Phương pháp và kĩ thuật ôn nhanh thi đại học đạt điểm cao môn toán nguyễn phú khánh

510.76
PH561P

n c u Y i n P I I U Kiirinii

PHUONO PHAP
on niinnii TIII oni

HOC

D A T DIEM CAO


NGUYEN PHU KHANH

PHUONG PHAP
&
KI THUAT
on niinnii TIII oni

HOC

DAT DIEM CAO

\SP/

NHA XUAT BAN DAI HOC SlT PHAM


Phucmg phdp vd ki thudt on nhanh thi Dai hoc dat diem cao mon Todn giiip
hoc sinh on tap, van dung sang tao kien thuc, tong hop, phan tich, kiem tra

nang lye toan dien, hucfng dan chien thuat giai de thi dai hoc. Sach trinh bay
noi dung trong tam on thi dai hoc gbm:
- Phucmg trinh, bat phuong trinh, he phucmg trinh dai so. He phucmg trinh mil va Idgarit.
- Bai todn to'ng hap. Bat dang thuc; cue tri cua bieu thuc dai so.
- 8 phuang phdp giai phuang trinh lugng gidc nhanh nhat.
- Van de lien quan den sophiec, dai sotohffp, xdc sudi.
- Cdc bai todn lien quan den ung dung ciia dqo ham va do thi cua ham so: chieu bieh
thien cua ham so; cue tri; gid tri Ion nhat va nho nhat cua ham so; tiep tuyen, tiem can
(dimg vd ngang) cua do thi ham so; tint tren do thi nhirng diem c6 tinh chat cho truac,
luang giao giua hai do thi (mot trong hai do thi Id dubng thang)...
- Tim giai han.
- Tim nguyen ham, tinh tich phan. Ung dung cua tich phan: Tinh dien tich hinh
phang, the tich khdi trdn xoay.
- Hinh hoc khong gian (tong hap): quan he song song, quan he vuong goc cua dubng
thang, mat phdng; dien tich xung quanh cua hinh non trdn xoay, hinh tru trdn xoay; the
tich khdi lang tru, khdi chop, khdi non trdn xoay, khdi tru trdn xoay; tinh dien tich mat
can vd the tich khdi cdu.
- Phuang phdp toa do trong mat phang: Xdc dinh toa do cua diem, vecta. Duong
trdn, ba dudng conic. Viet phuang trinh dudng thdng. Tinh gdc; tinh khodng cdch tit
diem den dudng thang.
- Phuang phdp toa do trong khdng gian: Xdc dinh toa do cua diem, vecta. Duong
trdn, milt cdu. Viet phuang trinh mat phdng, dudng thdng. Tinh goc; tinh khoang cdch
tie diem den dudng thang, mat phdng; khodng cdch giita hai dubng thdng; vi tri tuang
ddi cua dubng thdng, mat phdng vd mat cdu.
Mac du tac gia da danh nhieu tam huyet cho cuon sach, nhung sai sot la dieu
kho tranh khoi, rat mong nhan duac sy phan bien va gop y quy bau ciia ban doc
de nhung Ian tai ban sau cuon sach dugc hoan thien hon.
Tac gia

3

CAU T R U C D E T H I DAi HQC M O N TOAN'
I. P H A N C H U N G (7 d i i m )
Cau 1 (2 diem):
a) Khao sat su bien thien va ve do thi cua ham so.
b) Cac bai toan lien quan den u n g dung cua dao ham va do thi cua ham so: chieu bien
thien ciia ham so; cue t r i ; gia trj idn nhat va nho nhat cua ham so; tiep tuyen, tiem can
(dung va ngang) cua do thi ham so; t i m tren do thi n h i i n g diem c6 tinh chat cho truac,
tuong giao giira hai do thi (mot trong hai do thi la d u o n g thang)...
Cau 2 (1 diem):
Cong thiic lugng giac, phuong trinh lugng giac.
Cau 3 (1 di£m):
Phuong trinh, bat p h u o n g trinh, he phuong trinh dai so.
Cau 4 (1 diem):
- T i m gioi han.
- Tim nguyen ham, tinh ti'ch phan.
- u n g dung ciia tich phan: tinh dien tich hinh phang, the tich khoi tron xoay.
Cau 5 (1 d i i m ) :
Hinh hoc khong gian (tong hop): quan he song song, quan he vuong goc cua d u o n g
thang, mat phang; dien tich xung quanh cua hinh non tron xoay, h i n h t r u tron xoay; the
ti'ch khoi lang t r u , khoi chop, khoi non tron xoay, khoi tru tron xoay; tinh dien tich mat
cau va the tich khoi cau.
Cau 6 (1 diem):
Bai toan tong hop.
I I . P H A N R I E N G (3 diem)
Thi sink chi dugc lam mot trong hai phan (phan A hoac phan B)
A. Theo chucmg trinh

Chuan:

Cau 7a (1 diem):
Phuong phap toa do trong mat phang:
- Xac d i n h tpa dp cua diem, vecto.
- Duong tron, elip.
- Viet phuong trinh d u o n g thSng.
- Tinh goc; tinh khoang each t u diem den d u o n g t h i n g .
Cau 8a (1 d i i m ) :
Phuong phap toa do trong khong gian:
- Xac d j n h toa dp cua diem, vecto.
- Duong tron, mat cau.
- Tinh goc; tinh khoang each t u diem den d u o n g thang, mat phang; khoang each giiia
hai duong thang; v i t r i tuong doi cua duong thang, mat phang va mat cau.
5


Cau 9a (1 diem):
- So phuc.
- To hop, xac suat, thong ke.
- Bat dang thiic; eye t r i cua bieu thiic dai so.
B. Theo chuang trinh Nang cao:
C a u 7b (1 d i i m ) :
Phuong phap tpa do trong mat phSng:
- Xac d i n h toa do cua diem, vecto.
- D u o n g tron, ba d u o n g conic.
- Viet phuong trinh d u o n g thang.
- Tinh goc; tinh khoang each t u diem den d u o n g th5ng.
C a u 8b (1 d i i m ) :
Phuong phap tpa do trong khong gian:
- Xac dinh tpa dp cua diem, vecto.

- D u o n g tron, mat cau.
- Viet phuong trinh mat phang, d u o n g thang.
- Tinh goc; tinh khoang each t u diem den duong thSng, mat p h i n g ; khoang each giua
hai duong thang; v i t r i tuong d o i cua d u o n g thang, mat phang va mat cau.
C a u 9b (1 diem):
- So phuc.
- Do thi ham phan thue h i i u t i dang y =

+ bx + e
px + q

quan.

- Su tiep xiie cua hai d u o n g cong.
- He p h u o n g trinh m u va iogarit.
- To hop, xac suat, thong ke.
- Bat dang thiie. Cue t r i eiia bieu thiic dai so.

