luon
nhdn
gid tri duang : x'^ + 2. Khi do
/
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.
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
o
dong
bien k{x) sao cho k{x) = x <^ g{x) — 0,
muSn
vdy ta xet v. .
^ + ax - 15 = X <!::^ ^ + (Q - l)x - 15 = 0.
Do do chon a sao cho a - 1 = 2 a = 3, khi do k{x) = — + 3x - 15 fd
x3 ^
A;(x)
= y tuang dudny U(H
.•
— + 3x - 15 = y <^ x^ + 9x - 45 = 3y. TH
phiiang
tnnh
cuoi ndy thay x bdi y ta thu duac he doi
xilng
loai hai
x3 + 9x - 45 = 3y
y3 + 9y - 45 = 3x.
Til:
he tren, sU dung phep the ta thu duoc phuong
trinh
x3 + 9x - 45
+
9
/x3 + 9x-45\

- 45 = 3x
0fiOJ
i.'
I
9
<^ (x'' + 9x - 45)^ + 81 {x^ + 9x - 45) = 1215 + 81x.
Vay ta thu dUdc hai todn sau. ,
Bai
toan 10. Gidi phuong trmh ; V
(x^ + 9x-45)V81 (x^ +
9x-45)
= 1215 + 81X. (1)
ji
Giai.
Tap xac dinh M. Dat + 9x - 45 = 3y. Ket hdp v6i (1) ta c6 he
:^i^:,^.^7j
. / x^ + 9x - 45 = 3y (2)
••-•J\ + 97/- 45 = 3x. (3) ^
Lay (2)
tiif
(3)
theo
ve, ta
dUdc
x^ - + 9x ^ 9)/ = 3?/ - 3x ^ - + 12(x - ?y) = 0
.^(x - y)(x^ + xy + + 12) = 0 <^ X = y.
Thay vao (2) ta
diWc
x^ + 9x - 45 = 3x <=> (x - 3) (x^ + 3x + 15) = 0 ^ x = 3.
Phildng

tiiiih
da cho c6 nghiem duy nhat x = 3.
LuTu
y.
Phcp
dat x^ + 9x - 45 = 3y dittfc tun ra uhu sau: Ta dat
x^ + 9x - 45 = ay (vdi a
tini
sau).
Khi
do
x^ + 9x - 45 = ay ^ | xj^ + 9x— 45 = ay_
a'Uj^
+ Slay = 1215 + Six \ + Slay - 1215 = 81x.
, . , «3 81« 1215 81 „ „
Vay can
chon
a thoa man dieu kien — = — = —7- = — => a = J. Uo cto
••' • 1 9 45 a
dat x^ + 6x ~ 45 - 3y, ta so thu
ducJc
mot ho doi xi'mg loai hai.
Vi
du 5. Chon mot phMng trmh chi c6 hai nghiem /d0 vd 1
IdlV
= lOx+1.
Tii
phiCOng
trinh
nay ta thiel lap mot he doi xvCng loai hai, sau do lai quay

ve phuang trmh nhu sau :
f ll'-
= lOy + 1 ^ I y = logn (lOx + 1) - = log,, (lOr + 1)
\iry
= 10x + l ^\lF=10y + l ^ 10 logiiuux+ij.
Suy ra IF = lOlogn (10x + 1) + 1 ^ IF = 21ogii (10x+ 1)^ + 1. Ta c6 bdi
todn sau.
Bai
toan 11. Gidi phuang trinh IV = 21ogii (lOx + 1)^ + 1.
10
Giai.
Dicu
kiCm
> Dat y = logn (lOx + 1),
Ivhi
do IP = 10x+ 1. Ket
hdp vdi phUdng
trinh
da cho, ta c6 he | JJy ^ ^ | ^ "» "
•
- - w.,
Lay (1) tnr (2) tlieo ve ta ditdc • f-: , A. •
IF
-
ir^
= lOy - lOx <^
ir^
+ lOx = IP + lOy, (3)
Xet ham so /(/) = 11' + 10/ Ta c6
f'{f.)

= 11' hi 11 +10> 0, V/. G K. Vay ham
so /
dong
bien tren E. Ma (3) chinh la /(x) = /(y) nen x = y. Thay vao (1)
ta
difdc IL''= 10x+1 <^ IF - lOx-1 =0. .• (4)
Xet h;\ so f/(x) = IF - lOx - 1 tren khoang ( ; +00 . Ta c6
V 10 J
g\x)
=
ll-^nll
- 10, g'\x) =
lF(lnll)2
> 0.
/ 1
Vay ham so g c6 do thi luon 16m tren khoang ^-j^; +00 j , suy ra do thi cvia
ham g va true hoanh co vdi nhau khong qua hai diem chung, suy ra (4) c6
khong qua 2 nghiem. Ma g{l) = 0, y(0) = 0 nen x = 0 va x = 1 la tat ca cac
nghiem ciia (4). Nghiem ciia phUdng
trinh
da cho la x = 0 va x = 1.
Vi
du 6. Ta se su dung phuang phdp lap di sang tdc phuang trinh tit. he
phuang trinh doi xvtng loai hai. Xuat phdt tit \^^Z
^^^4^30
•^^
^^'^'"'(1
P^^'^P
the ta diMc phuang trmh Ax = ^30 + |v/x + 30. Til phuong trinh nay ta lai,
thu

dtWc
he doi xtCng loai liai
; .Zi nst,- 4
4M
= ^/30+ -v/aM^
4x= ^/30+
-v/^r+30.
Tir
he niiy, ticp tuc s'li dung phcp the ta thu diMc phuang trmh
\
4x =
Ta
CO bdi todn sau.
Bai
toan 12 (De nghi Olympic
30/04/2010).
Gidi phuang trinh
4x =
30
+ - W
30
+ - ^30 + - \/^T30.
11
Giai.
Do x la
nghifni thi
x > 0.
Dat
u = 30 + ^x +
30, tit phitdng tnnh

da
cho ta c6 ho
Gia s\t
X > u.
Khi
do
4u
=
J30+
-y/xT3Q
(1)
4x
=
A/30+
-v/w
+
SO.
4u
= \/30+ + 30 > ^30 +
-\/u
+
30
= 4x =^ u > x =^ x = ?i.
Vay tit
he (1) ta c6 a; = u va 4x = ^30 +
-yxTSO.
Dat
. =
\V^FT30,
tit

(2) ta c6 he
| ^J I
Gia sii
x> V.
Khi
do
(2)
(3)
4v
= Vx + 30 >
VtTTSO
= 4a: 4u > 4a:
=^ V
> a; =^ u =
X.
,
f T > 0 1 +
\/l921
Vay
r = .x va
4.T
= ^ | Jgp^^
^,
^ 30 — .
PhUdng
trinh
da cho co
nghiem
dny
nhat

x =
1
+ 71921
32
Vi
du 7. Vdi X = 8 thi
^/x-\-8-\- \Jx-l —
3, ia c6 bai
todn {ch&c chan
co
mot nghiem
dep x = 8) sau.
Bai
toan
13.
Giai phUdng trinh y/x
+ 8 +
\/x
- 7 = 3.
Giai.
Dieu kien
x > 7. Dat u = ^x + 8 > 0 va u = v'x - 7 > 0. Ta c6 he
u
+ r = 3 { V =
2)
— u
U.,V>{)
^ i 0/2 ,,2)(„2 +
,,2)^15
u4-t;'*-15

[u,(;>0
{
u
= 3- u ft; = 3- u
0
< i< < 3
(2u-3)(2u2_6u
+ 9) = 5
^ ro
< 1/- < 3 ^
ro
< u < 3
^ \4u^
- 18u2 +
36u
- 32 = 0 ^\ = 2-
Tit
do ta
thu
dudc
1= 2 <=>{^ + f=p ^x = 8
(thoa
man
dieu kien).
Vay
phitdng
trinh
da cho co
nghiem
duy

nhat
x = 8.
<
3
u2
+ (3 - uf
=
5 ^'
12
Ltfti
y- Doi
vdi phu'dng
trinh
- JJx) +
"\/b
+ /(x) = c, ta co
each
giai
:
Dat
u = 'ija - fix), v = '^h + f(x), dan den he
{^H
^+,n"s'^['^
Nhit
vay
dang
nay la
j)hn'dng
trinh
vo

ti,
infi
san
khi
dat an
phu dita
ve he, roi
dimg
phep
the dan
tdi phudng
trinh
da
thitc,
do do
khi
sang
tac de
toan
ta
phai
dac biet chii
y cac
chi
so
can. Chang han
d vi dn 7
thi
m = n = 4 nen ta yen
tam

rang
se dan
tdi phitdng
trinh
da
thite
bac 4 co it
nhat
mot
nghiem
dep.
Vi
du 8.
Vd'i.
x = -2 thi
2<y3x
-2 + 3^6 - 5x = 8, ta c6 bai
todn {chac
chdn
CO mot
nghiem
dep x = -2)
.sau.
1
"i;.,/
.;- .f
'-,1,;
Bai
toan
14.

