tai lieu on thi dai hoc mon toan sang tao va giai phuong trinh bat phuong trinh he phuong trinh bat dang thuc nxb dai hoc quoc gia 2014 2 4085

De y rang phuong trinh thu hai cua h?. la phuong trinh dang cap doi voi x,y... The, bie'n doi cdc phuang trinh vedqng tich,cgng trk cac phuang trinh trong he de tqo ra phuang trinh he

^ ^'^•^ dieu ki?n > 4P h§ phucmg trinh da cho tro thanh:

V^y h? da cho c6 hai cap nghi^m (x;y) = ( - 2 ; 3 ) , ( 3 ; - 2 )

2(a^+b3) = 3(a2b + b2a) c) Dat a = \/x, b = ^ h? da cho tro tharJi:

d) Dieu ki^n: xy >0

x , y > - l cho tro thanh:

S->/P=3

Dat ^ dieu ki^n >4P hf phuong trinh da

P = x.y

S>3;P = ( S - 3 f 2^S + ( S - 3 f + 1 = 1 4 - S

Giii:

a) Dgt Vx = a,7y = b dieu ki?n a,b > 0

phuong trinh tro thanh: Va^+b^+N/2ab = 8V2

a + b = 4 Ta viet Igi h^ phuong

trinh thanh: ^|ia + b)" - 4ab(a + b)^ + 2a^b^ + yflab = 8V2

a + b = 4

IS = a + b 19^ > 4P D^t -I „ , dieu ki?n <! thi h? da cho tro thanh

man dieu ki^n)

Vay h# da cho c6 nghi^m ( x ; y ) = ( l ; 0 ) , ( - 2 ; 3 )

xy (x + y)(x + y + xy) = 30 xy(x + y) + x + y + xy = 11

Dat x y ( x + y ) = a;xy + X + y = b Ta thu du(?c hf:

b)

d)

( x - l ) ( y 2 + 6 ) = y ( x 2 + l ( y - l ) ( x 2 + 6 ) = x ( y 2 + l )

Tdi li(u on III, ,t,u ho, s<i;rx t,u> v.) gtdi Fl, bat PI, hfPi, bdl i J i -?Wglf

Hay x^-2x + >/x=0<»x^+>/x=2xoVx|>/x-l||x + >/x-lj = 0 <=>

x = 0

x = l

X

=-Vgiy CO 3 c l p nghi?m: (x;y) = ( 0 ; 0 ) , ( l ; l ) , ( 3 - N / 5 3 - N / 5 ^

b) da cho o xy^ + 6x - y^ - 6 = yx^ + y

x = y = 3 + N e u x + y - 2 x y + 7 = 0 < » ( l - 2 x ) ( l - 2 y ) = 15

M|it khac k h i cpng hai p h u o n g trinh ciia h$ da cho ta dupe:

, ""V'i-i- 7X<! -r

a = x - l d) D|it

Ta can giai p h u o n g trinh: a + Va^ + 1 = 3* i a

Lay loga theo co so' e ca hai ve ta c6: , j rl

Ket lu^n: Phuong trinh c6 mpt nghi^m x = y = 1

1 1 7

He c6 Y^u T6 D A N G C A P D A N G C A P

+ La nhung he chua cac phuong trinh dang cap

+ Hoac cac phuong trinh ciia hf khi nhan hoac chia cho nhau thi tao ra

phuong trinh dang cap

Ta thuong gap dang h^ nay 6 cac hinh thuc nhu:

^ ax^ + bxy + cy^ = d ^

gx'^ + hx^y + kxy^ + ly^ = mx + ny

Mot so' h^ phuong trinh tinh dang cap dugc giau trong cac bieu thuc chua

can doi hoi nguoi giai can tinh y de phat hi^n:

Phuong phap chung de giai h^ dang nay la: Tu cac phuong trinh cua h? ta

nhan hoac chia cho nhau de tao ra phuong trinh dang cap bac n :

alx"+a,x"-^y^ + a „ y " = 0

Tu do ta xet hai truang hgp:

+ y = 0 thay vao de tim x

+ y 5^ 0 ta dat x = ty thi thu dugc phuong trinh: ajt" + a^t""'' + a„ = 0

+ Giai phuong trinh tim t sau do the vao h^ ban dau de tim x,y

Chii y: (Ta ciing c6 the dat y = tx )

Vi d\ 1: Giai cac h | phuong trinh sau:

De y rSng ne'u nhan cheo 2 phuong trinh ciia hf ta c6:

6(x^ + y'') = (8x + 2y)(x^ + 3y') day la phuong trinh dJing cap bac 3: Tu do

ta CO 16i giai nhu sau:

Vi X = 0 khong la nghiem cua h^ nen ta dat y = tx Khi do h? thanh:

xy|x^ + y^j + 2 = x^ +y^ + 2xy o |x^ + y^ j(xy -1) - 2(xy -1) = 0 •

x 2 + y 2 = 2 <=> < x 2 + y 2 2 NC'u ta thay x"^ + y^ = 2 vao phuong trinh (*) thi thu dugc phuong trinh d5ngca'pb|c3; 5 x ^ y - 4 x y ^ + 3 y ^ =|x^+y^j(x + y) Jt^r;

Tdi li?u on thi dai hQC sang tao vd giai PT, hat PT, hf fl, mi t>J -histuySiTrw^^^_

Tu do ta CO loi giai nhu sau:

Ta thay y = 0 khong la nghi^m ciia h? i , \

St^y^ - 4ty^ + 3y^ = 2 (ty + y) Xet y ^ 0 d§t X = ty thay vao h? ta c6: t2y2+y2 =2

Chia hai phuoiig trinh aia h? ta Avtqc:

Dlit Vy = ^ y = t^x^ thay vao (1) ta du(?c: + = "t^^

Riit gpn bien x ta dua ve phuong trinh an t: -r,,rs;,u

V|iy nghifm ciia hf (x;y) = ^/l7-3 13-3N/I7^

Vi dv 3: Gidi cac hf phuong trinh sau:

3 x 3 - y 3 = ^

x2+y2=l

x^^y + l -2xy-2x = l x^ - 3x - 3xy = 6

a) Ta CO the viet lai h? thanh:

Giii:

3x3-y3)(x + y) = l x2+y2=l (1)