PHlTdNG TRINH, BAT PHlTdNG TRINH, HE
I

PHlTOfNG TRINH

Phuong trinh, bat phuong trinh, he phuong trinh dai so. H( phuong trinh mil va Iogarit
M O T SO H E P H U O N G T R I N H C O BAN
1) He bac nhat hai an, ba an
2) He gbm mot phuong trinh bac nhat va phuong trinh bac cao

He ba p h u o n g trinh

-

He hai p h u o n g trinh

-

Phuong phap ehung: Su d u n g p h u o n g phap the

•

6


3) He doi xung loai 1
Phuong phap chung: Dat an phu a = x + y; b = xy .
4) He doi xung loai 2
Phuong phap chung: T r u tung vehai phuong trinh da cho nhau ta dugc: (x - y).f(x; y) = 0
5) He phuong trinh dang cap bac hai
Xet truong hop y = 0. V o l y ?t 0, ta c6 the tien hanh theo cac each sau:
- Dat an p h u y = t.x .
- Chia ca hai ve'cho y ^ , va dat t = —.
y
MOT SO P H U O N G PHAP G I A I H E P H U O N G T R I N H
1) Phuong phap the
• Phuong phap: Ta riit mot an (hay mot bieu thuc) t u mot p h u o n g trinh trong he va
the vao phuong trinh con lai.
• Nhqn dang: Phuong phap nay thuong hay su d u n g k h i trong he c6 mot phuong
trinh la bac nha't doi voi mot an nao do.
2) Phuong phap cong dai so
3) Phuong phap bien doi thanh tich

4) Phuong phap dat an p h u
5) Phuong phap ham so
6) Phuong phap su d u n g bat dang thuc
C/iM y: ting dung dao ham gidi todn phuong trinh, he phuong trinh
Su dung cac tinh chat ciia ham so de giai phuong trinh la dang toan kha quen thupc.
Ta thuong c6 ba huong ap dung sau day:
Huang 1: Thuc hien theo cac buoc:
Buac 1: Chuyen phuong t r i n h ve dang: f(x) = k .
Buac 2: Xet ham so y = f ( x ) .
Buac 3: Nhan xet:
•

V o i x = Xg <=> f(x) - f(xg) = k, do do X g la nghiem.

•

Voi X > Xg o f(x) > f(xQ) = k, do do phuong trinh v6 nghiem.

•

Voi X < Xg <=> f(x) < f(xQ) = k, do do phuong trinh v6 nghiem.

•

Vay X g la nghiem duy nhat a i a phuong trinh.

Huong 2: Thuc hien theo cac buoc:
Suae 1: Chuyen phuong trinh ve dang: f(x) = g(x).
Buoc 2: Dung lap luan khang d i n h rang f(x) va g(x) c6 n h u n g tinh chat trai ngugc
nhau va xac d i n h X g sao cho f ( X g ) = g(xg).

Buac 3: Vay Xg la nghiem d u y nhat ciia phuong trinh.
7


Huang 3: Thuc hien theo cac buac:
Buac 1: Chuyen p h u o n g trinh ve dang f(u) = f ( v ) .
Bwac 2: Xet ham so y = f ( x ) , dung lap luan khang d i n h ham so don dieu.
Buac 3: K h i do f(u) = f(v) o u = v .

Cac vi du|
V i du 1. Giai cac p h u o n g trinh sau tren tap so thuc:
1. V 5 x - l - 7 3 x + 13
3. ~ + . X
X V

3

=x+. 2x--

2.

V

+-=-2x-4 + X
x

4. 2 ' " ' + 2 ^ ' ' - 9 . 2 ' ' ' + 1 8 3 = 2^''*'+2''"+9.2^*"

X
Lcri giai

1. Dieu kien: x > — .

5

Phuong trinh da cho tuong d u o n g v o i

VSx - 1 - V3x+ 13 = -if VSx - 1 - V3x +13)(VSx - 1 + V3x + 1 3 ) .
Truang hap 1: V S x - l - %/3x + 13 = 0 o V s x - l = V s x T l S , binh phuong hai ve roi riit
gon ta dugc x = 7 (thoa man).
Trucrng hap 2: V S x - l + V3x + 13 = 6 (1)
Neu x > l thi V T ( l ) > >/4 + Vl6 = 6 , con neu x < l thi VT ( l ) < ^/4 + Vl6 = 6 .
De thay x = 1 la nghiem p h u o n g trinh (1).
Vay p h u o n g trinh ban dau c6 hai nghiem x = 1 , x = 7 .
2, Dieu kien: 1 + - ^ 0 o
x

X

> 0 o x < - 2 hoac x > 0 .

K h i do p h u o n g trinh da cho viet lai: x ^ l + - = -2x^ - 4x + 3 .
+ Vdi X > 0 , phuong trinh o > y x 2 + 2 x = - 2 ( x 2 + 2 x ) + 3 .
Dat: t = V x ^ + 2 x , t > 0 , ta c6 2t^ + 1 - 3 = 0 o t = 1 hoac t = - - , doi chieu dieu kien
ta dugc t = l , tuc la phai c6 Vx^ + 2 x = 1 o x^ + 2x = 1 , phuong trinh nay c6 nghiem
x = -l + \f2 thoa man dieu kien.
+ V d i X ^ - 2 , p h u o n g t r i n h o - V x ^ +2x = -2(x^ + 2x) + 3 .
Dat t = > / x ^ + 2 x , t > 0 , ta c6: 2t^ - 1 - 3 = 0 o t = | hoac t = - l , d o i chieu dieu kien
ta dugc t = ^ , tuc la phai c6 4(x^ + 2x) = 9 <=> 4x^ + 8x - 9 = 0 , phuong trinh nay c6
nghiem x = ~ ^

'^'^''^ man dieu kien.

8


Vay phuong trinh da cho c6 nghiem x =

-4-V52

; X =

-1

+

r/2 .

3.Dieu kien: x # 0 , x - i > 0 , 2 x - - > 0
X

X

Phuong trinh da cho dugc bien doi ve dang: x - — = . Ix - — - . p x - — (1)
X
V
X
V
X
A p dung cong thuc:

— - j = , v d i a > 0; b ^ 0; a ^ b va Va + Vb > 0 .
Va-vb
- X

Khi do ( l ) o x — =
X

^

4

(2)

x-UJ2x-^

4

* Neu X — > 0 thi — x < 0, khi do t u (2) ta c6 ve trai Ion hon 0, ve phai be hon 0, v6 li.
X
x
4
4
* Neu X — < 0 thi — x > 0, khi do t u (2) ta c6 ve trai be hon 0, ve phai Ion hon 0, v6 li.
4

X

X

2

Vay X — = 0 o x

- 4 = 0 = > x = 2 (thoa man dieu kien).

X

Phuong trinh da cho c6 nghiem duy nhat x = 2 .
4. Phuong trinh da cho tuong d u o n g v6i
(2'" +2
o

- 2 ( 2 " +2

(2"+2")'-2

<^(2" +2

- 7 2 ( 2 " +2 ") + 185 = 0

- 2 ( 2 " + 2 - " ) ' - 7 2 ( 2 " + 2 ") + 185 = 0

- 6 ( 2 " +2

- 7 2 ( 2 " + 2 ") + 189 = 0.