Gtdi phiMng trinh 2 v^3x
- 2 + 3^6 - 5x = 8.
Giai.
Dieu kien
x <^. Dat u = ^3x - 2, v =
^/G
- 5x > 0.
Khi
do
5
I"2
Z
t' 7
^
+
Sv'^
=
5(3x
- 2) +
3(6
- 5x) = 8. .
—
0,
ox
Mat
khac
ta
lai
co 2u + 3r -8 = 0. Vay ta co he
{^t +

'fv
=
8^
=^
+ 3 (^^) ' = 8 ^
15 ^
+ 4^2 -
32z.
+ 40 = 0
Phu'dng
trinh
nay c6
nghiem
duy
nhat
u = -2 nen
v'Sx
- 2 = -2 X- = -2.
Bai
toan
15.
Giai phiMng trinh
1
+
\/l
-
x2 [V(l
+
:r)-*
- ^(1 -

x)'A^
=2+ yjl - xK
ffif, t uM
Giai.
Dieu kien
-1 < x < 1. Dat ^l + x = a, \/r^ = vdi a > 0,
6
> 0.
Khi
do a' + l? = 2. Ta co he
sau
( \ . . S "" '
\l
+
ab{a-^
-b^) = 2 +
ab.
(2) ,
(1) =^
{a + bf = 2+ 2ab^
s/lT^=-^{a
+
b)
[do
a,b>0).
V2
Ket-hop
(2) ta co ' . , |
1
1 ' ( '

-7= (a
+ b){a -
b){a^
+
b^
+
ab)
= 2 +
ah =>
^(a^ ~
h'-)
= I.
v2
v2
Tit
do ta c6 he | ~ ^2
!l
2^
Cong
hai phitdng
trinh
ve
theo
ve ta co
2a2 =2+y2^a2
= l +
4=^l+a;=l
+ ^^x = 4=-
V2
s/2 V2 ,

Vay phitdng
trinh
co
nghiem
duy
nhat
x = —.
vj
/ j
13
Bai
toan 16. Gidi phuang
irinh
\/\/2 1 - x + =
v2
Giai.
Dini
kien 0 < x < \/2 - 1.
Dfit
\/\/2 - 1 - x = u va ^ = v. Khi do
0 < u <
\/s/2-l
va - 1.
Nhit
vay ta c6 he
.u:^
+ v^ = v/2 - 1
Ttr
phudng
triiih

thi'i:
hai, ta co
u = —^ V
(
1
- V
+
7-4 =
v/2-1.
1
\ 2v
+
+ i;' = \/2 - 1
v/2 v/2
1
±
- 3
,72
Bai
toan 17. G'jdv phifdng trmh v/l -
-x^
= Q - .
Giai.
Dieii kien | - ^' <^ 0 < x < 1. Dat u = sji: va v; = ^ - vdi
2 r 1 - x^ = 1 - «4
« > 0, v< Do do 1(1 Ta
CO
he
U
+ V = -

I
(«2+t.^)^-2»^.7>2^1 I [(i7,+ r)2-2n.r
2
\
2
- - 2u v
3
-
2u2.,;2
^ 1
W
+ U = -
2u^.i)-^ u.v =0
9 81
14
n +
J'
= -
^ < 8 - vfei h"^^
18
71
+ 7) = -
8+ vfei
Vay w, f la nghiem ciia
8 - yi94
18_
8 +
3' 18
nen nghiem duy nhat ciia phudng
trinh

la
/
18 •, ! i i
= 0 (1)
= 0. (2)
1
-2 +
^2(7194-6)+^^
/
Do (2) v6 nghiem
1.3 Phifcfng phap difa phifdng
trinh
ve phifdng
trinh
ham
1.3.1 Phu'dng phap giai.
Dita
vao ket ciua : Neu ham so y = f{x) ddn dieu tron khoang (a; b) va
x,ye (a; b) thi
/(^)
= /(y) a; = 7^
ta
CO
the
sang
tac va giai
dUdc
nhien phitdng
trinh
hay va kho, thudng gap

trong
cac k}' thi hoc sinh gioi. D6 van dung dildc phitdng
phap
nay, ta thirdng
bien ddi phiWng
tiinh
da cho thanh phitdng
trinh
ham f
{<f){x))
= f
(ipix)),
trong
do / la ham ddn dicu. Tfr day
diui
den mot phvtdng
trinh
ddn gian
lidn
4>{x)
=
tpix).
De giai
dUdc
cac bai toan
bang
phitdng
phap
nay thi nhftng
kien

thifc ve ham so
nlut
dao ham, xet sit bien thien va kl nang
doan
nghiem
la
cite ki ciuan trong, c6 nhitng bai
doan
dUdc
dap so la da
hoan
thanh den
hdn 90% Idi giai. Phitdng
phap
nay ditdc si't dung nhien,
chang
han d muc
3.6.3 d trang 191. Mot so
tritdng
hdp dac biet thitdng gap :
• Neu / la ham ddn dieu tren khoang (a; 6) thi plutdng
trinh
/(x) = k {k la
hang so)
CO
khong qua 1 nghiem tren khoang {n; h).
• Neu f yk g \h hai ham ddn dieu ngitdc chieu tren khoang (a; b) thi phitdng
trinh
/(x) = g{x) c6 khong qua 1 nghiem tren khoang
(o;6).

• Neu ta thay cum tit "/ la ham ddn dieu tren khoang (a; 6)" bdi cum tit
"/ la ham ddn dieu tren m5i khoang (a; 6),
{c]d)"
thi hai ket qua d tren se
khong dung,
ti'tc
la plutdng
trinh
co thc^ sc c6 nhien hdn mot nghiem. Ban
doc hay xein bai toan 20 d trang 16.
Bai
toan 18 (HSG Quang Ninh
2011).
Gidi phieang tnnh
1
1
= +
v^5x - 7 v/^^
=
0.
(1)
15
Giai.
Dicu
kicu
^->
T-
Klii
do
o

(1)
^
(5.T
-
6)2
- ^
1
Taco
f'{f) = 2t +
^ f{5x-6)
= f{x), vdi f{t) = t^-
1
(2)
_
, >
0,V/,
> 1. Vay /
doug bieu
trcu
(1; +oo),
2v/rn:(t-i)
tit
(2)
CO
5x - 6 = X X = 1,5.
Phifdng
trinh
c6
nghiem duy nhat
x = 1,5.

Bai
toan 19
(HSG
Lam Dong, nam
hoc
2010-2011). Gidi phuang trinh
Giai.
Dieu kien
x > 1. Dg
thay
x =
1 khong
la
nghigm ciia phirong
trinh
nen
ta
chi xet
x >
1.
Ta c6
^/^T6
+
x^
= 7 -
v/^^
^
s/^Te
+ +
\/x- 1

=
7.
(*)
Xet
ham
so /(t)
=
s/T+G
+ +
yTH:,Vi
>
1.
Khi
do
1
f'(t)
= - I ,
+ 2t
+
>0,Vi>
1.
Do
do
ham
so
naydong bien. Suy ra
(*)
c6 khong qua mot nghiem, mat khac
f{2)
= 7

nen phitdng
trinh
da
cho c6 nghiem duy nhat
la x = 2.
' Bai toan 20. Gidi
cdc
phiCcfng trinh
a)
5^8^
=
500
;
h)
3^8^ =
36.
Giai.
a) Dieu kicu
x 7^
0. Khi
do
5-8^
= 500 ^ 5^2^ =
5^2^
^
5-^2^-2
^ 1 ^ 5^^-^ -
=
1
^

logs
(5^-^2"?')
=
logs
1
^
logs
5""' +
log5
2^ = 0
^x-3
+
—log52
=
0^(x-3)
(l +
ilog52)
=0
x-3
= 0
l
+
-logs2
= 0 ^
X
.
X
logs
2
2^

= 3 , „
X
= -
logs
2.
16
Vay phUdng
trinh
da
cho
c6
tap nghiem
la 5 = {3, -
logs
2}-
•
,; ^ , ,
hiiu
y. Xet
ham
so /(x) =
5^.8"^,
Vx ^
0. Khi
do
/(3)
=
500
va ,. '
fix)

=
5^8^. In 5
+
^.5^8^. In 8
> 0, Vx ^
0.
' "
Suy
ra
ham
s6 /
dong bien
tren
moi
khoang (-CXD;
0), (0;
+00).
Tuy
nhien
nlu
ket
luan
3 la
nghiem
duy
nhat ciia phUdng
trinh
thi
se
mic phai sai lam.

Vay
ta
can nhd
chinh
xac
ket qua
"Neu / la
ham ddn dieu
tren
khoang (a;
b)
thi
phUdng
trinh
/(x) = k {k
\h hang
so) c6
khong qua
1
nghiem
tren
{a; b)".
b)
Tudiig tit cau
a). ''
Bai
toan
21
(Chon
doi

tuyen Ninh Binh nam
hoc
2010-2011). Gidi
phuang trinh
32^'-^+2 - 3^'+^^ +
x^
-
3x
+
2
=
0.
' ' (1)
Giai.
PhUdng
trinh
(1)
viet lai
tiino
fu.:':
32x^ +2
_
3x3+2x
^ _ (23,3 _ ^ + 2) + (x^ + 2x) (2)
^32x3-x+2
^ (2x^ - X + 2) = 3^'+2. ^ (^3 ^ 2x) h)\
(2x^
- X +
2)
= /

(x^
+
2x)
,
vdi
/(<) =
3*
+
i.
,,(3)
Ham
so /
dong bien
tren
E vi f'{t) =
3*hi3
+ 1 > 0,G E. Vay
(3)
^
2x^
- X +
2
=
x^
+
2x <^ x^
-
3x
+
2

= 0 <^
X
= -2
X
= 1.
Phitdng
trinh
da
cho
c6
hai nghiem
x =
-2,
x = 1.
Lirti
y.
Phep
phan
tich
(2)
difdc tim
ra
nhu sau
: Ta
can tim
sao cho
- (x^
-
3x
+ 2)=k

[(x^
+
2x)
-
(2x^
- X +
2)]
=^k = l.
Con phitdng
trinh
t6ng quat
a-^(^) -
a^^^)
= h{x)
dUdc giai tUdng tu. Thudng
thi
ta
dung plntdng phap
he so
bat (Hnh uhu tren
de
difa
vc
,^
hix) =
k[g{x)
^
fix)].
,
Bai

toan
22 (HSG
Thai Binh nam
hoc
2010-2011). Gidi phUdng trinh
2x-
1
(x
-
1)^
log3
,
=3x2-8x
+
5.
(1)
Giai.
Dieu kien
0,5 < x
7^
1.
Khi
do (1)
titdng ditdng
=
•
logs
(2x - 1) -
log3
(x -

1)2
= -
(2x
- 1) + 3 (x -
1)2
+
1.
(2)
^logg
(2x - 1) + (2x - 1) =
log3
(3(x -
1)2)
+ 3 (x - 1)^
4»/(2x-l)
=
/(3(x-l)2),
vdi fit) -
log3
t + t. (3)
THLT VIEN TiNHBINHTHUAN
r-\^ A
1 A
Q
, , A A
Vi
/'(/:)
= —^ + 1 >0, > 0 neii / (long
bicii
ticii

(0; +oo), tit (3) c6
/.In
3
\2 ^ o 2
2x - 1 = 3(a; - 1)' <^ 3x' -
8x
+ 4 <^ x e |2, ? | (thoa di^u kien).
Tap nghiem ciia phudng
trinh
(1) la 5 = |2,
Lifti
y. Pliep
phan
tidi
(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
(
ft = 3 ( a=-l
=>
^

2Q
- 2/3 = -8 ^\ S
[
-a + ft +
'y
= 5 I 7 = 1-
fix)
Dang tong
quat
log„
—T-T-
= h(x) dUdc giai titdng tu. Ta thitcing dung phudng
phap
he so bat dinh nhir tren dfi dua ve mot trong cac tnrdng hdp
h{x)
= -fix) + a'=g(x) + k, hix) = fix) -
a'gix)
+ k, /i(x) =
k[gix)
-
fix)].
Bai
toan
23.
Giai
phucing
trinh Sx^
- SGx^ +
53x
- 25 =