Ta thay ve trai ciia phuong trinh (1) la bac 4 De tao ra phucmg trinh dang

cap ta se thay ve phai thanh (x^ + y^ )2

Nhu vay ta c6:

3x3-y3 ^^^^y x2+y2 0 2x''+3x3y-2x2y2-xy3-2y''=0 '

o (x - y)(x + 2y)(2x2 + xy + y^) = 0 <=> x = -2y x = y

2x2 + xy + y2 = 0 Neu 2x2 + xy + y2 = 0 <=> —x2 +

Neu X = y ta c6 2x2 = 1 o x = ±

/ N2

X + y

2 = O o x = y = 0 khongthoaman

Ta thay cac p h u o n g trinh ciia h$ deu la p h u o n g trinh dang cap bac 3 doi vol

De thay y = - 1 khong phai la nghiem ciia he p h u o n g trinh

Xet y > - 1 Dat x = t ^ y + 1 thay vao h? ta c6:

= 1 t^ - 2 t

Vay h ^ CO 1 cap nghiem d u y nhat ( x ; y ) =

V i dy 4: Giai cac h? phuong trinh sau

V e t r a i ciia cac p h u o n g trinh trong h? la p h u o n g trinh d i n g cap bac 3 doi

vol x,yjy De thay y > 0 Ta dat x = t ^ thi thu dugc h?:

<=>

^ ( 2 t + t^) = 3 t 2 + 2 3 , 2

<=> = - o 2 t ^ - 3 t + l = 0<=> t = l

2 + Ne'u t = 1 thi X = ^ y o X = 1 => y = 1

l t h i x = l

2 2

+ Ne'u t = — thi x = — ^ y < : > y = 4x<=>x"'= — o x =

^ ^ ^ ^ ^ Tom lai he c6 cac nghiem: ( x ; y ) = ( l ; l ) 1 4

Dat J y = t./(x^ +2) ta thu duoc: 3t^ - 2t - 1 - 0 <=> 1

Khi t = 1 ta c6: y = x^ + 2 thay vao p h u o n g trinh t h u nhat cua h? ta thu duoc: X = - 1 => y - 3

Tom lai h ^ p h u o n g trinh c6 mot cap nghiem (x; y) = ( l ; - 3 )

Vi du 5: Giai cac phuong trinh sau

a)

X + y x~ 2x + • / x - \ x ^ y

x ^ l

,(-8;12)

De y rang phuong trinh thu hai cua h? la phuong trinh dang cap doi voi

x,y T a thay neu y = 0 thi tit phuong trinh thu hai cua h^ ta suy ra x = 0,

cap nghi^m nay khong thoa man h?

Xet y > 0 Ta chia phuong trinh thu hai cua h? cho y ta thu dug-c:

Phuong trinh thu nhat cua h^ tro thanh: x^ - 3x - 1 = 3V^(Vl - x -1)^

Dieu ki?n: 0 < x < 1 Ta thay x = 0 khong th6a man phuong trinh

Ta xet 0 < X < 1 Chia bat phuong trinh cho x^ > 0 ta thu du^c phuong

N h u vay ham so' f(t) dong bie'n tren [ l ; +oo) suy ra f(t) > f(1) = 3 T u do suy

ra phuong trinh c6 nghi?m khi va chi khi t = 1 o x = 1

Tom lai h^ phuong trinh c6 nghi^m (x;y) - ( l ; l )

Chu y: Ta cung c6 the tim quan h? x,y dya vao phuong trinh thu hai ciia h^ theo each:

Phuong trinh c6 d^ing:

PHLTONG P H A P B I E N O O l T L / O N G DLTONG

Bie'n dot titvng ditong la phteang phdp gidi he dua tren nhimg ky thudt ca ban

nhir The, bie'n doi cdc phuang trinh vedqng tich,cgng trk cac phuang trinh trong

he de tqo ra phuang trinh he qua c6 dang dqc biet >

* Ta xet cac v i d u sau:

5 + 2 y > ( x - ] ) ^ Xuat phat t u p h u o n g t r i n h (2) ta c6:

Tir h? phuong trinh ban dau ta nham dugc nghi?m la x = y = 1 nen ta se c6

h# nay c6 nghi?m khi: a = 2; b = 1

[(a-2)b = 2 ( l - b ) Dod6tasephantichheved,ng:|^^_^^,^^^^^^^^_^^,^^^^^^

^ 2 ( l - b )

Vi ta luon c6: b ^ 0 nen tu phuong trinh tren ta rut ra a - 2 = — - —

The xuong phuong tririh duoi ta dugc:

i ^ ^ ^ ( a +1) = (b - l)Hh + 2) o (b - if [4(a +1) - b2(b + 2)] = 0

b = l 4(a + l) = b2(b + 2)

Voi: b = 1 => a = 2 , suy ra: x = y = 1;

Vol 4(a + l) = b^(b + 2).Talaic6: ab = 2 o b ( a + l) = b + 2 o a + l = - ^

• The len phuong trinh tren taco:

b + 2

i ( ^ = b ^ ( b 2 )

b

b = -2 => a = -1 o X = 2; y = b^ = 4 (Khong TM) V^y h? da cho c6 2 nghi^m la: (x;y) = (1;1),

-x > - l d) Dieu ki^n:

y >0 Ta Viet lai h? phuong trinh thanh:

^ 2 ( x - y ) 2 + 6 x - 2 y + 4 - ^ = Vx+T

<=> ^ 2 ( x - y ) ^ + 6 x - 2 y + 4 = ^y + Vx + 1 Binh phuong 2 ve ta thu dugc:

2x^ - 4xy + 2y^ + 6x - 2y + 4 = x + y +1 + 2^y(x +1)

o2l{x + lf -2y(x + l ) + y 2 l + (x + l + y) = 27y(x + l )

<:i>2(x + l - y ) 2 + ( V ^ - V y ) ^ = 0 « | ' ' / ^ ^ ^ < » x + l = y

[ Vx +1 = ^ y Thay vao phuong trinh (2) ta c6:

1 2 3±2-\/3

+ Neu y = - thay vao phuong trinh (1) ta c6: 4x - 12x - 3 = 0 <=> x =

+ Neu y = X - 1 thay vao phuong trinh (1) ta c6:

- 2x^ - 3(x -1)2 = 0 o -4x2 + 6x - 3 = 0 V6 nghi^m

Ketluan: (x;y) = (^^;l),(-^/3;l), 3 - 2 V 2 1^

2 ' 2 3 + 2N/2 2 '2

» Cach 2: Phuong trinh thu hai phan richdu(?c: {2y^ + x)(x- y - 3 ) + l = 0

Phuong trinh thu nhat phan tich du(?c: (x - y)^ - 2{x + 2y2) = 0 ,

+ Neu X = 2:

y = 0

y = 3 + Neu y = x + l thay vao phuong trinh (1) ta thu dugc: l + 2y2+2y = 0v6

nghi^m

Ketluan:

H$ phuong trinh c6 cac c^p nghi^m la: (x;y) = (0;0),(0;-2),(2;0),

c) Truoc tien ta di bien doi phuong trinh (1) trong h? ta dug-c '4

= 0

x2(x2 - y2) + xy(x2 - y^) -9(x - y) = 0 , o ( x - y)[x(x + y ^ - 9

R6 rang vol x - y = 0 thi h^ v6 nghi^m khi do ta dua h? phuong trinh ban

dau ve h^ phuong trinh x(x + y)2=9 (3)

[x(y3-x3) = 7 (2)

Tu phuong trinh (3) ta suy ra dugc x, y > 0 Cung Kr (3) bSng phep rut an ta

thu duQ-c y = - = - x Thay vao phuong trinh (2) ta thu A\xqc phuong trinh

X = 1, y = 2 Vgy h? phuong trinh c6 mpt nghi?m duy nhat (x, y) = (1; 2)

d) dugc viet lai nhu sau:

( x y 2- y ) + ( 3 x 2 - 3 x 3 y j = 4 x 2 y ^ ( x y - y ) ( y - 3 x 2 ) = 4x2y 3x2y - y2 + 3xy + 1 = 0 3 x 2 - y 2 + 3 x y + l = 0>/-:

Xet voi y = 0 thay vao ta thay khong la nghi^m ciia h^ Vol y ^ 0 ta bien doi h? thanh :

t - 4 x t + 4x2 o ( t - 2 x ) 2 = 0 o t = 2 x o 2x = x y o

.2 2x = y - 3x^

V i d\ 3: Giai cac h f phuang trinh sau

Do do x + y'^ - 9 < - 1 < 0 nen x + y'' - 9 = 0 v6 nghi^m

Ta chi can giai truong hgp x = y The vao phuong trinh ban dau ta

• + — = ,

thay vao phuang trinh thu hai cua h | ta duQc:

4x^ x2 x 2 - 1 5 2(x2-15) 3 V x 2 - 1 5 24

Xet phuang trinh (*) 36x2 ^ _ j g j p ^ _ j

Vi x = 0 khong phai la nghi^m Ta chia hai ve phuang trinh cho x^ ta c6:

x^ 2x 4x 3 V

- + o — x = llx^

TH 2: x = 2y Thay vao phuang trinh thii hai ciia h$ ta c6:

<::>X = 0 (loai) (do dieu ki§n y^O)

( 2 x - 2 - y ) ( 3 x 2 + y - 2 ) = 0<:> y = 2 x - 2

y = 2 - 3 x 2 Voi y = 2x - 2 the vao phuang trinh (1) ta dugc:

- V o i y = 2 - 3 x ^ <.2 v 6 n g h i ^ m do dieu ki?n y > 3

V^y h§ da cho chi c6 1 n g h i ^ m (x;y) = (4;6) >,

d) The p h u o n g t r i n h 2 vao p h u o n g t r i n h 1 a i a h# ta d u Q C p h u o n g t r i n h :

- x^y - 8 y * + Sx^y = -2(2xy + y - y^) <=> (x^ - 8 y ^ + 3x^ ) y = (-4x - 2 + 2 y ) y

V i y = 0 k h o n g la n g h i ^ m a i a h? Chia ca hai ve cho y ta d u g c p h u o n g trinh

a) P h u o n g t r i n h dau a i a h f dupe viet lai n h u sau:

l o g j x + Vx + 4 +log2 yjy +i-y = l o g 2 4

xy = y^ + 3y - 3 415,17

51 ' 3

x = l ; y = l

x = — ; y = — ( L )

2 1 ^ 3

c) Tu phuong trinh (1) ta thay: 2x(l - y^) = sjl - y^)

THI: y = l thay vao (2) ta c6: -7x+ 6 = 0 <=> x = l;x = 3;x =-2

f2x + 2xy + 2xy2 =3 + 3y (•)

TH2: Ket h<?p voi (2) ta co h? moi: , , - • ov

[2x^-x^y = 2x^y^-7xy + 6 Phuong trinh (3) tuong duong voi: (xy - 2)^2xy + - 3J = 0

+ Neu: xy = 2 thay vao (*) ta c6:

2x + 4 + 4y = 3 + 3y X = => y(l + y) =-4

Phuong trinh nay v6 nghi^m nen h? v6 nghi^m

+ Neu 2xy = 3-x^ thay vao (*) ta c6:

2x + 3-x^ + y(3-x2) = 3 + 3y=>y = - ^ - l =>2x ^

V^y he CO nghi?m (x;y) = (l;l),(3;l),(-2;l)

d) Phuong trinh (1) tuong duong:

c = ^/5=>y = l

-+ X - 3 x 2 - x 3 = 1 0 o

: = -yfs => y = 1 + Vay h^ CO nghi^m

o 4 ( x + y - l ) (x-l)^-(x-l)y + y2 =3y(y2+x + l) • :

o 4 ( x + y l ) (xl)^(xl)y + y2 =3y(y2+xyyl + l)

-o 4 ( x + y - l ) ( x - l f - ( x - l ) y + y2 =3y2(x + y - i )

« ( x + y - l ) ( 2 x - 2 - y f =0 Voi y = 1 - X thay vao (1) ta du(?c: x^ - x + 2 = 0 (v6 nghi^m)

4x^ + 24x2 + 48x + 32 y^ + 3xy + 12y

o 4(x + 2)^ + 4y^ = 3y^ + 3xy + 12y

<»4(x + y + 2) (x + 2)^-(x + 2)y + y2 =3y(y2+x + 4) The X = xy + 2y - 4 vao VP ta dug^c:

Tai li$u on thi dai hoc sdng Uio va giai PT, bat PI, he PI bai Pi \^^„i,c,t intn^]^'l