Dat t = 2" + 2 ", t > 2 , phuong trinh da cho tro thanh: t^ - 6t^ - 72t +189 = 0
c = > ( t - 3 ) ' ( t ' + 6 t + 2 l ) = 0 o t = 3, v i t ' + 6 t + 21 > 0 , V t > 0 .
Voi t = 3 o 2 " + 2 - = 3 « 2 " = ^ ^ « x

= log/^*^^
/
/—
3±V5
Vay p h u o n g trinh da cho c6 nghiem duy nhat x = log^
V i d u 2. Giai cac phuong trinh sau tren tap so thuc:
1. log25(x^-8x + 15)^ = | l o g ^ ^ + log5|x-5
2
2. l o g , (4 - x ) ' + ^ log, (x + 2 ) ' = 3 + log, (x + 6y
3. 4" (V2.6"-4" + V4.24" - 3 . 1 6 " ) = 27" -12" + 2.8"

9


LOT gidi
1. Dieu kien: x > 1;x # 3;x # 5 . Phuong trinh da cho tuong duong v o i
logs x'' - 8 x + 15 = ' 0 g 5 ^ + l0g5 X - 5
X-1
o l o g g x - 3 +log5 X - 5 = l o g 5 - y - + l o g 5 | x - 5
o 2 | x - 3 = x - l < : = > 2 x - 6 = x - l hoac 2 x - 6 = l - x <:=>x = - (vi
3 ^

7
Vay p h u o n g trinh c6 nghiem x = - .
3

x^5)
'

2. Dieu kien: - 6 < x < 4 va x ;t - 2 (*)

I^huong trinh da cho <=> 3 1 o g ^ ( 4 - x ) - 3 1 o g ^ x + 2 = 3-31og^ {x + 6)
o

log^ (4 ~ x) + iog^ (x + 6) = 1 + log^ |x + 2|

o

(4 - x ) ( x + 6) = 4|x + 2

<=>4(x + 2) = ( 4 - x ) ( x + 6) hoac 4(x + 2) = - ( 4 - x ) ( x + 6) (vi (*) nen ( 4 - x ) ( x + 6 ) > 0 )
+ V6i x - + 6 x - 1 6 = 0c:>x = 2,x = - 8 .
+ V6i x ' - 2 x - 3 2 = 0<=>x = l + 733,x = l - V 3 3 .
Vay p h u o n g trinh c6 hai nghiem x = 2; x = 1 - V33 .
3. Dieu kien: 2.6" - 4" > 0 va 4.24" - 3.16" > 0 «

x > log, -

Ta C O cac danh gia sau: 4"^2.6" - 4 " = 2".2"^2.6" - 4 " < ^ 2 " (4" + 2.6" - 4 " ) = 12"
4" V4.24"-3.16" = 4" ^^4" ( 4 . 6 " - 3 . 4 " ) < ^ 4 " (2" + V 4 . 6 " - 3 . 4 " )
= - 8 " + i 2 \ 2 " V 4 . 6 " - 3 . 4 " < - 8 " + - 2 " ( 4 " +4.6" - 3 . 4 " ) = 12".
2
2
2
4
^
'
Do do 4" (V2.6" - 4 " + ^ 4 . 2 4 " - 3 . 1 6 " ) < 2.12", dSng thuc xay ra k h i x = 0.
2 7 " - 1 2 " + 2 . 8 " = 27" +8" -12" > 3N/27".8".8" -12" = 2.12", dau bang xay ra khi x = 0
Vay p h u o n g t r i n h nay phai c6 nghiem x = 0
V i d u 3. Giai cac phuong trinh sau tren tap so thuc:

1 . 2 x ^ + 6 x ^ + 6 x + l = 3| x + 2

2. 8 log4 N/X^ - 9 + 3^2 log4(x + 3)^ = 10 + log2(x - 3)^
Lai gidi

l . D a t 3|^i±?.=y + i c ^ ^ i ± i = y 3 + 3 y 2 + 3 y + i o x = 2 y ^ + 6 y ^ + 6 y

(1).

10


T h e o d e b a i ta CO p h u a n g t r i n h 2 x

+6x

+ 6 x + l = y + 1 <=> y = 2x

+6x

+6x

(2),

Ttr (1) va (2) s u y ra
2x^ + 6x2 + 6x - (2y^ + 6y2 + 6 y ) = y o

X

o

2(x^ - y^) + 6(x2 - y ^ ) + 7(x - y ) = 0

(x - y ) ( 2 x 2 + 2y2 + 2 x y + 6x + 6 y + 7) = 0

o x

= y h o a c 2x^ + 2 y 2 + 2 x y + 6x + 6 y + 7 = 0

(3)

P h u a n g t r i n h (3) v 6 n g h i e m v i
2x^ +2y^

+ 2 x y + 6x + 6 y + 7 = ( x + y)2 + 4 ( x + y ) + 4 + (x + l ) 2 +{y +

= (x + y + 2 ) 2 + ( x + l ) 2 + ( y + l ) 2 + l > l

lf

+1

Vx,y.

^i-#-Vay x = y , t h a y v a o (2) ta d u o c
2x'' + 6x2 + 6x = x o

+ 6x2 + 5x = 0 o

^^2x2 + 6x + 5) = 0

C5>

x = 0.

V a y X = 0 la n g h i e m p h u o n g t r i n h .
2. D i e u k i e n : x < - 3 hoac x > 3 .
81og4 7x2 - 9 + 3 j 2 1 o g 4 ( x + 3)2 = 10 + \og^(x
<=>

2log2(x2

-sf

- 9 ) + 3 j l o g 2 ( x + 3)2 = 10 + l o g 2 ( x - 3 ) 2

l o g 2 ( x + 3)2 + 3^/log2(x + 3)2 - 1 0 = 0 .
D a t t = 7 l o g 2 ( x + 3)2 > 0 . Ta CO p h u a n g t r i n h :
t^O
t2+3t-10 = 0

o

t = 2 o

l o g 2 ( x + 3)2 = 4 o

(x + 3)2 = 16 <=> X = - 7 h o a c x

1.

D o i c h i e i i d i e u k i e n , ta c6 n g h i e m p h u a n g t r i n h la x = - 7 .
V i d u 4. G i a i cac p h u a n g t r i n h sau t r e n t a p so t h u c :

1. x^ + 3x2 + 9x + 7 + ( x - i o ) V 4 - x = 0

2. V x ^ + yjx^ +x^ +x + \ 1 + Vx'* - 1
Lai

giai

1. D i e u k i e n : x < 4 .
Bien d o i p h u a n g t r i n h ve: (x +1)"' + 6(x +1) = V 4 - x . ( 4 - x + 6).
Dat u = x + l , v = \ / 4 - x ^ 0 . T a c 6 p h u a n g t r i n h : u ' ' + 6 u = v'' + 6v
o

( u ' ' - v^) + 6 ( u - v ) = 0 <=> u = V hoac u2 + u v + v2 + 6 = 0 .

Trudmg hQfp 1: u2 + u v + v2 + 6 = 0 <=> ( u + —)2 +
2
Trumtghgp

2: u = v <=> V 4 - x

= x + 1 <=>

4

+ 6 = 0 , v6 nghiem.

x + 1^0
<=> X =

-3+N^

4 - x = x2+2x + l

11


2. Dieu kien: x > 1.
Dat a = V x - l , b = Vx^ +x-^ + X + 1 , voi a > 0 , b > 0 , t a c 6
V x ' * - l = \ / ( x - l ) ( x ^ + x 2 + x + l ) = V ^ ^ . V x ^ + x^ + X + 1 = a.b .
Phuong trinh da cho tro thanh:
a + b = l + a b < » ( a - l ) ( b - l ) = 0<=>a-l = 0 hoac b-l=0<::>a = l hoac b = l .
+ Vdi a = l t h i V x - 1 = l < : i . x - l = l<=>x = 2.
+ V6i b = l thi Vx^+x^+x + 1 = l » x ' ' + x ^ + x = Oc=>x^x^+x + l j = 0<=>x = 0(loai,do
\2

1 + - > 0, vol
X > 1) hoac x + X +1 = 0 (phuong trinh nay v6 nghiem vi x + x +1 =
x+—
4
2,

moi x )

V

Vay phuong trinh da cho c6 nghiem duy nhat x = 2.