\/3x
- 5. (1)
Giai.
Ta
CO
(1)
^
8x^
-
36x2
+
54x
- 27 + 2x - 3 = 3x - 5 +
N/3X^
^
(2x -
3)^
+ 2x - 3 =
3x
5 + s/3^^
^/(2x-3)
=/(sy3^^),
vdi/(«)
= + (2)
Vi
fit) = 3/2 + 1 > 0, Vi e R nen / dong bien tren R, vay tir (2) ta c6
2x - 3 = ^ (2x -
3)^
= 3x - 5
<^8x^

-
36x2
+ 51x - 22 = 0
X G {2;
^-^^}.
Lifu
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^
-u + 5=
\/3x
- 5.
18
Dong
nhat
he so vdi ve trai cua (1) ta dildc
-12u = -36 : *'
6w2
_ 1 = 53 u = 3. s
-u^-u
+

5
= -25
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
thi
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 9x2
_ 28^; + 21 = sjx - 1.
3

1
Giai.
Dicu
kien x > 1. Neu 2 3x - 5 > Ta co
(3x
-
5)2
+ 3x " 5 = X -
1
+ v/x - 1 '
:
:
^/(3x
- 5) =
/(^F^),
v6i J{t) = e + t
<^3x
- 5 = \/x - 1 ^do ham / dong bien tren ( ;
+00)j
(1)
^
r 3x-5>0 ^
^
1
(3x - 5p = X - 1 ^
<^
X
=
2
(^thoa

a: >
3 1
Neu l<x<-=>4-3x> thi
(1)
<^ (4 -
3x)2
+ 4-3a; = x- l + \/x - 1
^
fii - 3x) =
fi^/^i^)
(vdi fit) =t^ + t)
4 - 3x = i/x - 1 ^do
f{t)
dong bien tren (-^; +c)o)^ •
r
4 - 3x > 0 " - 3 25 - \/l3
1
(4-3xf
= x-l
4
,25±\/l3.
«>x
=
18
18
Vay
(1)
CO
tap nghiem S'=
{2;

^^-j^^}.
Lxiu
y. Ta xay dvtng ham fit) =
mt^
+ nt. Dg y rftng
ha,ng
tii v/x - 1 d v6
phai
CO
bac
thap
nhat
nen n = 1. Vi
9x2
_ 9
(^.2^
^
l.(3x)2
nen ta phai xet
2 tritdng hdp m = 9, m = 1.
•
Ngu m = 9
thi/(/)
=
9/2
+/. Can dua (1) ve
dang
9(x
-
'u)2

+ X - u = 9(x - 1) + v/x - 1
-»9x2
+
x(-18u-8)+ti2-u
+ 9 =
v^x-1.
19
Dong
nhat he so ta
dUdc:
/-18w-8
= -28
^jii
= — f s
• Neu
771
= 1 thi fit) =
t'^
+ t. Ta can diia (1) ve dang
(3x - uY + 3x — u = {x
—
1) + \/ X — 1
<^9x^ + x{-6u + 2) + - u + 1 = y/x - 1.
Dong
nhat he so ta
duoc
|
^^^1'^^'^'^"
= 5. Den day c6 le bai
toan da

ditdc
giai quyet, nhung that ra
"chong
gai" con ci phia trudc.
(1) <^ - 30x + 25 + 3x - 5 = X - 1 + \/x - 1
^/(3x-5) = /(\/^^), vdi/(() = f2 + i. (2)
Luu
y rang f{t) = t^ + t chi
dong
bien tren (-^; +oo) va nghich bien tren
1
1
(-oo;
),
hon niTa \/x - 1 > 0 > Nhu vay ta chi c6
(2) 4^ 3x - 5 = ^f^^
13 3
khi
3x-5> <(^x> Con 1 < x < - thi sao ? Lai de y rang ham so
bac 2
cung
c6 cai hay cua no, do la (-/)2 — tren, dua vao he so bac cao
nhat la 9, ta chi mdi xet t = 3x - u nen bay gid ta se xet i = u - 3x. Can
dUa
(1) ve dang
(u - 3x)2 + u-3x = x- l + \/x - 1
^Six^
+ x(~67/ - 4) + + 77, + 1 = Vx - 1.
D6ng nh^t he s6 : { "2*^J " "f^ ^ 7i = 4.
3

1
Kiem tra lai : Ta c6 x < - <;=^ 4 - 3x > do do
chon
u = 4. Den day bai
z z
toan mdi thuc sU
dudc
giai quyet. Nhu vay ta can linh,hoat trong
viec
xay
dung ham so, nhat la 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.
Bai
toan 25. Giai phuong trinh
3x^
- 6x2 _ 3^. _ 17 = 3 y9(-3x2 + 21x + 5). (1)
Hu'dtng dan. Nhan 9 vao hai ve ciia (1) ta
d\tdc
(3x
- 3)'^ + 27(3x - 3) = 9(-3x2 + 21x + 5) + 27^9(-3x2 + 21x + 5)
20
<=>/(3x
- 3) = /(^/9(::3x2 +
21.7?T5))
vdi i{t) = + 27t. ^ ^
^3x
- 3 = >y9(-3x2 + 21x + 5) o 3x^ - 6x2 _ -8 = 0.

Day la
phUdng
trinh
da thUc bac 3,
dUdc
do cap d bai 2.2.1 (d
trang
117).
Liiu
y. Nhan 9 cho 2 ve
ciia
(1) ta
dUdc
27X'''
- 54x2 273. 153 = 27^9(-3x2 + 21 + 5). ' (2)
Do bieu thiJc chita can c6 he so la 27, hang tii bac cao nhat la 27x^ = (3x)^
nen ta se difa hai ve ciia (2) ve dang f{t) = t^ + 27t. Ta ])ien doi (2) thanh :
(3x
- uY + 27(3x - 77) = 9(-3x2 + 21x + 5) + 27 V9(-3x2 + 21x + 5)
^27x^ +
x2(-277/+
27) +
x(9772
~ 108) + (-u^ - 2777 - 45)
=27^9(-3x2 + 21x + 5).
wi-iart.R
f
-2717 + 27 = -54
D5ng nhat he so ta dirdc : { 9u'^ - 108 = -27 <^ 77 = 3. '
I

-2777
-
7x^-45
= -153
Co the ban doc se
thac
mac tai sao lai nhan 9 ma khdng phai la so
khac.
That ra dieu nay da
dUdc
de cap den roi. Khi xay
dUng
ham f{t) = mt^ + 3t,
ta
thudng nghi tdi 3x'^ = 3(x^) nen cho m = 3 ma
quen
ring
ngoai ra con c6
3x'' =
Q(3.X)^
(trudng hdp nay that ra hiem gap). Nhu vay f{t) ciing c6 the
la
- +3i.
Viec
nhan 9 chi ddn gian la
khii
mau so.
Si'ssi;:
,,, f;,?
Bai

toan 26 (De nghi Olympic
30/04/2011).
Giai he phxcang trinh
30.7:2011 ^ 4^^,^2010 ^ 30y4022 ^ 4y2012 , , „
1627/2
+ 27 ^3= (8x^-^3) ^ (2)
Giai.
Thay 7/ = 0 vao he thay khong
thoii
man, vay chi xet y 7^ 0. Ta c6
™2011
„
(1)^30.^+4 =
30^2011 + 47/. (3)
y y
Xet ham so f{t) = 30<20ii ^
4^^^^
^ y/^^^ ^ 30.20iu20io + 4 > Q nen
ham /
dong
bien tren M. Do do tif (3) ta c6
'x\, , X o
•
Mki
f(-)=fiy)^- = y^x = y\
\y/ y
Vay (!) <^ x =
7/2.
Thay vao (2) ta
dUdc

162x +
27V3=
(Sx^ - ^/3)^ • (4)
(
21
Then l)ni loan 9, cJ trang 8, cac nghiem cua
(4)
la
cos-;^.
cos^-i^. cosi^-
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 ^ ,
Bai
toan
27 (De nghi Olympic
30/04/2011).
Gim phuang trinh
Hu'dng dan. Xet,
/(,;:)

=
Vx'-^
+
3.7-2
+ g^. _ 13 _
^^332^.
Ta c6
3.7:2 + 6^. + 0
+
1
2\/jr'^
+
3a;2
+
9.7:
- 13 v/3 - 2x
>
0, Vx e
04
Vay
ham / rlong bion trcn
4
,
mat
khac
/
/4\ \/543 - \/27
Uj
~ 9
nen - la

3
nghiem duy
nhat
cua phitdng
trinh
da cho.
Bai
toan
28 (De thi hoc sinh
gioi
cac trtrdng Chuyen khu vu'c Duyen
Hai
va Dong Bang Bac Bo nam 2010).
Gidi
phiMng trinh
2.x'* - x2 + v/2x^ - 3x + 1 = 3x +
1
+ v/x2 + 2
Hu'dng dSn. Tap xac dinh D =
M.
Bien d5i phildng
trinh
ve •
2x'* - 3x +
\/2x'-^
- 3x + 1 =
x2
+ 1 +
\/x2
+ 2

Xet
ham ,s6 /(/) = t +
<yt
+ I tren R. De chitng minh ham so dong bien tren
R. Tvt gia thiet suy ra
/(2.T'*
-
3.7-) /(.r2
+ 1)^
2x-^
- 3.r =
x^
+ 1.
.
, .
,
•
1 1 ± f,
rhirdng
trmh co nglnOm x = x = —-—. ^
Bai
toan
29.
Gidz
phuang trinh (x +
5)\/x
+ 1 + 1 = \/3x +
4.
Giai.
Dicu

kiOn
x > -1.
Dtlt
a = 4- 1. = v/3xT4.
Klii
do x = _ ^
3a2
+ 1 = 6'^ Ta c6 he |
^^^2^^^" "^3^
" ^ Cong ve fheo ve ta co
a'* +
30.2 _^
2 = 1/ + (a +
I)'*
+ (a + 1) = +
b.
(*)
22
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.
I'
1.3.2 Phu'dng phap sang tac bai
toan
mdi. -' ^
Vi
du 1.
Xudt
phdt
tic mot
phuong
trinh
cd
each
gidi
rat cd ban, do la :