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

(x + y + 2)(^4(x + 2^ -4(x + 2)y + y2 = 0

Voi y = - X - 2 thay vao (1) ta du<?c: x^ - 5x + 8 = 0 (v6 nghi^m)

Voi y = 2x + 2 thay vao (1) ta duQ-c: 2x^ -7x + 4 = 0<::> X =

Phuong trinh (2) tuong duong: y + 2 - - = 0«>xy = - - 2ci>y = 1 2

Thay vao (1) ta dugc:

Vay nghi|m ciia hf (x; y) =

d) Dieu ki?n: x + y 5^ 1 Phuong trinh (2) tuong duong:

x^ -4y^){x + y - l ) + 2xy = -(x + y - l )

Phan tich nhan tu ta duc?c: (x + 2y - l)(x^ - 2y^ - xy + y +1

THl: x + 2y - 1 = 0 thay vao (1) de dang tim dug-c:

/ V f-l-2>/l4 3 + N/I41 f2Vi4-l S-N/TI'

= 0

Cty TNHH MTV D W H miang Vift

TH2: Ket h(?p vol (1) ta c6 h# moi: • ''^ xy + y +1

[x +y^+x = 3 / Giai bang each:

PT(l)-PT(2)«3y2+xy + x - y - 4 = 0<»(y + l)(x + 3 y - 4 ) = 0

Vay nghi^m ciia h§

(x;y) = -l-27l4 3 + N/14' 2^Ju-l 3-y[U '_10_17'|

, l l ' l O ,(i;i),(l;-l ),{-2;-i)

Vi d\ 6) Giai h? phuong trinh voi nghifm la so thyc:

a) ^4x^ + (4x - 9)(x - y) + ^ = = 3y a)

De thay x = y = 0 khong phai la nghi^m ciia h# Ta xet x^ + y^ ?t 0

Nhan lien hc?p (*) ta c6: ^(4x-9)(x-y)-4y^ ^ yOj-y) Q ^ ^

473x(x + 2) = 3(x + 3) <=>

x = l

27 Tom lai c6 nghi?m duy nhat: (x;y) = ( l ; l )

b) Dieu kifn: x,y > 0 Ta viet lai phuong trinh (1) cua h? thanh:

^ x y - ( x - y ) ( ^ - 2 ) - y + >^-7y =0 (*) De thay x = y = 0 khong thoa

man h^ T a xet x^ + y^ 0

N h M e n h ^ p n t a c o : - J - V ) ^ - ! ^ - ^ ) ' ^ ^ = 0

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

Tir phuong trinh thu hai ciia h^ ta c6:

y + J ^ - 2 = — + x2-x-2 ^^ ^>0

^ ^ ^ x+1 x + 1 suy ra X = y thay vao phuong trinh thu hai cua h^ ta c6:

'x = l

l±Vi7 (x + l ) ( 3 x - x ^ ) = 4 o

X =•

Ket h(?p dieu ki$n ta c6: (x;y) = ( l ; l ) / ' l + ^/l7 1 + N/ I7

4 ' 4 c) Dieu ki^n: x > 0,y > 5 Ta viet Igii phuong trinh (1) cua h^ thanh:

^ x y - ( x - y ) ( 7 x y - 2 ) - y + > / x - ^ = 0 (*) De thay x = y = 0 khong thoa

man h^ Ta xet x^ + y^ ;t 0 Nhan lien h^p (*) ta c6:

<=>(x-yj , = + - T = ^

[Vx^ + i 6 ( y - x ) + 7 ^ Vxy + y

T u phuong trinh (1) ta c6: y - 5 - ^ y - 5 + x + 3 - 3Vx + 3 + 2 = 0 T a coi day

la phuong trinh b$c 2 an yJy-5 Dieu ki?n de phuong trinh c6 nghi^m la:

Vi dv 7) Giai phuong trinh v6i nghi^m la so thyc:

a) +2y^ +2x + 8y + 6 = 0 + xy + y + 4x + l = 0 b)

2x^ + 2xy + y - 5 = 0

y2 + xy + 5x - 7 = 0 Giai:

* Cachl:Dat x = u + a thay vao phuong trinh (1) cua h? ta c6:

y = v + b (u + a)2+2(v + b)2+2(u + a) + 8(v + b) + 6 = 0 v * >;

o +2v^+2(a + l)u + 4v(b + 2) + a^+2b2+2a + 8b + 6 = 0

Ta mong muon khong c6 so h^ng b^c nhat trong phuang trinh nen dieu

ki?n la: a + l = 0 a = - l b = -2

b + 2=:0

Tu do ta C O cac h dat an phu nhu sau: Dgt X y = = u - 1 v - 2 thay vao h$ ta c6:

u2+2v2=3 day la h$ dSng cap

u^ +uv = 2

Tu h? ta suy ra 2(u^ + 2v^ j = S^u^ + uvj <=> u^ + 3uv - 4v^ = 0 <=>

Cong vif c con lai la kha don gian

Cach 2:Ta cong phuang trinh (1) vai k Ian phuang trinh (2)

+2y^+2x + 8y + 6 + k x^+xy + y + 4x + l =0

<=>(l + k)x^ +(2 + 4k + ky)x + 2y^ +8y + ky + k + 6 = 0

Ta C O

A = (2 + 4k + ky)2-4(k + l)(2y2+8y + ky + k + 6)

= (k^ - 8k - 8)y2 + (4k2 - 32k - 32)y + Uk^ - 12k - 20

Ta mong muon A c6 d^ng (Ay -hB)^ o A = 0 c6 nghi^m kep:

o (4k2 - 32k - 32)^ - 4(k2 - 8k - 8)(l2k2 - 12k - 2o) = 0 o k = - |

Tu do ta C O each giai nhu sau:

Lay 2 Ian phuang trinh (1) tru 3 Ian phuang trinh (2) cua h? ta c6:

2(x2+2y2+2x + 8y + 6)-3(x2+xy + y + 4x + l J = 0

u = V

u = -4v

<=> -x^ - 3 x y - 8 x + 4y^ +13y + 9 = 0<:> x^ + (3y + 8)x - (4y2 + 13y + 9J = 0

Ta C O A = (3y + 8)^ + 4(4y^ + 13y + 9) = 25y^ + lOOy +100 = (5y + lO)^

Tu do tinh du^c:

3y + 8-(5y + 10)

x = ^ = - V -1

2 3y + 8 + (5y + 10)

x =

y

= 4y + 9 Phan vi?c con lai la kha don gian n , u

b) Lay phuang trinh (1) tru phuang trinh (2) ta thu dugc: s ^

2x*^ + 2xy + y - 5 - ^y^ + xy + 5x - 7j = 0 o 2x^ + (y - 5)x - y^ + y +12 = 0

x = - y + 2 Nhan xet: Khi gap cac he phuong trinh dang:

a j X ^ + ajxy + a3y^ + a4X + agy + = 0

b j X ^ + b2xy + b3y^ + b 4 X + bjy + b^ = 0 + Ta dat x = u + a,y = v + b sau do tim dieu ki?n de phuang trinh khong c6 so' hang bac 1 hoac khong c6 so' hang tu do

+ Hoac ta cpng phuang trinh (1) voi k Ian phuong trinh (2) sau do chpn k sao cho C O the bieu dien duQfC x theo y De c6 dugic quan h^ nay ta can dya vao tinh chat Phuang trinh ax^ + bx + c bieu dien du^c thanh dang:

(Ax + B)^ci>A = 0 Doi voi cac d^i so bac 3:

Ta C O the van dung cac huang giai + Bie'n doi h^ de tao thanh cac hSng dang thiic + Nhan cac phuang trinh voi mpt bieu thiic d^i so' sau do cpng cac phuang

Vi dy 8) Giai h^ phuang trinh vai nghi^m la so' thyc:

a) i x^ + 3xy^ = -49

c) x^ + 3x^y = 6xy - 3x -49 x^ -8xy + y^ =8y-17x c) x^ -6xy+ y^ =10y- 25x-9 b) x 3 - y 3 = 3 5 d) x^ + y^ = (x - y)(xy - 1) 2x^ +3y^ =4x-9y d) x^ - x^ + y +1 = xy(x - y - 1 )

a) Phan tich: Ta viet lai nhu sau:

G i i i : +3xy^ +49 = 0

y2+8(x + l)y + x2+17x = 0

Nh?n thay x = - 1 thi tro thanh: -3y2+48 = 0

y2-16 = 0 <=>y = ±4 iity

Tir do ta CO loi giai nhu sau:

Lay phuong trinh (1) cpng voi 3 Ian phuong trinh (2) ciia h? ta c6:

x^ + 3xy^ + 49 + 3(x2 - 8xy + y^ - 8y + 17x) = 0

o ( x + l)r(x + l)2+3(y-4)2] = 0

Tu do ta de dang tim dugc cac nghi^m cua h?: {x;y) = ( - l ; 4 ) , ( - l ; - 4 )

b) Lam tuong ty nhu cau a

Lay phuong trinh (1) cpng voi 3 Ian phuong trinh (2) thi thu du(?c:

(x +1) (x +1)^ + 3(y - 5)^ = 0 Tu do de dang tim dugc cac nghi^m cua h^

c) Lay phuong trinh (1) tru 3 Ian phuong trinh (2) ta thu dugc:

( x - 2 ) 3 = ( y + 3)^<::>x = y + 5

Thay vao phuong trinh (2) ta c6:

2(y + 5)2+3y2 ^4(y + 5 ) - 9 y o 5 y 2 + 2 5 y + 30 = 0 o y = -3

y = -2 Vay h? phuong trinh c6 cac nghi^m la: (x;y) = (2;-3),(3;-2)

d) Lay 2 Ian phuong trinh (2) tru di phuong trinh (1) ta thu du(?c:

Trir hai phuong trinh cho nhau ta c6: y = - 1 thay vao thi h? v6 nghif m

KL: Nghi^m cua h# la: (x;y) = 1 3 + 3N/5^

2 ' 4

1 3-32/5'

2 ' 4

PHLTONG PHAP DAT A N P H U

D|t an phu la vi?c chpn cac bieu thuc f(x,y);g(x,y) trong h? phuong trinh

de d$t thanh cac an phy moi lam don g i ^ cau true cua phuong trinh, h^ phuong trinh Qua do tao thanh cac h? phuong trinh moi don gidn hon, hay quy ve cac d^ng h^ quen thupc nhu doi xung, dla\ cap

De t^o ra an phy ngudi giai can xu ly linh ho^it cac phuong trinh trong h? thong qua cac ky thuat: Nhom nhan tu chung, chia cac phuong trinh theo nhung so'hang c6 sin, nhom dya vao cac hang dSng thuc, doi bien theo dac thii phuong trinh

Ta quan sat cac VI dy sau:

Vi dy 1: GiAi cac h^ phuong trinh sau

a) < 2 x 2 - 2 x y - y 2 =2

2 x ^ - 3 x 2 - 3 x y 2 - y 3 + l = 0 b) <

x * - 4 x ^ + y ^ - 6 y + 9 = 0 x^y + x^ + 2 y - 2 2 = 0

G i i i : a) Ta viet lai h? phuong trinh thanh:

[3x3+3x2y-(x + y ) 3 - 3 x 2 = - l [3x^{x + y)-(\ y f = - 1 \"

.2 D§t a = 3x ,b = x + y ta thu du^c h$ phuong trinh: o a-b^ =2

( x ^ - 2 f ( y - 3 f = 4 x^y + x^ +2y-22 = 0

b) Ta viet l?i phuong trinh thanh:

D|t a = x^ - 2; b = y - 3 Ta CO h? phuong trinh sau:

a 2 + b 2 = 4

a2 + b 2 = 4

<=> < j(a + b ) 2 - 2 a b = 4 (a + 2)(b + 3) + a + 2 + 2(b + 3) = 22 [ab + 4(a + b) = 8 [ab + 4(a + b) = 8

ab = 0

+ Neu: a = 0 , b - 2

+ Neu a = 2,b = 0=>

Vi d\ 2: Giai cac phuang trinh sau

a) x2 + y2j + xy + 2y^ + x - (x + y + l) = 25(y + l) 8y = 9 b)

D l t 2x + \/l7-4x^ = a;3x + y-y/l^ - 9y^ = b H | da cho tuong duong:

xy + x2y2 + i + x^y^ =4y^

Ta thay y = 0 khong thoa man h^.Chia phuong trinh dau cho y^, phuong

trinh thii 2 cho y^ ta dug-c:

a) Nhan thay x = 0 khong la nghi^m ciia h^

Chia hai ve phuong trinh cho x^ ta c6:

thanh: —+—=4 1 5 a b c ^ a = - , b = - « 1 5 1 R

b + 5a = 5 ^

( x ^ - y ) = x (x + y2) = 5y

(1) o x^y^ + 6xy + 9 + x^ + 2xy + y^ = 8 o x^y^ + x^ + y^ +1 = - S x y

o ( x 2 + l ) ( y ^ + l ) = - 8 x y

Nhan thay x = 0, y = 0 khong la nghi^m cua h?