Vi du 5. Giai cac phuong trinh sau tren tap so thuc:
1. X + V 4 " X ^ = 2 + 3xv'4-X-

2. 2 x ' - 6 x + 10-5(x-2)Vx + l = 0
3. 4 ( N / X + T - 3 ) X ' + ( I 3 V X + 1 - 8 ) X ~ 4 V X - 1 - - 3

-0

1. Dieu kien: -2 < x < 2.
Dat t = x + \/4-x' => t" = 4 + 2x\l4~x- => wli-x'
=~
^,
2
Phuong trinh da cho tro thanh: t = 2 + 3 - — ^ c > 3 t ' - 2 t - 8 = 0ci>t = - -4h o a c t = 2.
3
+ Voi t = 2, ta c6: x + V 4 - x ' = 2 <=> ^Ji-x'
4

/

4

= 2-x

/

x<-i
3
-2±Vl4

o

2-x>0
4~x^ = 4 - 4 x + x^ o

x=0
x = 2'

4

+ Voi t = — , ta CO x + v 4 - x ^ = — <=> v 4 - x ^ =

<=> <

x . - i
3
9x'+12x-10 = 0

=> X =

3

-2-Vl4

x=

-2-Vi4

Vay phuong trinh da cho c6 ba nghiem x = 0; x = 2; x
2. Dieu kien: x > - 1 .

Phuong trinh da cho tuong duong 2(x ~ 2)" + 2(x +1) - 5(x - 2)>/x + l = 0
12


2(x - 2) - v/x + 11 ll^jx + l - (x - 2)1 = 0 «

2(x - 2) - Vx + 1 = 0

hoac 2>/x + l - ( x - 2 ) = 0 .

+ Voi 2 V x + l = x - 2 <=>
„

x^2

x>2

+ Voi V x + 1 =2(x-2)<=>

4 x ' - 1 7 x + 15 = 0
x>2
,

„

[x'-8x = 0

X =

x>2

3; X

50x

= 3

= —

4

<=> X = :

[x=0;x = 8

Vay phuong trinh da cho c6 nghiem x = 3; x = 8 .
3. Dieu kien: x>l.
4X'N/>O^-12X'

Bien doi phuong trinh da cho, ta dugc:

+13x>/)rri-8x-4Vx^l - 3 = 0

x4x + -[{24x + l-3j

+i\-2^Jx^J

=0

x V ^ ( 2 V ^ - 3 ) ' = 0
De v e t r a i bang 0 thi ta phai c6:

5
<=> x =•

Vay X = ^ la nghiem cua p h u o n g trinh da cho.
V i du 6. Ciai cac he p h u o n g trinh sau tren tap so thuc:

1.

2x + 3 y = 5

2. <

3x2-y2+2y = 4

x ^ + 2 x ^ y + x2y^ = 2 x + 9
x^ + 2 x y = 6x + 6
Lai

1.

2x + 3 y = 5

(1)

3x^-y2+2y = 4

(2)

Tij phuong trinh (1) suy ra x =

5-3v

5-3y

3. <

3'

+6.2' = 11 + 4''

2>'+4.3^ = 7 + 9"

gidi

, khi do (2) tro thanh

-y2+2y-4 =0

o 3 ( 2 5 - 3 0 y + 9 y ' ) - 4 y ^ + 8 y - 1 6 = 0 o 2 3 y 2 - 8 2 y + 59 = 0 o y = l hoac y = —
23
Vay he phuong trinh c6 nghiem la ( x ; y ) =

2.

(l;l);

31 59

'23'23

x ' * + 2 x ^ y + x^y^ = 2 x + 9(l)
X ' + 2 x y = 6x + 6

(2)

13


Trumig hap 1: x = 0 khong thoa man (2)
Trudng hap 2: x^O, (2) suy ra y =
(

6x + 6 - x
\.x
)

2 ^

6x + 6 - x
2x
2 ^2

(

+ x' 6x + 6 - x
V
2x
J

<:»x'*+x^(6x + 6 - x ^ ) + ^^^"^^
Do X

, the'vao (1) ta dugc ,

= 2x + 9

^ ^ = 2 x + 9 o x ( x + 4)^ = 0 c i > x = -4hoac x = 0

0 nen he p h u o n g trinh c6 nghiem d u y nhat

4

3.Dat 3^ =a; 2'' = b ( a ; b > 0 ) .
He tro thanh:

a + 6b = n + b'

b + 4a = 7 + a~

<=> \

a-2 = (b-3)-

(1)

b - 3 = (a-2)'

(2)

Lay ( l ) - ( 2 ) ta dugc: ( b - a - l ) ( a + b - 4 ) = 0.
+ V6i b - a - l = 0 => b = a + l thay vao (1) ta dugc:
a = 2 => b = 3 => ( x ; y ) = ( l o g , 2 ; l o g , 3) hoac a = 3 => b = 4 => ( x ; y ) = ( l ; 2 )
+ Vol a + b - 4 = 0 v6 nghiem (do t u (1), (2) ta rut dugc a ^ 2; b ^ 3)
Vay nghiem cua he la: ( x ; y ) = ( l o g , 2;iog, 3 ) , ( l ; 2 ) .
V i du 7. Giai cac he p h u o n g trinh sau tren tap so thuc:
1.

1

2x^-9y^ = ( x - y ) ( 4 x y - l )

X - — =

2.

x"^ - 3 x y + y^ +1 = 0

1

y-

r

x^

( x - 4 y ) ( 2 x - y + 4) = -36
Lai giai
1.

' x-"^- 3 x y + y^
^ =-1

x2-3xy + y 2 = - l

J2x^ --99yy' '3 = ((x--y)(x^
+ xy + y^)
x

2x^ - 9 y ^ = ( x - y ) ( 4 x y - l )

2x^-9y^ = x ^ - y ^
<=> i
x^ - 3 x y + y^ = - 1

x = 2y
<=>

x = 2;y = l
x = -2;y = - 1

Vay he p h u o n g trinh c6 nghiem(x;y) la: (2;1); (-2; -1).
2.