-rx-i
= 6x - 5. Xet mot
ham.
so dOn
dicu
(j){t)
= t + 6 logy t. Khi do
0(7 ^)=0(6x-5)
^7-^-'
+ 6 log7 7^-^ - (6x - 5) + 6 logy (6x - 5)
^r~^ + 6 (x - 1) = (6x - 5) + 6logy (6x - 5) i: '-ih ; /
^7^-^
= 1 + 2 logy (6x - 5)^ . •[ ', x <
,,,,,,
^,
Ta
duoc
bai
toan
sau.
Bai
toan
30.
Gidi phuang trinh
7'^''^
= 1 + 2 logy (6x - 5)^ (1)
Giai.
Dieu kien .x > |. Ta co
6
(1)

<^ 7^-^ - 6 logy (6x - 5) = -6 (x - 1) + (6x - 5). (2)
7-^-^ + 6(x - 1) = (6x - 5) + 61ogy (6x - 5)
<^
0 (7^-1) =
(;6
(6x - 5),
vcii
(/>(«) =
«
+ 6 logy t. (3)
Ham
so 0 dong bien tren (0;
+00)
vi
(i>'{t)
= 1 + >0, ^t > 0. Vay
(3)
7^-1 = 6x - 5
<(=>
7"^-' - 6x + 5 = 0. (4)
Cach
1. De thay
ring
x = 1, x = 2 thoa
(4).
Xet ham f{x) = T'^ - 6x + 5
tren R. Ta co /'(x) = 7^-iln7 - 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}.
Cach
2. Ta co /'(x) = 0 7^-' = — x = xo = 1 + logy (6. logy e). Vi
v6i moi x
G
R thi /"(x) > 0 nen suy ra /' la ham dong bien tren R va
/'(x)
< 0, Vx
G (-00;
xo) ; /'(x) > 0, Vx €
(XQ; +00).
,
23
Vay ham / nghich bien
tren
{-OO;XQ) va dong bien
tren
(xo;+oo),

do do
(4)
CO
khong qua 2 nghiem. Vay x = 1, a; = 2 la lat ca cac nghiem ciia (4).
PhUdng
trinh
(1) c6 tap nghiem la S =
{1,2}.
Lifu
y.
Phep
phan
tich
(2) diidc tim ra
nhil
sau : Can
chon
a, /3, 7 sao cho
l=a(x-l)+/3(6x-5)=.{ - +
6/?j0^
^ { ? = T.'
Cac phiTdng
trinh
long quat
' ' ' " ' - klog,jj{x) =
h{:r.)
(vdi a > 1, A; > 0
a-''^^)
+ klog^gix) = hix) (v6i 0 < a < 1, A: > o)
Dudc

giai tudng tif
nhil
tren
bang phitdng phap he so bat dinh, phan
tich
=
^(x) -/c/(V), (doi vdi (*))
h{x)=!jix)
+ kf(x), (d6ivdi
(**)),
Vi
du 2. Xet ham .so ddiig hiev. tren M Id f{t) = 3' + t. Tic phuang trmh
ham f = /(^ ~ 1)' ^" ^-'^
1
or-i , f /^V^ f^V~" 2x2-2x-l
2x(v/3)'
2x -
1\
l-x
=
2x^-2x-l.
Ta CO
bai toan sau.
Bai
toan 31.
Gidi
phudng
trinh
2x (v/3)
^

- 2x ( -
1\
=
2x2 _ 2x - 1.
Hu'dng
dan. Tifdng tif bai toan 21 d trang 17. PhiTdng
trinh
c6 hai nghiem
l
+ V^ l-\/3
X = —-—, X = —-—.
Vi
du 3. Xet ham so nghich bien tren khodng (0;
+00)
la f{t) = logi t — t.
Tii
phuang
trinh
ham j (\{x - = f {2x + \) ta cd
\ /
'Ux - if] = logi (2x + 1) - (2x + 1)
^3 + log! ((x - 1)-) - logi (2x + 1) = (\{x - lf \ (2x + 1)
2 2 \ /
logi
(\{x-\f
24
X - 1)2
5
2^^ '
^8 logi = x2 - 18x - 31.

i
Ta
diCcfc
bdi toan sau. ^ 1 ' •••
Bai
toan 32.
Gidi
phuang
trinh
8 logi
-~—-j-
= x'^ ~ ISx - 31.
Hii'ding
dan. Tudng tit nhxi bai toan 22 d trang 17. Phitdng
trinh
c6 hai
nghiem x = 9 -
2^22,x
= 9 +
2\/22.
/^y.
i^ ;^.
.y) Q V,.
Bai
toan 33 (De nghi Olympic
30/04/2011).
Gidi
phuang trmh
9x2 + (ix + 3126
x^ - 2x = 1 +^/^ + in ^,^3^^^ . , ^ • •

Htfofng dan. PhiJdng
trinh
viet lai
'• '
'' "' <f
1
- J ''
/(x) = /(A/S^TT), vdi /(i) = + i + ln(^« +
3125).
Tac6/'(i) = 3i2 + i +
-^J^i_,
ma
g«l.>lifii|
5(t*^
+ 3125) = Si** + 5*^ >
6^56^30
= 30
+
3125
<
1
nen ham c6 /'(<)> 0, suy ra ham / dong bien
tren
E. -i •
Bai
toan 34 (De nghi cho ki thi hoc
sinh
gioi cac tru'dng Chuyen
khu
vyc Duyen Hai va Dong Bang Bac Bo nam 2010).

Gidi
phudng
trinh
(6^' - 3^) (19-^- - 5^) (10^ - r) + (15^ - 8^) (9^ - 4^') (5^ - 2^) = 23F. (1)
Giai.
Ta c6 cac nhan xet sau :
Nhan
xet 1. Vdi a > 6 > c> 1 thi > 6^ neu x > 0 va < 6^ neu x < 0.
Nhan
xet 2. Vdi a > 6 > 0 cho
tritdc
thi ham so /(x) = - 6^ xac djnh
dong bien va lien tuc
tren
tap D = fO;
+00)
do
/'(x) =
a"" In
a - 6^
in 6
> 0, Vx > 0. '' '
Nhan
xet 3. Tich hai ham so dong bien, nhan gia tri 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 thi 's.ji. - •< 'i t ,
(6^ - 3^) (19x - 5^) (10^ - 7^) + (15^ - 8^) (9^ - 4^) (5^ - 2^) < 0,
25
trong khi 231'^
> 0,
auy ra
phitdng
trinh
khong
c6
nghi^m khong ditdng.
V6i
X
>
0,
chia
hai
vg
phitdng
trinh
cho 23F
=
(3.7.11)^
dudc
+

iv
)
(2)
Goi
y la
hani
so d ve
trai
ciia
(2). Tit
nhan
xet
2
va
nhan
xet
3,
suy ra
y
dong
bien tren
D =
(0; +oo)
va
5(1)
=
(2-1)({^
9
_ 4\5 _ 2\
7

7) [3 3)
_
14 3 7_b3_
11
7 ^
11'73
~ •
Vay
(1)
^
g{x)
=
y{\)
o x =
1. Suy ra
x =
1 la
nghiem
duy
nhat ciia (1).'"'I
1.4
Mot so
phep
dat
an
phu
cd
ban khi
giai
he

phufdng
trinh.
Co
rat
nhieu
each
dat an
phu khi giai
he
phitdng
trinh.
Dat an
phii
nhit
the
nao
con tuy
thuoc
vao
tifng
he
phitdng
trinh
cu
the.
Bai nay
se
neu ra mot
so
phcp

dat an phu cd
ban, thit6ng
gap. Nam
ditdc
cac
phep
dat nay ta
se
CO
dinh hudng
tot hdn
khi giai
he
phitdng
trinh.
1.4.1
Phep
dat
« = x +
y,
v
—
x - y.
Khi
do
u
+ v = 2x
9
9
uv

= X - y
+
=
2x (x"
+
10x2y2
+
5y4)
u-v
= 2y
+ 2
=
2 (x2
+
y2)
u^-v''
= 2y (3x2 ^ ^2)
u'-v'
= 8xy
(x2
+
y^)
- v'^
= 2y
(5.7;'*
+ 5x2y2 ^
^^4^
Bai
toan
35.

Giai
he
phiCOng
trinh
2x
+
2x2
_2y2 = 7
2
(.r2
+
y2)
^ 5
Hvfdng din.
Dat u
= x +
y,
v = x - y.
Khi
do
u
+
v
=
2x,uv
= x2 - 2/2^^2
^.
^2 ^
2{x'^
+

xf).
Thay
vao he ta
ditdc
he doi
xiing loai
1
doi vdi
u
va
v. {
""2"^^
t ^
26
Bai
toan
36.
Gidi
he
phitang trinh
| ^ J ^2 _ ^_
^^^^
Hu'o'ng din.
Dat
u
=^
x + y, v = x -
y-
Khi
do

u
+-V =
2x,
uv
=
x2
-
y2, ^2
+ 1-2 =
2(x2
+
y^).
Thay
vao he ta
ditdc
|
^f+^;^
I
Tit (1) suy ra
u
thav
vao (2)
dUdc
:
4r^
- ,-2
3
4v'^
-
t;2

+
3^;2
= 4v^
4v^
+
8t'2
- 12u = 0
{4V^
+
8v
-12)
=0<^v {v
- 1)
(4i;2
_^
4,^ _^
^2) = 0
V
= 0
V
=
1.
Bai
toan
37
(Dl
nghi Olympic
30/04/2011).
Gidi
he

phxMng
trinh
4431
.,. ,
(*)
Giai.
Dieu kien
x 7^ 0
va
y 0.
Dat
u = x + y, v = x - y.
Khi
do
M- +
V
W.
- I
X
= —y =
2 " 2
1
1
4y
2x
2
{u
-
-t)) z(,
+

w
2 (^2
-
7/2)'
Thay
vao he
(*)
ta
du'dc
uv
(^2
+
t'2)
^
u
+
01;
[uvY
= -5
=»7/7;
{u^
-
v^)
=
u
-
{uvfv.
(3)
Neu
u = 0

thi
y =
T,
the vao
phitdng
trinh
thit
hai ciia
he
thay khong
thoa
man.
Vay xet u
0.
Tit
(*)
ta
c6
V
{u^
-
(/)
= 1 -
uS;"
<=>
u'^v{l
+
v'')
=
1

+
If'
^
Khi
1 + = 0
ta
CO
v;
= -1,
suy ra u
=
v^S.
Vay
s/5
-
1
V5
+ 1
1
+ = 0
u'^v
= 1.
X
=
y
=
27
Khi
7/.^?; = 1 ta c6
1'

=
—r,
thay
vao {v.ny = -5 ta
ditdc
1
5
V-5
Do
do X = —-—z ) y =
(x;
y)
=
2
' " 2
,
(x;y)
=