Phuang trinh (1) khi do la: ^ ^ ^ - ^ " ^ = •

V^y h? CO nghifm (x;y) = (-1;2 - ),(-1;2 + >/3),(2 - VS; -l),(2 + V S ; - l ) ,

b) Phuang trinh (2) tuong duong:

(2x - y2 )(y - 9x2 ) = i8x2y2 ^ g^l^l ^ jg^3 ^ y 3 ^ 2xy

9x2y2+18x^+y3 „ ^ ISx^ y^ >'= 5

^ = 2<::>9xy + + — + 2 = 4 c,

o 9 x

Dat a =

xy 2x^

H +

y J

9x + ^ x>

b) • (x + 2 ) 7 y + l = ( x + l ) ' Gi4i

rat Hf« on mt aat nQesangratrva^ttj n, mi m, J i,mri vi -Nguyen irung^*^;^

t r i n h ciia h? ve d?ng: f [ u ( x ; y ) ] = £[v(x;y)] trong do h a m so dac trung

f(t) d o n di?u tang, hoac d o n di?u giam t u do suy ra u(x; y) = v(x; y)

+ De phat h i ^ n ra f [ u ( x ; y ) ] = f [ v ( x ; y ) ] ngoai vi^c thanh thao cac k y nang

bien d o i hang dang thiic, n h o m nhan t u chung d o i k h i ta can chia cho m p t

bieu thuc g(x;y) ho?c the m p t bieu thuc t u p h u o n g t r i n h t h u nhat vao

p h u o n g t r i n h con lai de t^o ra p h u o n g t r i n h c6 cau true h a m so'

Ta xet cac v i dy sau;

Vi dv 1: G i i i cac h$ phucmg t r i n h sau

2

Phucmg trirJi (1) t u o n g d u o n g :

( 2 - x ) V r ^ + V 2 ^ = ( 2 y - l ) 7 2 y - l + 7 2 y - l o f ( V 2 x - l ) = f ( 7 2 y - l )

Xet h a m so f ( x ) = x^ + x t a c6 f (x) = Sx^ + 1 > 0 suy ra h a m so f(x) d o n di$u tang -

T u do suy ra f ( V 2 - x ) = f ( 7 2 y - l ) < » ^ 2 y - l = V 2 - x < » x = 3-2y thay

vao ta c6: 3/5-2y+27y + 2 =5<»D|[t a = ^ 5 - 2 y ; b = ^ y + 2 ta c6 h?

phucmg t r i n h sau:

a + 2 b = 5 a^+2b2 =9

(x + 2)N£27l=(x + l ) ' ^ ( x + 2 f ( x 2 + l ) = (x + l ) '

V^y h? CO n g h i ^ m ( x ; y ) = (±^/3;s)

c = N/3,y = 3 : = -V3,y = 3 (thoa man)

V i dv 2: GiAi cac h ^ p h u o n g t n n h sau j j j , j j , v

2x3 _ 4^2 + 3x _ 1 ^ 2x3 ^2 - y ) ^ 3 - 2 y

a) x 5 + x y ' ' = y l ° + y ^

V4x + 5 + 7 y ^ + 8 = 6 b) yf^ = ^U-Xyl3-2y + 1

G i a i a) D i e u k i ^ n : x > — F, ;

4

Ta thay y = 0 k h o n g la nghi?m ciia h? chia hai ve ciia (1) cho y^ ta dupe:

Ta CO f ( l ) = 0 => x = 1 la nghi?m duy nha't Tu do tinh dugrc y = ±1 i-' •

Vay hf da cho CO nghi^m (x;y) = ( l ; ± l )

b) Dieu ki?n: x > -2; y < | Ta thay khi x = 0 thi h# khong c6 nghi§m

Chia phuang trinh (1) cho *0:

W -2-^4-4- = ( 4 - 2 y ) 7 ^

1 - i ^3 / 1 - 1

X

Xet ham so f(x) = x^ + xta c6 f ( x ) = 3x^+1 >0suy ra ham so f(x) dan

di^utang Taco f ( j 3 - 2 y ) = f [ ^ l - - J yJS-ly ^1

Thay vao (2) ta dupe: x + 2 - sJ\5-\ 1

Ta thay ve'trai la ham dan di^u tang nen phuang trinh c6 nghi^m duy nha't

Tru theo ve'hai phuang trinh tren ta duQC

u + Vu^ +1 + 3" = v + V v 2 + l + 3 ' ' Xet ham so f(t) = t + Vt^ + 1 + 3 ' Ta c6 f'(t) = 1 + + 3' l n 3 > OVt suy ra ham so f(t) dong bieh tren R Ta c6

-f(u) = f(v) <» u = V Thay vao phuang trinh dau ciia h? (*) ta c6:

u + Vu^ +1 = 3" o I n f u + V u ^ T l = u l n 3 ; Xet ham so f(u) = ln(u + Vu^ +1) - u In 3 ta c6

1 + u .4 (i&'^i'yV f'(u) = - 3 = / - l n 3 < OVu => f(u) la ham so nghjch bie'n

u + Vu^ +1 Mat khac f(0) - 0 u = 0 la nghi^m duy nha't ciia phuang trinh

Tu do suy r a u = v = 0<=>x = y = l

b) D^t z = 75 - 2y => z = - y - thay vao phuang trinh (1) ta c6: ^' ( '