1
x - - = y-

(1)

( x - 4 y ) ( 2 x - y + 4) = -36
1
X—- = y

(2)

1
/
X (y-x)(y^+xy+
- c ^ ( x - y ) = ^^
^^-TT^
xY

^

, - y ^ + x y + x^
x = y hoac ^
^
= -1
xY

14


Trumg hap 1: x = y , the vao phuong trinh (2) ta dugc
+ 4 x - 1 2 = 0 o X = - 6 hoac x = 2 .
2

Trumtg hop 2:

2

V + xv + X

= - 1 => xy < 0 .
y
Khi do (2) o 2x^ + 4y^ ~ 9xy + 4x - 16y = -36 o 2(x +1)^ + 4(y - 2f - 9 x y = -18 .
Truong hop nay khong xay ra do xy < 0 => 2(x +1)^ + 4{y - 2)^ - 9xy > 0 .
Vay nghiem cua he p h u o n g trinh la ( x ; y ) = (2; 2); ( - 6 ; - 6 ) .
V i du 8. Giai cac he phuong trinh sau tren tap so thuc:
2-

f x ^ + y ^ + ^ ' ^ y =16
x+y
1.
^ x + y = x^ - y

1-

y
1=2
X

Lai giai

1.

x ^ + y ^ + ^ = 16
x+y
/x + y = x

,2

-y

(1)

(2)

Dieu kien: x + y > 0 .
Phuong trinh (1)
<=>(x^+y^)(x + y) + 8xy = 16(x + y) <=> (x + y ) ^ - 2 x y (x + y) + 8xy = 16(x + y)
<=>(x + y) (x + y ) ^ - 1 6 - 2 x y ( x + y - 4 ) = 0 <=> (x + y - 4 ) [ ( x + y)(x + y + 4 ) - 2 x y ] = 0 .
Truong hap 1: x + y - 4 = 0 the vao (2) ta dugc:
x ^ + x - 6 = 0 c ^ x = - 3 = > y = 7 hoac x = 2 => y = 2 .
Truong hap 2: (x + y)(x + y + 4) - 2xy = 0 <r> x^ + y^ + 4(x + y) = 0 , v6 nghiem do dieu kien.
Vay nghiem cua he la (x; y) = (-3; 7); (2; 2 ) .
2. Dieu kien: x^-

,y> - .
2^
2

T r u ve hai phuong trinh ta duoc

Vy->/x

Vx

= 0 o

-

+
Vy

V

2-1-.12-1=0
y V x

y-x
Vxy(^/^ + ^/y)

y-x
xy

= 0.

2-UJ2-'

15


Trucntg hop 1: y - x = 0<=>y = x the vao (1) ta duoc —j= +. 2 — =2.
Vx V X
t<2
Dat t =
t > 0 ta duoc V2-t^ = 2-1 <=> 2 - t ^ O
<=>

t^-2t +l = 0
2-\} = 4 - 4 t + t^
Vx

<

<=> t = 1 = > X = l,y = 1 .

Truang hap 2:

7^(V^ + 7 y )

xy

2-UJ2-^

= 0 ,v6 nghiem do dieu kien.

Vay he c6 nghiem duy nhat (x; y) = (1; 1).
Vi du 9. Giai cac he phuong trinh sau tren tap so thuc:
2x^ + xy = y^ - 3y + 2
(xy+ l)x^ +(x + l)^ = x^y + 5x
4x^y + 7x^ +2x^^y + l =2x + l
1. <

2.<

x2-y2=3

Lai giai

1. 2x^+xy = y ^ - 3 y + 2 c : . y ^ - ( x + 3)y + 2-2x^ = 0.
Coi day la phuong trinh bac hai an y tham so x, ta c6:
A = (x + 3 ) ^ - 4 ( 2 - 2 x ^ j = 9x^+6x + l = ( 3 x + l f , s u y r a : y = 2x + 2 hoac y = -x + l.
Truang hap 1:

[3X^+8X + 7 = 0(A'

x2-y^=3

Jy = 2x

y = 2x

+2

+2
= -5<0)'

x=2
y = -x + l
|y = _x + l
Truang hap 2:
x 2 - y 2 = 3 <=> i x = 2
o y = -l
Vay he phuong trinh da cho c6 nghiem (2;-l).
(xy + l)x^+(x + l)^ =x^y + 5x (1)
2.
4x^y + 7x2 + 2x2 ^9+1 = 2x + 1 (2)
Dieu kien: y > - 1 .
v

Tu phuong trinh (1) ta c6:
x2(xy +1) + (x - \

-x(xy

+1) = 0 <=> x(xy + l)(x - 1 ) + (x-1)^ = 0

o ( x - l ) ( x 2 y + 2 x - l ) = 0 o x = l hoac x2y + 2 x - l = 0 .
+ Voi x = 1 thay vao (2) ta duoc:
4y + 4 + 2^y + l = 0 o 2,/y + l(2,/y + l +1) = 0 o y = - 1 .

+ Voi x^y+ 2 x - l = 0 o y

— ^ (vi x = 0 khong thoa man) thay vao (2) ta duoc:

16


4x^

+ 7 x 2 + 2 x ^ f ' - ^ + l =2x + l
V

X

y

o(x-l)^-2x^

x-1

= 0<=> x - 1

- 2 x(x-l) =0

X

Tniang hap 1: x - 1 = O o x

= l=>y = - l .

x - l = 2 x o x

Trumghop2:

1
= - l = > y = 3 ; x = — =>y = 3 .

Vay nghiem cvia he da cho la: (x; y) = ( l ; - l ) ; ( - l ; 3 ) ; ( - ; 3 ) .
3
Vi du 10. Giai he p h u o n g trinh sau tren tap so thuc:
^xy + (x - y X T x y - 2 ) + V x = y +

^

(x + l ) ( y + .Jxy + X - x^) = 4
Lcri giai
Dieu kien: x y + ( x - y ) ( , y x y - 2 ) ^ 0 va x ; y ^ O .
Phuong trinh dau <=> ^xy + (x - y)(^^(x-y)(y + V ^ - 2 )

x-y

Jxy + ( x - y ) ( 7 5 ^ - 2 ) + y

^/^ + ^/y

= 0

o(x-y)

= 0(*)
^xy + ( x - y ) ( ^ - 2 ) + y

Ta

CO

y + Jxy = x' - x +

x +1

V^ + >/y^

4 ^
= (x -1)^ + x + 1 +•
-2>2
x+ l

y+

>0

7xy + ( x - y ) ( 7 ^ - 2 ) + y

V ^ + N/Y

P h u o n g t r i n h (*)<=> x = y, thay vao p h u o n g t r i n h t h u hai ciia he, ta d u g c
x^ - 2x^ - 3 x 4 - 4 = 0 <=>x = l hoac x = ^ - ^ ^ ^

hgp v o i dieu kien ta dugc

l + \/l7
x = 1, x =

.
2

Vay he da cho c6 hai nghiem (x; y) la: (1; l};__Lt.^^^^y^ ^


V i d u 11. Giai cac he phuong trinh sau tren tap so thuc:

1.

i

+

+ x + y = 18

^

xy(x + l ) ( y + l ) = 72

x"^ + y^ + 2(x + y) = 7
y ( y - 2 x ) - 2 x = 10

Lai giai
1. H § p h u o n g trinh da cho <=>

Dat \

( x 2 + x ) + { y 2 + y ) = 18
( x 2 + x ) ( y 2 + y ) = 72

x ^ + x = a, a > - 4
y ^ + y = b, b ^ - i
4

Ta duoc

Trumghap

a + b = 18
ah = 72
1:

<=>

a =6
b = 12

a = 6 , b = 12
a = 12, b = 6
X

+x = 6

y ^ + y = 12

X =

2,

X=

-3

[y = 3 , y = - 4

Truong hap 2: Doi vai tro cua a va b trong truong hop I, ta dugc

X =

3,

X=

-4

y = 2,y = - 3 '

Vay nghiem cua he la
(x;y) = (2; 3); {2;-4);

(-3; 3); (-3;-4); (3; 2); (-4; 2); (3;-3); ( - 4 , - 3 ) .