Cac
nghiem ciia
h§
1^
'^f"
•i>
(SI ofiv
"/,fii!
Bai
toan
38. he

phuang
trinh
| ^3 :t
|^2^
?!
17^
5.
Vi.
Ui*o
Hifdfng dan.
Dat u = x + y, v = x - y.
Khi
do
Thay
\ao he ta
dvtdc
{ JJ3 | J^3 ^ 35
Bai
toan
39.
Giai
he
phUOng
trinh
| ^2 3y2^_""i^
Hu'dng dan.
Cach
1. Dat Dat u = x + y, v = x - y.
Khi
do

'*
' \ + - UV = 1 ^
<^u^ + = .,j3 _ ,^3
_^
2ur2 - 2u^u <^ 2v^ + 2u^v - 2uv^ = 0
r
^'
= 0 2 r u = 0
<^t; (t;^
- ur +
u^)
=
0 _|.
"j'^
4.
^ =
0
{"-0
Cach
2.
Dua
ve
phildng
trinh
ding
cap bac ba
doi vdi
x va y.
1.4.2
Phep

dat u = u ^
Khi
do
x + l
u
+
u
= r +
y
+1
22:y
- 2
x+1
y+1
(x+l)(y+l)
28
u-v=
^~
^
_ 2/- 1 = 2 (x - y)
x+1
y+1 (x +
1) (y+1)
x-ly-1
xy-(x
+ y) + l
uv
—
x+l'y
+

1 (x+1) (y+1)
2xy
+ 2
,
_ xy - (x + y) + 1
'^""^
(x +
l)(y
+ l) (x +
l)(y
+ l)
\-uv^\
^2/
- (x + y) + 1 ^ 2 (x + y)
(x+1)
(y+1)
(x + 1)
(y+1)
,.
"
- ^ _ X - y ,
x,,ff,ij
11
1
—
uz)
X + y ,
x-y
^r:^g''\-^i-iu W.a
1

- xy 3 - X
3^
+ y ^ 1 - 2y
'.\>
giib-Jii-i
1
+ xy 2 - y
• Jiftwrfs
,ijtjt.3,i
VB
Bai
toan
40.
Giai
he
phuong
trinh
<
Hu'dng dan.
Dtit
x = ^^—4- V =
-—r- Khi
do
x-y
u+1
u
—
V X + y
t;
+ 1

uv —
\
I
- xy u + v' I + xy uv
-\-1
3u
- 3
1
-3x
3-x
Ta
thu
dUdc
he
1
-
3-
u+\ 4-2^ 1 -2y
"
-
1
2it
+ 4' 2-y
u
+ 1
1
-
2w
- 2
v+l

^-v
2
-
V
- 1
V
+ 1
u
+ 3'
I
u
- V _ 4 - 2u
u
+
•()
~
2u
+ 4
uv
—
\
— V
kb
.QJTi
\.
uv +1 7' + 3
^1
2^2 +
4u
-

2uv
- 4v =
Au
+
4(;
-
2u'^
-
2uv
I
uv^
-v + 3uv - 3 =
3uz;
+ 3 -
uv'^
- v
^
/ 4u2 -
8u
= 0 ^
^\2ut;2
= 6 ^
/
tx2 = 2z;
I
= 3.
1-4.3
Phep
dat u ^ x + v = ij +
X y

Khi
do = +
1-
+ 2,
v^
= r/ + \ 2 vk
1
"
+
w
= (x + y)
1
+ — )
u'-*
+ =
(x^
+ y'
\ '
+
4
29
uv
= xy
-\
xy
xy
n
_ y
x'^
+ 1

V
x'
y'^
+ 1'
Bai
toan 41. Giai
he
phudng trinh
•
/
1 A
{x
+ y)
1
+ —
\
1 x2 +
xy
+ h —
xy
xy
=
4
=
4.
=
4
Hifcing dan.
Dat u = x + i, u = y + ^, ta
tlm dudc

he { ^4.
Bai
toan 42. Giaijie phmng Irinh
| ^
^2) _^ 3.2^2^
^
208x2y
Hvfdng dan.
De
thay (x;
y) =
(0;
0) la
nghieni cua h§.
Tiep
tlieo
xet xy 0.
Ho
titdng
ditrmg
vrJi
1 \
= 208.
Dat
u = X + ^, v^y^-K ta
tlm
diOTc
he |
"2"t|.\,2
^212.

1
y
(x2
+ 1) = 2x (y2 + l)
Bai
toan 43. Gidi
he
phMng trinh
\ .2 _^ 2^ ^ ^ = 16.
Uvtdng dan. Dieu kieu
xy ^ 0.
Dat
u = x + -
,
7; = y + ^, ta
tlm ditcfc
he
^
y
+ i;2
=
20.
Bai
toan 44. Gidi
he
phmng trinh
=
1
r
2x'^y

+
y^x
+
2y 4-
x = Qxy
Bai
toan 45. Gidi
he
phUdng trinh
I J_ +
4_
E = 4.
I
xy X y
Hifdng dan.
He
phimng
tiiiih
tucing diMiig
/
1
X
+ -
2
(x+ -
x)
(
y
+ -
V

V/
=
4
=
6.
30
1 4.4
Phep
dat a — x + i, v = y + -•
y
• X
Klii
do
u
+
t)
= (x + y) fl + —)
uv
= xy
H
1-2
xy
1^-11
=
(x-y)H
V
xy
V
xyy
f

(x + y)
1
\
1
+ — =4
V
xyy
Bai
toan 46. Gidi
he
phiMng irinh
\
xy
+ — = 2 /
^-
xy
Hifctng dan.
Dat u = x +
-i,
v = y + -
,
ta
thu dudc
he | ",+
2'r
i
y
X • ut — 4t.
Bai
toan 47.

G?d?
he
phuong trinh
V
xyJ 4
Hu-dng dan. Dat
u = x + ^,
i;
= y + ^, ta
thu dUdc
he |
4("2"^_j_^]2y
f
Bai
toan 48. Gidi he. phudng trinh
45.
/ •! •2\ 1 \ 25
Hifdng dan. Dat
u = x + J, ^ =
y
+ ^, ta
thu dudc
he | +
JJ^j
I ^35
{
xy (2x
+
y
- 6) + y +

2x
= 0
^^2 _|_ y2^ ^2 _)_
^ _ g
Hifdng din.
Dat u = x + -
,
v = y + -
,
ta
tlm dUdc
he
y
X
2u
+
w
= 6
u2
+
t;2
^ 8.
1-4.5
Mot so
phep
dat §n phu
khac.
\
Cac phep
dat fiu

phu
rat da
daug
va
phong
phii.
Ta
can
kliai
thac
cac dac
fiipm
rieng,
cac
tinh
chat dilc biet cua
tftng
he
phUdng
trinh
de
dita
ra
phep
<5§.t
phii
hdp.
f.
31
Bai toan

50.
Gidi
he phUdng trinh
Giai. Bien
d6i he da
rho,
ta
tlni ditrJc
+
x/ - xy{x + y) = 3
x^
+ xy{x + y) = 15.
do.
ta CO
x-'^
+ = 9 >^
<
. .
xy{M
'+y) = 6 • = 2
y
= 3
X
= 1
y
= 2
.X
= 2
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
1
2x
+ y + - =
y
1
Giai. Dieu kieii
y
7^
0,
x + - >
0,

x +
y
>
3.
Dat
a
=
^x+i, 6
=
v^x
+ y -
3, a, 6
> 0.
He
da
oho viet
lai la {
"J'^^t
5
•
V6i a = 2 va 6 = 1, ta c6
xJr-
'= 2 va Jx + y-i =
l<!=>x+-=4vax +
y
= 4
y
V
X
+ = 4

a
= 2 va 6 = 1
a
= 1 va 6 = 2.
1
4
- X
[
y = 4 - X
Vdi
a = 1 va
6
= 2, ta c6
f
x2-8x
+ 15 = 0
<^
x^4 <^
I
y=4-x
X
= 3 va y — I
X
= 5 va
?y
=
-1.
X•
+
-

=
1
va Jx
+
y
-
3
=
2<s^x
+ - =
lvax
+
y
= 7
j x+-^ = l ^ r
x2-8x
+ 6 = 0,x^7
ly=?=J
^ly=7-x
32
x
= 4- y/W va y = 3 + /lO
X
= 4 + 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
x2
+
y2
_ 1
+
=
1
V
2x
x'
+ i/ =4
y
Giai. Difiu kien:

xy ^
0, x^
+
y^
7^
1. Dat a = x^ +
y^
-
1, 6
=
06
7^
0. He
da cho
trd
thanh
a-2b
=3
3
2 _
26
+ 3 6 ~
rt
= 26
+ 3
6=-lvaa
= l
h
= 3 vh a = 9
• V6i

a= l,h=
-1,
ta c6 x^ +
y^
= 2 va x = ~y, ta
tini ditdc
hai
nghiem
la
(x;y)
=
(l;-l),(x;y)
=
(-l;l).
^ ,
Vdi
n = 9,b = 3, ta c6 x^ +
y^
= 10 va x =
3y,
ta
tini dUdc
hai
nghiem
la
(.x;y)=:(3;l),(x;y)
=
(-3;-l).
Thii
lai ta deu

thay thoa man.
Vay he
dii cho
co
4
nghiem phan biet
la
(x;y)
=
(l;-l),(x;y)==(-l;l), (x;y)=
(3;l),(x;y)
=
(-3;
-
1).
,,,,
'
x^ +
y2
+ x^ = 4y - 1
Bai toan
53.
Gidi
he
phuang trinh
x
+ y =
x2
+ l
+

2
Giai. Nhan thay
he
khong c6 nghiem dang (x;
0).
Gia siit
y
7^
0. He (*)
tUdng
^^
+ x + y = 4 f x' + l
y
Dat ^ ^ = -y- Ta
ditdc
x
+ y =
—r,
+2.
I
i; = X + y.
duong
x^
+ 1
x2
+ l
=
1
Giai he nay
ta

thu ditdc
u =
1,
u =
3.
Do
do
l
X + y = 3
Vay
he c6
nghiem (x; y)
=
(1;
2), (x;
y) =
(-2; 5).
33
Bai
toan 54.
Gidi
h$ phuang
trlnh
2x + —!— = 3.
x + y
Giai.
Dieu kien x + y 7^ 0. He viet lai £ /Of v
=
7
ill

4<
3(x2+y2)
+ (x2-y2) +
(.r + yY
=
7
X + 2/ +
x + g
+
x - y = 3.
(1)
1
Dat
u = X + y +
_^
^ ^, = x - y, dieu kien u > 2. Thay vao (1), ta ditoc
{ a + 6 =^3""
^^^"i^^
^) = 1)' '^"y '•'^ (^; 2/) = (1; 0).
Bai
toan 55.
Crdi
he pliMng
trlnh
sau ( + '^-V
^^^J'
" 2 Si!
\ + 3x^y - + y^ =0. (2)
Giai.
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 + - +y = 3
x2 + ^ + 3y = 5
x''
(x+^)+y = 3
Dat
2 = X + -, ta thu ditdc he
y\) =
(4;-l)
y;z)
=
\l-2).
/
f +
y = 3
I
z' + y = b
khi
(y; z) = (4; -1) : I ^ ; I ^ _1 ^ { ^ 4 ^ 0 (vo nghiem).
Khi(y;z)
=
(l;2),tac6{;;;L2
^ { + I = 0 ^ { y = }.
Vay he c6 hai nghiem (x; y) = (0; 0), (x; y) = (1; 1).
Bai
toan 56.
Gidi

he phuang tfinh | +
^^^2^)
= ^
Giai.
Do x^ + x/ = I nen ton Lai a e [O; 360"] sao cho x = sinn, y =
coso.
Do do (1)
trcl
tlianh
\/2(sina
-
co.sa)(l
+
2
sin
2a)
= \/3
34
<^v/2.y2sin(rv -
45").2
Q +sin2a^ = ^3
<^4sin(a - 45")(sin2a + sin30") = \/3 f
^f^
I'^i •;
<t=>8sin(rv
- 45"). sin(a + 15")
cos(a
- 15") = \/3
<:^4cos(a
-

15")[cos60"
-
cos(2a
- 30")] = ^fi fti:^^i*i,3>
«i.2cos(a
- 15") -
4COS(Q
- 15").COS(2Q - 30") = N/3
""'^'^
.^.7^;
^ - 2
cos(3a
- 45") = ^3 ^ [ I
^^^^^^Q"
^ ^ ' '
Vay lie da clu) CO nghiem v*/ »» • .m^. i>^'
iifii-f.
(sin65";
cos65"),
(sin 185"; cos 185"), (siii305"; cos 305"),
(sin85";
cos85"),
(sin205"; cos 205"), (sin325";
cos.325").
. •„
•
^,
,
1.5 Phu'dng phap cong, phifdng phap the. - |
Day la phitdng phap cd ban nhat. Tit bai hoc "vfl long" ve he phudng

trinli
da
CO
phitdng phap nay. Tuy nhien phitdng phap nay van
thilclng
xuat hien ci
nhrtng
ky thi Idn, nhifng ky thi chi danh cho
nhitng
hoc sinh xuat s;lc.
Sach
nao viet vc he phitdng
tiinh
cung c6 pluldng phap nay, do vay sau day ta chi
trinh
bay mot so bai toan kho va di sau
lidn
vao
viec
phan
tich
ky
thuat
giai
cung
nhif
ky
thuat
sang tac.
Vi

du 1. Xuat phdt tit mot bien doi tiMng dudng do ta chon '"^ ,
(x - 2)'* = (y + 3)^ ^ x'^ - 6x^ + 12x = y'* + 9y^ + 27y + 35. (1)
Khi
(x; y) = (3; -2) thi (1) dung. Cung vdi (x; y) = (3; -2) thi ,
x-'-y^ = 35. -nf. ,4. (2)
Tf(
(1) vd (2) fa dmr 2x2 ^ 3,^2 ^ 4^,, (3)
Tii
(2) vd (3) /.a c6 bdi toan. sau.
Bai
toan 57. Gidt he phuang
trinh
|
l[2f~^yi
_ cjy. (2)
Giai.
Nhan hai ve ciia (2) vdi -3 roi
cong
vdi (1), ta dUdc
x^ -x/ - 6x^ - 9y2 = 35 - 12x + 27y
- 6.r2 + 12.T - 8 = + 9.1/2 + 27?/ + 27 •
^ (x - 2)^ = (y + 3)^ X - 2 = y + 3 ^ X = y + 5.
Thay vao
(2) ta
dUdc
»' ~
2 (y
+
5)'-^
+ = 4

{y
+
5)
-
9y 5(/^
+
25y
+ 30 = 0
<^
Nghiem cua he
la
(x; y)
= (3;
-2), {x; y)
=
(2;
-3).
Lvfu
y.
Trong 15i giai tren, quan trong nhat
la
biet nhan hai ve phitdng
trinh
(2) vdi
-3
roi
cong
vdi phiTdng
trinh
(1), tai

sao
phai la so
-3
ma khong phai
la
so
khac,
tai sao
lai
cong
vdi phudng
trinh
(1)
ma lai khong
trit
?
Dicu
bi
mat
do
nam
d
"phudng phap
gia
dinh" (hay con gpi
la
phudng phap
he s6
bat dinh) nhu sau:
Xet (1) +

Q.(2)
ta
ditdc
, , -
-
/ +
2ax^
+
3a?/^
=
35
+
4ax
-
9ay.
T\i (3)
ta
chon
a, a,b sao cho
(3)
+ 2ax^
-
4ax
- +
3ay'^
+ 9ay - 35 = (x - af - (y - bf
(
2a= -3a
a =-3
-4Q

=
3o2
3a
= 3b
9a
=
-362
L
-35 =
63_a3
a
= 2
6=
-3.
Di
theo
hudng nay
ta
cung tim dUdc Idi giai
: Tii (3) ta c6
x^
+
2ax^
-
4ax
= i/ -
3ay'^
- 9ay + 35
3
, r,„2

-^x^
+
2ax^
- 4ax -
2'^
=
y-^
- 3m/ - 9ay +
3^.
Vay
ta can
chon
a sao cho
2nx'^
=
-3.x^.2
=^
a = -3.
Qua day thay rang
nhiing
he c6
chiia
cac
hang
tiit
x^, x^,
x va
y^, y^, y
ta c6
the diing

he so bat
dinh
de du:a ve
cac
hSng
dang thulc. Chu
y
rang
viec
xet
(l)+a.(2)
van khong
giam t6ng quat hdn
so
vdi
viec
xet ft.{l)
+
a.(2),
vi khi giai phudng
trinh
ta
CO
quyen
chia
ca
hai ve cho mot
so
khac
0.

Vi
du 2.
Xuat
phdt
tit mot
bien
doi
tMng diCdng
do ta
chon
ix-2f =={y
+ lf -6x^ +
12x
=
y^-\-3y^
+
3y
+
9.
(1)
Khi (x; y)
=
(2;
-1)
thi
(1)
dung.
Cung
v6i
(x;y)

=
(2;
-1) thi
x^-y^
= 9. (2)
Ti( (1)
va (2) ta
dicac
2x2
+
^^2
_
4^
^
Q
Ta (2)
va (3) ta cd bai
todn
sau.
Bai
toan
58 (De thi
chinh
thiJc
Olympic
30/04/2012).
Giai
he
f
x^-y^=9

\2
+
y2
-
4x
+
y
= 0.
36
Vi
du 3.
Xuat
phdt
tii mot
bien
doi
tUOng
ductng
do ta chon
{x-2)'
= {y-At '
^x^
- 8x^ +
24x2
- 32x =
y^
-
16y^
+
96y2

- 256y + 240. (1)
Khi
(x; y) = (4; 2) thi (1)
d,ung,
vdi (x; y) = (4; 2) thi x^ -
y^
- 240. (2)
Tii
(l) va (2) ta
ducic
- Sx^
+
24x2
_ ^
_;igy3
_^
9gy2
_
256j^
'
^x'^
-
2y3
= 3 (x2 -
4y2)
- 4
(x
-
8y).
(3)

Tii
(2) va (3) ta c6 bai
todn
sau. ' ' '
Bai
toan
59
(HSG
Quoc
gia-2010).
Giai
he
phuang trinh
/
x4-y4
= 240 (z)
\3 - 2y3
= 3 (x2 -
4y2)
-
4 (x
-
8y).
{li)
Giai.
Nhan phudng
trinh
(ii) vdi
-8
roi

cong
vdi phudng
trinh
(i) ta
ditdc
x"*
-
y4
-
8x3
+
16y^
=
240
-
24x2
+
96y2
+
32x
-
256y
•^x'*
-
Sx^
+
24x2
_
32^
+ 16 =

y''
- m/ +
9Q>if
-
256y
+ 256
^(x-2)^
=
(y-4)^^[-2z|:J
^[%Zr-l
• Khi
y =
X
+ 2,
thay vao phudng
trinh
thii
nhat ciia
he ta
dUdc:
f'
<<
x^
-
(x^
+
8x3
^
24x2
+

32x
+ 16) = 240
<^x3
+
3x2
+
4a,>
-I-
32
= 0
<!:^
(x
+
4)
(x2 - X +
8)
x = -4.
• Khi
y = 6 -
X,
thay vao phitdng
trinh
thi'r nhat cua ho
ta
ditdc
' '
x*
-
(1296
-

864x
+
216x2
_
24^3
^
^4^
^ 240
<^x3
-
9x2
-|_
36a;
_ 64 = 0
<t:»
(x -
4)
(x2 -
5x
+
16)
= 0
<^
X = 4.
Vay
he da
cho chi
c6
hai nghiem
la

(4; 2)
va (-4; -2).
LuTu
y.
Vi
sao lai
nhan (ii) vdi
-8
roi
cong
vdi
(i) ?
Dicu nay ditdc
li
giai
titdng
tir nhvt luu
y d
ngay
sau Idi giai bai toan
57 d
trang
35.
Mot dieu
lvfu
y
nita
la
mot
so

thay
giao
cham
thi
d ky
thi HSG
quoc
gia
nam
2010 cho
bigt,
CO
rat
nhieu thi sinh
"bo
tay" trudc
bai
nay,
do
khong
xac
dinh dung
plntdng
phap, do sii; dung "dao to biia Idn", nhftng
cong
cu qua manh
(chang
ban nhit dao ham), nhflng
cong
cu

manh
do
khdng thich hdp vdi
bai
toan
"ay. Con ve phan
sang
tac
dc toan, khoug biet ngitdi thiet
kc
ra bai toan nay
lam nhit
the
nao
? c6
tildng tit nhit
each
sang
tac da
trinh
bay d
vi du
3 d
trang
37
hay khong
?
37
Nhaii
xet 1. Ddii liieii. dO

diCa
ve
Jiang
ddiuj tlnic la timi.y kc cd
chda
./•^:t.
// ,;:•* ± //•',.,:•' ±
Vi
du 4. XadI
pfiat
li!.
tnoi
bihi
do/.
tUdnf/
dudnij do fa
chon
• ••
(v + 5f + (a - :if - 0
•(/•*
+ -
9v'^
+ 15o' = -98 - 27u - 75v. (1)
f
Kill
((/.: r) -5) //;/ (1)
dmui.
Cfmcj
v&i [u: o)
•=