2. He phuong trinh da cho <=>

x^ +y^ +2(x + y) = 7
y ( y - 2 x ) - 2 x = 10

<=>

(x + l ) ^ + ( y + l)^ = 9
(y-xf-ix

+ lf

=9'

Dat a = x + l , b = y + l = > b - a = y - x , t a dugc he
(b-a)^-a^ =9
= > a ^ + b ^ = ( b - a ) ^ - a ^ o a ^ = -2ab<=>a = 0 hoac a = - 2 b .
+ Vdi a = 0 => b = ±3 =:> X = - l , y = 2 hoac x = - l , y = - 4
+ V 6 i a = - 2 b ^ 5 b 2 = 9 o b = 4-,a
3

= — ^ hoac b = -4=,a

= -^

u 1 6
3
hoac X = - 1 + - 7 = , y = - 1 -

Vay nghiem cua h § la (x; y) = (-1;2); (-1; - 4 ) ; ( - 1 — ^ ; - 1 + A ) . (_i + ^ . _^

V5

V5

3
sl5

V5

V i d u 12. Giai cac he phuong trinh sau tren tap so thuc:
( l + y 2 ) + x ( x - 2 y ) = 5x

1.

2.

( l + y ^ ) ( x - 2 y - 2 ) = 2x

2>/x^+3y-Jy^+8x-l=0
x(x + 8) + y ( y + 3 ) - 1 3 = 0

18


Lai gidi
1. Neu X = 0 thi he da cho <=>

l + y2=0
,
, he nay v6 nghiem.
(l + y 2 ) ( 2 y - 2 ) = 0
^ '

Neu X ^ 0 thi chia ca hai ve ciia ca hai phuong trinh da cho x ta duac:
1 + y^
1 + y^

+ ( x - 2 y ) = 5^
(x-2y-2) = 2

l.y^
1+

+ (x-2y-2) = 3

^(x-2y-2) =2

1+y
X ' '^ = ^ " 2 y ~ 2 , t a d u o c h e phuong trinh:
u +v =3
uv = 2

+ Voi

u=l
v =2

[u = l

<=> •!v = 2 hoac
l.y^

u=2
v

= 1

x-2y-2 =2

=l
[x = l + y2

rx = 2y + 4

[^=2y + 4

[y^-2y-3 =0

x = 2y + 4

x =2

y = - l ; y = 3 ^ ly = - l
+ Voi

u =2

1+

^

=2

_ J l + y2=2x

„ fx = 10
hoac
y =3

[x = 2y + 3

v =l

x-2y-2 =l

>l^ = 2y + 3

[y2-4y-5 = 0

x = 2y + 3

fx = l
^ o •{
hoac
y =- l ; y=5
|y = - l
"

x = 13
y =5

Vay he da cho c6 4 nghiem ( x ; y ) = ( 2 ; - l ) ; ( 1 0 ; 3 ) ; ( l ; - l ) ; ( 1 3 ; 5 ) .
2. Dieu kien: x^ + 3y ^ 0, y^ + 8x > 0 .
Dat u = J x 2 + 3 y , v = J y 2 + 8 x ( u , v > 0 )
He da cho tro thanh:

2u-v = 1

v = 2u-l

u^+v2=13

|u2+v2=13

<=>

v = 2u-l
[ u 2 + ( 2 u - l ) 2 =13
v =2u-l

fv=2u-l

5u^-4u-12 = 0

Vol

u=2
lv = 3

V x ^ 3 y =2

x^ + 3 y = 4

i/y^Tsx = 3

y^+8x = 9

y=-

u = 2;u = —
5

u =2
lv = 3

4-x^

4-x^

+ 8x=9

19

<=>

x^-8x^+72x-65 = 0
4-x^
<

X

= -5; X = 1

x= l
y= l

hoac

y =

4-x^

(x--l)(x + 5)(x^ - 4 x + 13) = 0
x = -5
y = -7

Doi chieu v d i dieu kien ban rfau ta thu dugc nghiem ciia he p h u o n g trinh la:
(x;y) = ( - 5 ; - 7 ) , ( l ; l ) .
V i du 13. Giai cac he phuong trinh sau tren tap so thuc:
1.

^Ux-y

-y]y-x

7yjy-x

=1

2.

+ 6y - 26x = 3

3y^

3x

2x

1

-+-

x+y
2x^+y^

(x>0,y>0).

^2y + 2 0 = x ^ + x - 5
Lai giai

1.

7 ^ y - x - 2 ( l l x - y ) + 4(y-x) = 3

7 ^ y - x +6y-26x = 3

^llx-y-^y-x

^Ux-y

-yjy-x

=1

=1

Dat u = ^ l l x - y , v = ^ y - x ; u , v > 0.
He p h u o n g trinh da cho tuong duong:

<=> i

u = v+1

V l l x - y =2

5

V =

1

u - V =

7v-2u^+4v^ =3

' u = 2;v = l

u = v+1

3
u = —;
2

2v^+3v-5 = 0

4v^-2(v + l)^+7v = 3
llx-y = 4
-X

—

2

+y=1

1
X = —

2

^ = 2

Vay he phuang trinh c6 nghiem (x; y) = r i . 3 ^
2'2
2. Phuong t r i n h t h u nhat ciia he tuong d u o n g v d i

^

^

3xy^

= ^ P 4 « ( 2 x ^ + y ^ ) ^ = 3 x y ^ ( x + y)
2x^ + y^

2x2+y2^

3xy

x+ y

<=>

2+

y

2^2

2

= 3 y

1+y

\

Dat t = ^ , ta CO p h u a n g trinh: (2 + t2)2 = 3 t 2 ( l + t) c=> t ' * - 3 t ^ +1^+4 = 0
o (t -2)2(t2 +1 +1) = 0 o t = 2 » y = 2x.
Thay y = 2x vao p h u a n g trinh t h u hai cua he ta dugc: 2Vx + 5 = x^ + x - 5.

20


Dat z = -J^^

> 0 . Ta CO he phuong trinh

z = x+ 5
2z = x^ + X - 5

<=>

z^ = x + 5
x^ = 2 z - x + 5

<=>

z =

z^ = x + 5

= z

X

x^ - z ^ = 2 z - 2 x

5

X +

x + z + 2 = 0.