(3; -5) th'i
""""" = -98. (2)
Tif
(1) vh (2) ta diMc Su^ - 'Trv- - + 25o. (3)
Dal 11
—
.i: I //. = J- - 1/
{Ill/if
ph.rji d/ri hirn, ro' hdii),
iJiay
vdo (2), (3) d,iM(:
• \-
h-ii^
= -49
./•- - 8x/y + - 8y - 17x.
7);, rci bdi U>uii saa
Bai
toan
60 (HSG Quoc Gia nam
2004,
bang B). Cidi lir phiMiui trhili
•r'*
4 3.,/y- = 49
(1)
.(•-
- 8ry + y~ = Sy -
17.;:.
(2)
Giai.
Cach

1. Dat
% ^-'y— [! => ~ " ,^ ' . /; - '') • Tliay vao (1). ta dudc
u + I
n - V
(v + vf
-(•
3((/ -f- r) ((/ - v)- -= -392
^ti-^
+ :hi'r -I 3(/,//- + (••'» f 3
(/»••'
- (/-^u -
(.''-'»
+
);••')
= -392
^4 (ir^ +
!'•'')
^. 392
//,••*
-I-
= 98.
Thav vao (2), ta ihuk-
iv {-n>f - 8 ((/'^ - r^) + (,/ - vf = 16 (,/ - (•) - 34 (v + v)
-^lOr'
Cm- = 18u oOr O 3(/- oc' r= 9a+ 25r.
Ta CO lu)
iiK'ii
I [[.^/^-^.^
^'Jj],
^25r. (4)

1^^"^'"'^
""'1'
^'''i
r6i
coiig V(n
jjlurdng
trhili
(3). ta clu'dc
-f (;•' - 9(/;' + 15(/- = -98 - 277; - 75(;
^a-^ - da' + 27u - 27 = - (r'^ + 15r- + 7ov + 125)
[r
+ 5)'' = (3 - 44> 0 + 5 = 3 -
i/
t» = -u - 2.
38
Thay van (3) difdc vr* - (u + 2)^ = -98 <4. (i?/'-^ + 12« - 90 = ()<!=>
)i
= 3
lA
= —5.
Vay
(u;t')
= (3:-5
((/.;
= (-5: 3
=
(-1;4)
'.•;y)
= (-i;-4).
Ngliiem

c.ua he l;i
(x;/y)
=
(-1;4),
(;r; =
(-1:-4).
s
Cach
2. Nlian phiWng
triiili
(2) vdi 3 loi cpng v6i plutciiig
tiiiih
(1). ta dudc
+
3xy^ + 49 + 3x'^ - 24xy + [h/ - 2Ay + 51x = 0
<^(,T
+ l)(x^ + 2x 4- 3?/^ -
24?y
+ 49) = 0
^(:r+
l)[(.,;
+
l)'-^
+ 3(y-4)'^] =0
.r
+ 1 = 0
.r
= -1 va y = 4
.r
+ 1 = 0 va

;//
- 4 = 0
•
'
• '
>
• Neu x = -1
thay
vao phitdiig tnuh thi't iihat cua he
dUcJe
x/ = 16 y = ±4.
• Neu X = -l.y = 4 thay vao he thay thoa. • , \
Vay he
i^hitdng
trinli
da cho c6
iiglnciii
{x. ij) la ( 1.4), (-1. 4).
Lifu
y. Cach
giai
2 rat
ugaii
goii
va de
hiCni,
nliitug
dang
sail su
iigan

gpii
de
hiCu
dp ta da phai rat vat va de tim ra
IcJi
giai nhit sau : DP phirpug triidi
thti' nhiU cua he c6 hac cao uliat iicii ta .se de nguyen, nhaii philPng trinh thit
hai
cho n roi
cpiig
vc3i
pliitPiig
trinh
thif
iihat
ta ditdc : •-' . .
+
3xy^ + 49 + n{x^ - 8xy + if + 17x - 8y) = 0.
(*)
Mat khac ta c6 thfi nhaiii
ditrtc
nghiem ciia he la (x; y) =
(-1;4),
do do ta
nioug inuoii (1) phan ti'ch dUdc thauh :
(x +
l)(rtx^
+ hx + cy^ + dy + 49) = 0 (he so ciia xy
trong
ngpac bang 0)

<^nx'^ + bx^ +
cxy"^
+ dxy + 49x + ax^ + bx + ry^ + dy + 49 = 0
<^ax^ + (o + b)x'^ +
CXA/
+ dxy + cy^ + (6 + 49)x + dy + 49 = 0.
Tit
(*) va (**) doug uhc\ he so ta ditdc :
(**)
a = 1
a + h = a.
C — 6
d= -8a
c = a
b+A9^
Ha
d = -8a
I
c = a = 3
^ \ = 2
[ d = -24.
Bai
toan
61. Gidi hf. pfixMng trinh
Gx^y + 2f + 35 = 0
5x'-^+5y^+2xy + 5x+13y = 0. (2)
(1)
39
Giai.
Dat

suy
ra x = y = Thay vao (1). ta duac
•V
6.i!i±i:l!.!i^
4-2^^ + 35 = 0
4 • 2
^6
{u^ + 2uv + v^) (u -v) + 2 (u'^ -
Su'-^'i;
+ Sww^ - t^^) + 280 = 0
44-3
(?/^ -
i/,^r
+ 2v'^v - 2uv^ +
77,7;^
- v^) + -
37/,''^v
+
377,7;^
- 7/' + 140 = 0
<^-(/;'*
- + 35 = 0.
Titdrig
tut, thay vao (2), ta dufdc
77 - 7'
77, + 71
XI.
— V
U
+ V

U
- V
<^107x^
+
IOT;'^
+ 2 (7?
•••
7;^) + lOit + lOv + 26u ~ 26L' = 0
+ 27'^ + 9?i - 47; = 0.
Ta CO
= - 35
(3)
3w^ + 97J = -2x? + 47;. (4J
Nhan (4) vdi 3 roi cong vdi (3), ta cd
7/^ +
977,2
_,_ 27,^ = ,,3 __ g,,2 ^ 247) - 35
^{u
+
3)'^
= (7. - 2)^ ^ u =
: 7; - 5.
Thay vao (3) ta ditdc
(7-
- 5)'^ - 7/^ + 35 = 0 <^ T;^ - 57; + 6 = 0 i' e {2,3}
V 2'^2j
• Vdi V = 3, siiy ra u = -2, ta cd { ^ + ^ 3 ^ ^ (x; ?;) = (^2' '
Ket
luan
: He cd hai nghiem la (x; y) =

^"^i
""^
-
(^5 v) = '
Lvfu y. Vi sao ta lai
nhiin
(4) vdi 3 roi cong vdi (3) ? Ta cd
Vdi
u = 2 ta cd u = -3, dan tdi | ^ t
j{
~ 2 <^ (x; y) =
u
+ A
(37i''^
+ 977) = - 35 + A
(-27;^
+ 47,')
(5)
+
SATi^
+ 9A7/ = - 2\v~ + 4At' - 35.
Ta can chon A sao cho (5) cd dang
: (7/ - of = (7) - hf ^ 77;^ - 3a77,2 + 3a277, - = 7'^ -
Zhxp-
+
3//'^7;
- \? (6)
I
-36=-2A
I

362 = 4A
( -3a = 3A
3fl2
= 9A r a = -c
So sanh (Gy va (5) suy ra { -36 = -2A ^ l h = 2
I
A = 3.
-35
40
g^i toan 62 (Olympic cac trtrdng Chuyen khu vi^c Duyen hai va
Dong
bang
Bac bo-2011). Gidi he phuang trinh |
^l~^2
I^"^
Bai
toan 63. Gidi he phuang trinh
Giai.
He viet lai \7
x2+y2^i
1
(1)
57
4^' + 3:r = -y(3x+l). (2)
(1)
4x2 +
3x +7/(3.7-
+1)=0.
(2)
.1

,
•
Nhan phifdng
trhih
(2) vdi 2 roi cong vdi phitdng
trinh
(1) ta dUdc
IF].
x2 +
7/2
+ 8x2 + 6x - ^ = A - _ 2y
<(4>9x2 + 6xy + 7/2 + 6x +
27/
- ^ = 0
25
(3x
+ y) V 2 (3x + y) -
119
25
Khi
y = - - 3x, thay vao (1) ta diTdc
5(3x + y) = 7
5(3x + y) = -17. ,,,,
X
/7-
15x\ _ 1
V 5 ;
Vay(.;rt=(?;i),(x;„ = (li;|^
• Khi y = - —- - 3x, thay vao (1) ta dUdc
5

x^ +
-17-15x\ 1
102 284
5 j " 5 T""^ + ^ ° nghi$m).
Heed
hai nghiem (x;y) = . (x;y) =
(^;^)-
Lifu
y. Vi sao nhan phifdng
trinh
(2) vdi 2 roi cpng vdi phitdng
trinh
(1) ?
^et (l) +
/;(.(2)
ta dudc
2 2 1\
5/
57
+
/3
4x^ + 3x-
—+y(3x
+ l)
\o
(1 + 4/?) x2 +
3/3xy
+ y^] + /3 (3x + y) - - -
1
57/?

25
= 0.
,.1,
= 0. '
(i)
41
Ta union cho (i) la niol phircJng
trinli
bac hai theo {3x +
y),
vay t.hi (i) phai
dUdc viet lai thanh
Tir (i) va (ri)
dong
nhat
he so ta
duoc
(ii)
"•:m
u'
Ta chu y rKng, nhfrng he phitdng
trinh
chvta hang tit x'^.xy,y'^ phan Idn c6
tho dira vfl phudng
trinh
bar hai theo ax + by.
Bai
toan 64. Gicit he phuang trinh / + +J^y^ + = ^ (1)
• ^ ^ \ + y^ +
iy+l

= 0. (2)
Giai.
Lay phUdng
trinih
(1) cong vdi plntdng
trinih
(2) nhan 2 ta ditdc :
x'^ + 4xy + 4y^ + 3x + 6y + 2 = 0
^{x
+ 2yf + 3(.x + 2y) + 2 = 0 (x + 2,y + l)(x + 2y + 2) = 0
".r
+ 2y + l = 0
X
+ 2y + 2 = 0.
Neu X +
2i/
+ 1 = 0 thi x = -2y - \, thay vao (2) ta dUdc :
-y^ + 2y + 1 = 0
jy
= 1 + v/2 =!> X = -3 - 2v/2
y = 1 - v/2 =^ X = -3 + 2\/2.
* N6n X + 2(/ + 2 = 0 tin x = -2y - 2, thay vao (2) ta dUdc :
1
+ ^5
-y^ +
;i;
+ 1 = 0
V
=
y =

2
1 - \/5
=^ X = -3 - V5
=> X = -3 + \/5.
He phitdng
trinh
da cho c6 4 nghicin (-3 - 2\/2; 1 + v/2); (-3 + 2v/2; 1 - \^);
-3 + —-— , -3 - v^5; —-— .
^ / \
J
Lvfu y. Tai sao lai lay
phUdng
trinh
(1) cong
vcii
phitdng
trinh
(2)
nhan
2 ?
Y
tUcing
la ta se
bieii
doi de dua ve
phitdng
trinh
bac hai theo mx + ny. D6
lam
difMi do ta

nhan
phudng
trinh
(1)
vc'ii
a va
phitdng
trinh
(2) vdi (3 roi
cong
lai:
a{x^ + 2xy + 2y^ + 3x) + (3{xy + + 3y + 1) = 0
<i=>a
4 2^ xy+(- + 2] 7/
Xa can chon a va /3 sao cho :
H.
•
f
.r^+('^+2).x,;+(^+2),y2=(x
+ ^?y
/
^x^
+ { ^ + 2 )
x(/
+ ( - + 2
?y'^
= x^ +
2^x:iy
+
J^/y^.

"
a2
D6ng nhat he so. ta ditdc
-••nil
D6 cho ddn giaii, ta chon a = \ i3 = 2.
Tiep thto ta st phdt tritn them ky thudt yiui dd duoc de cap J each 2 cua Idi
giai bai loan 60 d trang 38. ' '
Bai
toan 65. Gidi he
phUdng
trinh
/x4 +
2(3v/+l)x2
+ (52/2 + 4y + ll)x-y2+i0y + 2 = 0 (1)
\t/
+ (x - 2)2/ + x2 + X + 2 = 0. (2)
•
^-•
Giai.
Khi y = -1 tin (2)
trci
thanh x^ + 3 = 0, v6 nghieni. Vay he khong c6
nghiem dang (x; -1), do do c6 the gia si'l y + 1 7^ 0. Nhan phitdng
trinh
(2)
cho y + 1, loi lay phitdng
trinh
(1),
trit
phitdng

trinh
vita nhan ditdc, ta c6 :
(x + y)(x - y + 2)(x2 - 2x + y2 + 3y + 5) = 0. ^ , \
Vdi
X = -y. tliay vao (2) ditdc (y + 2)(y - 1)^ = 0 <^ y G
{-2,1}.
Vdi
X = y - 2, thay vao (2) ditdc (y -
\ + 4) = 0 <^ y £
{-4,1}.
De thay x^ - 2x + y^ + 3y + 5 = (x - l)^ + (y^ + 3y + 4) > 0.
Tint
lai ta thay he c6 nghiem (x, y) =
(-1,1),
(2, -2), -4). ,
Lu\
y. Day la bai toan kho, de co difdc IcJi giai ngfln gon
nhit
tren ta da
Pliai phan
tich,
tim Idi giai nhu sau :
Budc 1: Tim nghiem cua he. Neu biet ditdc nghieni thi y titdng ciia ta se
ro rang hdn
nhifiu.
Lan
lifdt
thit
x = -2,-1,0,1.2. 3, ta
tiin

dildc 2 nghieni
cuahela(x;y)
=
(-l;l),(2;-2).
Bu-dc 2: Tim quan he tuyen tinh gii?a 2 nghiem nay. De thay do la
y ~ -X hay X = - y.
Bu-dc 3: Thay vao he va phan tich thanh nhan tu". Ta thay x bdi -y
43
hoftc
?/
bcii
—X (tuy
trirdng
hop
xeni
each nao c6 loi), vdi bai nay ta
thay
y
=
—X
vao hai
phifdng
trhih
ctia
he va thu
clUdc
x'*
+ 2(-3x- + l)x^ + (5x-2 - 4x +
1
l)x - -

lOx-
+ 2 = 0
-x^
- (a; - 2)x + x2 + X + 2 = 0 : •
^
,,.,;
+l)2(x-l)(x-2)=
0
f(.x-
\(x
+
lYix
- 2) = 0.
Bai
Viec
phan
tich
tren
la
khong
kho vi ta da
biet tiudc nghiem
x =
-1,
x = 2.
Biidc
4.
Li.ta
chgn bieu thiJc
thi'ch

hdp.
Nhit
the, so vdi
phitdng
trinh
thii
nhat
vita
nhan diWc thi j)hirdng
trinh
tint
hai
thieu
di mot
bieu
thitc
la x - 1,
nhung
thu y
rftng
bieu
thiic
nay
cung tUdng dUdng
vdi -y - 1. Ta se
chon
mot trong
hai
bidu
thiic

nay de
nhan
vao. Ro
rang
neu
chon
-y - 1
thi
viec
nhan
vcii
(2) se tao ra mot da
thitc
c6
chita
y'^
dong
bac vdi x'* d
phudng
trinh
(1). Vay ta se
nhan phitdng
trinh
sau cho -y - I.
1
toan
66.
Gidi
he
phudng

trinh
|^2+^
+J Z {2)
Giai.
Xet x = 0, x = 2 ta
thay
he c6
nghiem
(x; y) = (0; 0), (x; y) = (2; 2).
Vdi
X ^ {0; 2}, xet
(l).[x(x
- 2)] + (2) ta c6
phitdng
trinh
x(x
-
2)(x''^
- 2xy +
X
+ (x"^ - ix^y + 3x^ +
y'^)
= 0
<^(x
-
y)(2x''*
- x^ + X - 2/) = 0.
• Khi
X =
thay

vao (1) : 2x - x^ = 0.
Tritdng
hdp nay
loai
do x ^ {0; 2}.
•
Neu
2x'^
- x^ + X - y = 0 ta
CO
he
phitdng
trinh
2x^
- x^ + X - ?/ = 0
x^
- Ixy + X + y = 0.
Cong
hai
phitdng
trinh
ta c6
2x^
- 2xy + 2x = 0
2x(x^
+
1
- y) = 0 x^ + 1 = y (do x 7^ 0).
Thay
vao (1)

taco
-
2x(x2
+ l)+x + x2 + l = 0-» -2x^ + 2x2 - X + 1 = 0 <^ X = 1.
Ttt
do
tim ditdc
(x; y) =
(1;
2).
Vay
he da cho c6
nghiem
(x; y) =
(1;
2),
(0;
0),
(2;
2).
Lvhi
y. Ldi
giai
tren
dildc
tiui
ra
nhil
sau : Dau
ticn

ta
thay
(l)<i^x2-2xy
+ x + y = 0.
Co
the
tim dUdc
3
nghiem nguyen ciia
he nay la (x; y) =
(0,0), (2;
2),
(1;
2).
•2x
= 0
-2)p = 0.
44
Tit
2
nghiem
dau ta
thay
x = y.
Thay
vao he ta
dudc
^%M2^_
Nhit
vay ta se

phai nhan
x{x - 2)
hoax-
y(y - 2) vao (1).
jyici
rong.
Nlnt
<ta
de cap,
neu
biet cang nhieu nghiem thi
ta c6
Idi
giai
cang
(iop.
Sau day
la
mot
cacli
phfui
tich
khi ta
biot ca
3
nghiem.
Ta .so Ian
litdt,
(y
= X

lap
3
quan
he
tuyen Mnh gnla
x va
y. Trong
bai nay, do la
|
y = 2x Do (2)
CO
bac ca.o
luJn
nen ta xet /
^iiunj
>s!stvijii<'(
,<«,>•
(l).tt
+ (2) O +
x'-^(-4.v
+ a + 3) + x{a - 2ay) + ay + y^ =
(),
(3)
Ta
,so
chon
(|uan ho nao de sir
dung nhat,
do
chinh

la y = 2. Nhu vay (3) c6
nghiem
y = 2.
nghia
la
^4 +
x2(a-5)+.r(-3a)
+
2a+4
= 0^o =
-^(,^.+i)(^^
+ 2)
Khi
do (3) iva
thanli
2.T''
+
x'^
+ x + y = 0. Ta
tiep
tuc
kliai
thac;
mot
trong
hai
(inan
he con
l;ii,
de y

rang
nhir
da
phan
tirh
d
tren,
vdi n = x{x - 2) ta
thu
ditdc
2;;;''
- x~ + x - y = 0. Vay ta di den
Idi
gitii
ng;ln
gon
nhu
sau :
Ldi
giai.
Thtt
vdi
x
G
A/ = {0; 2: 1; 2} ta
tim dUdc nghiem
(x;y)
=
(0;0).(2;2),(l;2).
m imi:

^
) ^
Xet
X ^ AI. ta cd
(l).(v
+ (2) y- + y{a - 2nx - 4x~) + ax +
3:'\a
+ 3) + x'' = 0.
Chon
a =
.;;(.;•
- 2)
la
cd
2x''^
- + x ~ y =
t).
Chon
a = + l)(x + 2)
ta,
cd
2j''*
+ + X + y = 0. r\
Cong lai
ta cd 4,r-* + 2x =
tJ
(sal do x ^ M). Vay
Iru-dng
hdj) nay
loai.

'
Tom
lai he cd
nghiem {x;
y) =
(t); {)).
(2: 2).
(1;
2) •
Bai
toan
67 (HSG
tinh
Ba Ria
Vung
Tau nam hoc
2010-2011).
Gidi
he phmmj l.rhih
/
+ V2^
= 4 y.7
\5
+ +
5
G.
Giai.
Dieu kien:
x >
I),

y > 0.
Cong
tftng
ve
hai
phuVing
trinh
ciia
he,
ta,
cd
[V2x
+ 5
4-
V^) +
(\/2y-f
5+ = 10. ' V
^ru
phitdng
iriiili
I
hit
hai cho
phU(Jng
trinh tint
nhal.
ve
thco
ve. ta
difde

(727/^13
v/27) + (/2?; + 5 - ^ 2
5 5
^
j - 2
v/27T5-f v/2y
+
5 -f
V%j ' . , K;
45