Truong hop x + z + 2 = 0 khong xay ra v i x, z > 0.
Voi X = z > 0 ta duoc x = ^ ^ ^ ^ , y = 2x = 1 + V2I.
Vay nghiem cua he da cho da cho la: (x;y) = (

;2x = 1 +

V i d u 14. Giai he phuong trinh sau tren tap so thuc:
x^y^ + 4x^y - 3xy^ + x^ + y^ = 12xy + 3x - 4y +1
3x^ - 2 y ^ = 9 x + 8y + 3
Lai giai
He phuong trinh da cho tuong duong:
y^^x^ - 3 x j + 4y^x^ - 3 x j + ^x^ - 3xj + y^ + 4 y + 1 = 2

( x ^ - 3 x + l)(y2+4y + l ) = 2

3(x^-3x)-2(y^+4y = 3

3

x^-3x

y2+4y) = 3

Dat u = x^ - 3x; v = y^ + 4 y , he tro thanh:
(u + l ) ( v + l ) = 2
3u-2v = 3

+ Voi u = 1; V

+ Voi u =

=

- —; V

3u~3

<=>

2
(u + l ) ( 3 u - l ) = 4

0, ta c6:

=

u = l;v = 0

V =•

o

x^-3x-l=0

X =

y^ + 4 y = 0

- 4 , ta c6:

5
u = —;
3

V =

-4

3±Vi3
2

hoac

y2 +4y + 4 = 0

3±N/I3

y = -4

y = 0

x^ - 3 X + - = 0
3

X =

9±N/21
« x =
[y = -2

Vay he da cho c6 6 nghiem (x; y) la:
( ^ ; 0 ) ;

3+

V13 ;0

3-N/T3

,

;(^^^;-4);

r 3 + Vl3

;-4

'

9 + V21

;-2

( 9-V2I

;-2

V i d u 15. Giai cac phuong trinh sau tren tap so thuc:
1. l o g 3 ( x - 2 ) = l o g 4 ( x ' - 4 x + 3)

2. log3 |x^ + X + 1 j - log3 X = 2x - x^
Lai giai

1. Dieu kien: x > 3 . Dat l o g 3 ( x - 2 ) = t > 0 C5> x = 3 * + 2 .

21


Khi do t = log4

Ham so f(t) =
v9y

Mat khac f

=9' - 1

«

9)

\

=

1(1)

nghich bien tren M nen phuong trinh (1) c6 toi da 1 nghi^m.
v9,

= 1 => t = — la nghiem duy nhat ciia phuong trinh (1).
v2/

1
Voi t = - 2

=:> X

= 2 + N/3

Ket hop voi dieu kien ta c6 nghiem phuong trinh da cho la x = 2 + Vs .
2. Dieu kien: x > 0
Phuong trinh da cho tuong d u o n g voi logj ^x^ + x +1 j - logj x - 1 = 3x - |x^ + x +1 j
log3( X

+X +

<=> Il o g j ^ x ^ + X

1 - logj 3x = 3x - ^x^ + X +1

+ 1J+ x^ + x + l j = log3 3x + 3x

Xet ham so t(t) = logg t +1 tren (0; +00) c6 f'(t) =

(*)
+1 > 0, Vt > 0
t In 3

f(t) dong bien tren (0; +00). Do x^ + x +1 > 0 va 3x > 0
Phuong trinh (*) <=> f(x^ + x +1) = f (3x) <=>x^+x + l = 3 x o x = l .
Vay phuong trinh da cho c6 nghiem x = 1 .
V i du 16. Giai cac phuong trinh sau tren tap so thuc:
1.21og(^ >/x + N/X j = log4 X

2. l o g j ^ l + x - N / x

=(l-x)V3x

Lcri giai
1. Dieu kien: x > 0. Phuong trinh da cho tuong d u o n g voi logg \[x + \/x j = log^ \/x .
= log4 >/x

Dat t = log(,

N/^ + ^

=

6'(1)

V^ = 4*(2)
1A

The (2) vao (1) ta duoc: 4* +2^ = 6' ci>
^2 Y
—

+

^ 11 A*

v3j

= 1.

v6.

v6y

Xet ham so f ( t ) =

v3y

3j

, ta thay f(t) nghjch bien tren # , ma f ( l ) = 1 nen t = 1

la nghiem duy nhat, thay vao (2) ta c6 x = 16.
Vay phuong trinh da cho c6 nghiem duy nhat x = 16.
2. Dieu kien: x > 0 . Phuong trinh da cho tuong d u o n g logj

= N/SVx ->/3\/x^

22


l o g 2 ( l + V^) = V 3 ( l + ^/^)-^/3 1

+ Vx

«log2

1

Olog2

1 + Vx

+ Vx

V3fl + ^ / ^ ' j = log2(l + ^/5^) + ^/3(l + V^) (*)

Xet ham so f ( t ) = log2 t + Vst, v d i t > 0 => f(t) dbng bien tren (O; + 0 0 ) .
KhidoH o

f|^l + V > ^ j = f ( l + - y x j « l + \ / x ^ = l + Vx « x = 0 hoac x =1.

Vay phuong trinh da cho c6 nghiem x = 0, x = 1 .
V i du 17. Giai cac phuong trinh sau tren tap so thuc:
1. V 2 x 3 + 3 x 2 + 6 x + 1 6 - V 4 ^

= 2>/3

2.

^ ^ +tT-x=2

Lot giai
2x3 + 3x2 + 6x +16 > 0
U ^ 2)(2x2 + x - 8) > 0
1. Dieu kien: •
o
<=> \
4-x>0
4-x>0
at f ( x ) = V2x-'+3x2
Dat

+6X

4.

+ I 6 - V 4 - X , ta c6

f ( x ) = - ^ - ^ - ^ "^^^"^^^

+ , i
V 2 x 3 + 3 x 2 + 6 x + 16 2V4^;
f (1) = lS>

-2 < x <

> 0,Vx 6 (-2; 4)

f ( x ) d o n g bien tren | - 2 ; 4 J . Ma

nen x = 1 la nghiem d u y nhat cua phuong trinh.

2. Dat f ( x ) =
Taco f ' ( x ) =

+ V 4 ^ ,v6i 2 < x < 4 .

ir-^=L==-—,
Ht/O^

^

-

t/(4-xf

Vxe(2;4):f'(x) = 0 o x - 2 = 4 - x o x = 3.
Lap bang bien thien suy ra: f ( x ) > f (3) = 2 Vx G [2;4]
=> Phuong trinh f (x) = \/x-2 + N / 4 - X = 2 C6 nghiem d u y nhat x = 3 .
V i du 18. Giai cac phuong trinh sau tren tap so thuc:

47X+21

1.2x2

2. 7 1 3 ' ^ ' - 7 1 3 x ^ 3 x 2 .

- 2 x ^ ^ = l - l
2
X

280-21X-7X
x^ +3x

Lot giai
1. Dieu kien: x ^ 0 .
Nhan thay

l-x2

]-2x

nhu sau:

2

-2

l-x^

2

= 1 — = 2(

l-x2

1 1

2

l-x^'

^
) , v i the p h u o n g trinh da cho c6 the viet lai
X

l-x^

«2

l-2x

1-x

= 2x^^.1

l-2x

2V x

y

23


X e t h a m s o f ( t ) = 2 ' + ^ t , ta c6: f (t) = 2 ' l n t + ^ > 0 , V t .
Ham f(t) dong bien tren M nen phuong trinh (*)
= f

l-x
l-2x
ri-2x^ o —
— = — — <=> X = 2 thoa man dieu kien.
V X

^

,

-

-

i.v,

,

y

Vay p h u a n g trinh da cho c6 nghiem duy nhat x = 2 .
2. Dieu kien: x # - 3 , x

0
40

7

Phuong tr inh da cho duoc viet lai:
7
0

713X

+7

713'<^

40

1- --- -

-+1

- 713"^

7 ^
l^x^

x^+^x = 7

40

— 1
^x + 3x

40
x^+Sx

Xet ham so f(t) = 713' + 7 ' , ta c6: f (t) = 713' In 713 > 0, Vt e ^ nen ham f(t) dong bien
tren

, do do p h u o n g trinh (*)
7
r7
^ +1 = f — +

Ix^

;

.x^

40

^

x2+3x.

+ 1 = -4+

0 ^
X

x^

-^^

x2+3x

o x ^ + 3 x - 4 0 = 0<z>

^x = 5
x = -8

Vay p h u a n g trinh da cho c6 nghiem x = - 8 , x = 5 .
V i du 19. Giai cac phuang trinh sau tren tap so thuc:
1. (6x + l)log^(x + l ) + 6 ( x - l ) l o g 2 ( x + l ) - 7 = 0

JM_>i|__L

2. e^'^ " ' - e l '

^

2x-5

x-1

3. log,„,,(2014-l''"^'+4'""!-4)

=
Lai giai

1. Dieu kien: x > - 1 . Ta thay x = -- khong phai la nghiem ciia p h u a n g trinh.
6
Voi X ^ - - , p h u a n g trinh da cho tuong d u o n g
6
l o g 2 ( x + !) = - ! hoac log2(x + l ) =

6x + l

l o g j (x + l ) = - l < = > x = - — thoa man dieu kien.
7

1

1

, dieu kien x 6 ( - l ; - - ) u ( — ; + o o ) ( * )

log2(x + l ) =

6x + 1
6
6
Xet x e ( - 1 ; - i ) , t h i V T (*) la h a m so d o n g bien, VP (*) la h a m so nghjch bien
ma V T

' 3^
^ 4 ;

= VP

(

3^

."4.

nen tren ( - 1 ; - i ) thi x = - ^ la nghiem duy nhat thoa man ( * ) .

24


Xetx 6 ( - — ; + o o ) , thi V T ( * ) la ham so dong bien, V P ( * ) la ham so nghich bien
6
ma V T ( I ) = V P ( l ) nen tren (-~;+co)
6

thi x = 1 la nghiem duy nhat thoa man ( * ) .

Vay phuang trinh da cho c6 3 nghiem:
2. f)ieu kien: x

1; x

- . Phuong trinh da cho viet lai: e^^""''^ - ^ ^ ^ = e^"
2x-5

Xet ham so f(t) = e' - - , t a c 6 f (t) = e'
nc:>f(|2x-5|) = f ( | x - l | ) o x - 1

^— (*)
x-1

> 0,Vt ^ 0, nen ham f(t) dong bien V t # 0

2 x ^ 5 <=> x = 2 hoac X = 4 .

Vay phuang trinh da cho c6 2 nghiem: x = 2 , x = 4 .
3. Dat T = 2 0 1 4 ' ' " " ' + 4 " " ' - 5
Phuong trinh da cho t r o thanh: log,,,,^ (T + 1) = 2014^ - 1 .
Xet ham de thay p h u o n g trinh tren luon c6 nghiem T = 0 <=> 2014^^^'""^ + 4 ' " " ' = 5 .
Data=|sinx| ( a 6 [ 0 ; l ] )

2014-'+4^^'

= 5 (1)

Ta c6: 2014'' ^ 4= ( 2 ) . Dau " = " xay ra k h i a = 0.
Dat f(a) = 4 ' + 4 ^ ' ( 3 )

Xet f'(a) = Oc=>

f (a) = l n 4 . 4 ' -

ln4.4^'^

( 4 ) . D a t g(z) = — , ( z e [ 0 ; l ] ) .
4^:

Khi do xet g(z) trong m o i khoang

^
1 U 1
0;
In 4
In 4

Tiep tuc xet f(a) trong m o i khoang

^

(4) •» a = V l - a ' <=> a =
f(a) = 5 k h i a = 1 hoac a = 0

0;-

Giai phuong trinh sin x = 1 hoac sin x = 0 danh cho ban doc.
V i d u 20. Giai cac he phuang trinh sau tren tap so thuc:
( x - 3 ) ( x + 4) = y ( y - 7 )
1.

^

X

iogx-i(2-y) =

x-1

.ln(y +

2.

^ + 1)

e 2 y _ i = 2e>'.ln(x + \/x^+l)
Lai giai

( x - 3 ) ( x + 4) = y ( y - 7 )
1. i

iogx-1 ( 2 - y ) =

x-1

(l)
(2)

. Dieu kien:

00<2-y,y7^0

l[O ^ y < 2

25


Phuong trinh ( l ) <=> (x - i f + 3 ( x - 1 ) = (2 Xetham so f ( t ) = t ^ + 3 t , t e ( 0 ; + o c ) .

+ 3(2 - y ) (3)
'

Ta c6; f ( t ) = 2t + 3 > 0,Vt e ( 0 ; + « ) , suy ra ham so dong bien tren (0;+oc) .
Laico: x - 1 , 2 - y 6 (O;+oo) nen (3) <=> f (x - 1 ) = f (2 - y ) <=> x ^ 1 = 2 - y <:> x = 3 - y ,
Thay vao p h u o n g trinh (2), ta dugc:
X (3-y)-i
9
log2__y (2 - y ) =
y
c : > y ^ + y - 2 = 0<=>y = - 2 hoac y = 1
+ V a i y = 1 => X = 2 ( k h o n g thoa m a n dieu kien).
+ V o i y = - 2 => X = 5 (thoa m a n dieu kien).
Vay he p h u a n g t r i n h c6 n g h i e m d u y nhat ( x ; y ) = ( 5 ; - 2 ) .
2.Tac6 > / t ^ l + t > > / t ^ + t = t + t > 0
Ta viet he d u d i dang

Vte^.

e" - e " " =21n(y + ^/y-+l)
gy-e

>' = 2 i n ( x + V x ^ + l ) .

T u do suy ra: e" - e"" + 2 ln(x + Vx^ +1) = e^' - e"^ + 2 ln(y + ^y^ +1).
Xet ham so f(t) = e' - e " ' + 2ln(t + 7 t ^ + l ) .
2
Ta CO f'(t) = e' + e ' +

> 0 . Bo do f(t) dong bien tren R .
Vt^+1

Ket hop vdi f (x) = f(y) ta suy ra x = y.
Thay x = y vao he ta dugc:
e ^ - e - " =21n(x + \ / x ^ + l ) o g ( x ) = e ' ' - e " ' ' - 2 1 n ( x + V x ^ + l ) = 0
Taco g'(x) = e''+e

— — ^ 2 V e \ e " — p ^ =0

Suy ra g(x) dong bien tren

(*).

Vx e

. Do do phuong trinh (*) c6 khong qua mot nghiem.

Mat khac x = 0 la mot nghiem cua (*).
Vay phuang trinh (*) c6 nghiem duy nhat x = 0.
Vay he c6 n g h i e m la (x;y) = (0; 0).
Vi du 21. Ciai cac he phuong trinh sau tren tap so thuc:
p-2y.l=0
1.
,
,
(3-x)>/2^-2y72y-l=0

2.

L4x-.26x-42.^"-^^-^^^ =
2 x - - 1 3 x + 19
(x-iy

^ y " ' ) ^
6

=(y + i r '

26