4x + X = z 3 - 5 - z ,2> o Sx'' + 2x = z^ + z

Xet f (t) = t^ +1 ^ f'(t) = 3t2 +1 > 0 suy ra ham f (t) luon dong bieh

Tir do suy ra f(z) = f ( 2 x ) o z = 2 x o 7 5 - 2 y = 2 x o y = - the vao

phuang trinh (2) ciia h# ta dupe: g(x) = 4x2 + ' 5 - 4 x 2

Vi dv 4: Giii cac h? phuong trinh sau

M^t khac f (0) = 0 s = 0 la nghi?m duy nhat cua phuong trinh

Tuc la X - 1 = 0 Suy ra nghi^m duy nhat cua h^ phuong trinh da cho la

'•'-2

(x;y) =

b) Dieu ki?n: x - 2y +1 > 0

Phuong trinh (1) tuong duong voi 2x - 4y + 2 = 2y^ +1 + 2y^y^ +1 (3)

Thay vao phuong trirUi (2) ta dupe

(y + ^jy^+lf = 4 o y + ^ y 2 + l = 2 o

5 3 Thu lai thay x = - ;y = - thoa man

2 4 V|y h? phuong trinh c6 nghi^m (x;y) =

y + Vy2+9

(1)

(2)

b) x^-y^+2(y-x) = 61n y^ =x^-2x + l x +

Dieu ki^n:

Giai

x<5

y < 4 2x + y + 5 > 0 3x + 2y + 11^0 Bien doi phuong trinh (1) ta c6:

3(5 - x) + 2] = [3(4 - y) + 2 ] 7 4 ^ ( 3 ) Xet ham so: f(t) = (31^+2)t voi t>0 taco: f'(t) = 9t2+2>0

Do do f(t) la ham so dong bien tren R

Tir phuong trinh (3) ta suy ra:

1 ui nnu un ini ««»wpc yung luo vu giui yi, QUI fi, i-i,oui rrr- Nfuyju Tning^ferr

b) Phuong trinh dau tien tuong duong voi:

x^-2x + 6lnfx + \/x^ +9l = y^ -2y + 6lnfy + ^/y^ +9 \ V

Vay ham so f(t) dong bieh tren R nen f(x) = f(y) o x = y

Thay vao phuong trinh thii hai ta c6: x'^ - x^ - 2x +1 = 0

— J I - X T X I IVITV UVVH KHafig'VifT

Phuong trinh nay c6: f (-2)f (o) < 0 ;f (o)f (l) < 0 ;f (l)f (2) < 0 Vay phuang trinh c6 3 nghiem thupc doan (-2;2) nen ta dat x = 2cost voi xerO;n Thay vao ta c6: • ' ' '

8cos^t-4cos^ t-4cost + l = 0=>sint|8cos''t-4cos^ t-4cost + lj = 0

I di u?u un ini uui nyt sung

Thay vao (2) ta du^c: x^ - 5x^ + 14x - 4 = 6\/x^ - x + 1

Bien doi p h u o n g trinh da cho hiong duong:

K h i trong h? phucmg trinh c6 chua phuong trinh bac hai theo an x hoac y

ta CO the nghi deh cac huong x u ly n h u sau:

* Neu A c h i n , ta giai x theo y roi the vao p h u o n g trinh con lai cua h f de

giai tiep

* Neu A khong c h i n ta thuong x u ly theo each:

+ Cpng hoac t r u cac p h u o n g trinh cua h? de tao dug-c p h u o n g trinh b^c hai c6

A chan hoac tao thanh cac hang dang thiic

+ D i i n g dieu k i ^ n A > 0 de t i m mien gia trj cua bien x,y Sau do diing ham

so de danh gia p h u o n g trinh con l ^ i tren mien gia t r i x, y vua t i m du^c:

Ta xet cac v i sau:

V i 1: G i i i cac h f p h u o n g trinh sau

a) • xy + x + y = x2 - 2 y 2 (1) a) •

x 7 2 y- y V x - l = 2 x - 2 y (2) ' ''' '\'

b) • 2x2 + y2 _ 3xy + 3x - 2y + 1 = 0 4x2 - y2 + X + 4 = + y + ^ x + 4y

Giii

Xet p h u o n g trinh (1) ciia h f ta c6:

xy + x + y = x 2 - 2 y 2 « x 2 - x ( y + l ) - 2 y 2 - y = 0 Ta coi day la phuong

trinh bac 2 cua X thi ta c6: A = (y +1)2 + 8y2 + 4y = (3y +1)2 T u do suy ra

; ^ _ y + l - ( 3 y + l ) _

2 ^ ,

X = _ y + l + (3y + l )

= 2y + l

Truang hgp 1: x = - y T u phuong trinh (2) ciia h? ta c6 dieu ki?n:

suy ra phuong trinh v6 n g h i f m Truang hop 2: x = 2y + 1 thay vao phuong trinh t h u hai ta c6:

(2y + l)V2y - y V2y = 2y + 2 o y 7 2 7 + 72y = 2(y +1)

o (y + l ) ( 7 2 y - 2) = 0 « y = 2 => X = 5 Vay h? CO mpt cap nghi^m: (x;y) = (5;2)

) Xet phuong trinh (1) ciia h? ta c6:

2x2 + y 2 _ 3 x y + 3 x - 2 y + l = 0 o 2 x 2 + x ( 3 - 3 y ) + y 2 - 2 y + l = 0

Coi day la phuong trinh bac 2 ciia x ta c6:

A = ( 3 - 3 y ) 2 - 8 ( y 2 - 2 y + l ) = y 2 - 2 y + l = ( y - i ) 2

[ x > l len: <

V4x + 1 2V5X + 4 + 3 > 0 M^t khac f(0) = 0 => x = 0 la nghi?m duy

Ket lugn: phuang trinh c6 2 c|p nghi?m: (x;y) = (0;1),(1;2)

Vi d^ 2: G i i i cac phuong trinh sau

(1) a)

Phuang trinh (1) tuang duang (x + 3)^ = 4(y + l)(3y - x)

o x^ + 6x + 9 = 12y2 + 12y - 4xy - 4x o x^ + 2x(5 + 2y) - 12y^ - 12y + 9 = 0

Coi day la phuang trinh b^c 2 aia x ta c6:

A'= (2y + 5)^ + 12y^ + 12y - 9 = (4y + 4)^

•x = - 5 - 2 y - ( 4 y + 4) = - 6 y - 9

x = - 5 - 2 y + (4y + 4) = 2 y - l

Truong hgfp 1: x = -6y - 9

Do X > -3 => -6y - 9 ^ -3 o y ^ - 1 suy ra phuong trinh v6 nghi?m

Truong h^p 2: x = 2y - 1 thay vao phuang trinh 2 aia h? ta c6:

Ta viet phuang trinh thu nha't dudi dang:

^ 2 y 2 - 7 y + 1 0 - x ( y + 3) = x + l - 7 y ^

De binh phuang dugc ta can dieu ki?n: x +1 > ^ y + 1 o x^ + x ^ y ,

Ta binh phuang hai ve du^c:

2y2 _8y + 8 - X(y + 3) = x2 + 2 x - 2 ( x + 1 ) ( 1 )

Ta dua phuong trinh (2) ve d^ing: (x +1) ^y + 1 = x^ + x + 2xy + 2y - 3 (2) The (2) vao (1) ta du<7c:

2xy + 2y - 3J 2y2_8y + 8 - x ( y + 3) = x 2 + 2 x - 2 ( x 2 + x +

I 2y2 - 4y + 2 + 3xy + x^ - 3x = 0

2 „ / ,\/ ^2

x 2 + 3 x ( y - l ) + 2 ( y - l ) = 0 o ( x + y - l ) ( x + 2 y - 2 ) = 0 X + y - 1 = 0

x + 2 y - 2 = 0 V6i x + y - l = 0 o y = l - x , t a c 6 them x < 2 thay vao phuang trinh (2) ta c6: (x + l ) V 2 - x = - l + x-x2 o x 2 - x + l + (x + l ) > / 2 ^ = 0

Vi -1 ^ X < 2, ta de thay: VT > 0, nen suy ra phuang trinh v6 nghifm

Voi x + 2y-2 = 0 o y = - ^ ^ , thay vao phuang trinh (2) ta du(?c:

1 ^ x + Y = 2 D^t u = X + 1 khi do ta thu dug^c phuang trinh:

Ta Viet p h u o n g trinh (1) thanh: 7 4 x - y = 1 + J 3 y - 4 x Binh p h u o n g 2 ve ta

thu duoc: 2^3y-4x = 8x - 4y - 1 Thay vao p h u o n g trinh (2) cua h? ta c6:

4 x ^ - 4 x ( y + 2) + y ^ + 4 y = 0 Ta coi day la p h u a n g t r i n h bac 2 ciia x tlVi

T r u o n g h o p 1: y = 2x thay vao phuong trinh (1) ta c6: \f2x = -12 v 6 nghiem

T r u o n g h o p 2: y = 2x - 4 thay vao phuong trinh (1) ta thu dugc:

X e t h a i t r u o n g hgip: - f - A iti- fK^i^f <!

, N e u y = thay vao p h u a n g trinh t h u hai ciia h? ta duqyc:

M a t khac f (Vs) = 0 x = >/3 la n g h i f m d u y nhat

Vay h? da cho c6 d u y nhat m p t nghiem la (x; y) = {S) 2\/3 j

Vi 4: Giai cac phuong trinh sau

Ta thay d m o i h ^ p h u a n g trinh deu c6 chua p h u a n g t r i n h bac 2 n h u n g A

j^khong CO dang ( A x + B)^ nen ta khong the t i m x theo y Tir d o ta nghi den

luong t i m m i e n gia t r i x, y de l a m ca s6 danh gia p h u o n g t r i n h con lai

a) Xet phuong trinh thu 2 o i a h$: - x + + y - - = 0 Phucmg trinh c6

T h u lai ta thay chi c6 cap nghi?m: (x; y) = ' 32' 2 _r thoa man

b) Xet phuong trinh thu 2 cua h^: x ^ - x + y ^ + y - - ^ = 0 Phuang trinh c6

nghi^m khi A = 1 - 4 y + y - 2 = _4y2_4y + 3 > o o - | < y ^ - i

Cty 77VHH MTV DWH Khang Vift

Ta viet l^i phuang trinh thanh: + y+ -x + -— = 0 Phuong trinh c6

x 2+ 2 x ( y - l ) + y 2 - 6 y + l = 0, A = ( y - l ) ^ - y ^ + 6 y - 1 = 4y >0 o y >0

Ta viet lai:

y2 + 2 y ( x - 3 ) + x^ - 2 x + l = 0,A' = x 2 - 6 x + 9 - x 2 + 2 x - l = 8-4x>0<=>x<2 Phuong trinh (1) duQ'c viet lai nhu sau: Vx + 1 + ^ x - 1 = ^y"* +2 +

Xet ham so f(t) = t + Vt* +2 tren [0;+oo) ta c6 f'(t)=:l+ >0 nen

71^+2 ham so' f(t) dong bien Lai c6 f ( x - l ) = f(y*) o x = y^ +1 Thay vao phuong trinh thu hai cua h^ ta thu du^-c:

Tdi lifU on thi aai ho, s.hig tao vu xidi fl, bUtl'l, H( fl, mi Ul- K^uycn Tniflg^^

y | y ' ' + 2 y * + y - 4 ) = 0> y = 0

y^ + Zy* + y - 4 = 0

Xet g(y) = y^ + 2y* + y - 4 ta c6 g '(y) = 7y^ + 8y'^ +1 > 0 vai mpi y > 0 nen

ham so g(y) dong bien, ma g(l) = 0 => y = 1 la ngifm duy nhat

Tir do tim dugc 2 c^p nghi^m ciia h?la: (x;y) = (l;0),(2;l)

b) Ta viet lai phuong trinh thii 2 ciia hf th^nh:

The xy = -x^ - y^ + 3x + 4y - 4 tu phuong trinh (2) vao phuong trinh (1) ta

thu du(?c: 3x^ + 18x^ + 45x - 3y^ + 3y^ + 8y -108 = 0

+ Xet ham so f(x) = 3x^ + ISx^ + 45x tren -I

nen ham so f(x) dong bien Suy ra f(x) < f f-1

ta CO f'(x) = 9x^ +6x + 45>0

892

9 + Xet ham so g(y) = -3y^ + 3y^ + 8y -108 tren

g'(y) = 0<=>y = — , t u d 6 d e dang suy ra g(y) < g

taco g'(y) = -9y +6y+8

892 + Suy ra f(x) + g(y) < 0 Dau bang xay ra khi va chi khi x = y = —

Thir l^i ta tha'y cap nghi?m (x;y) = (4_ 4)

3'3 thoa man hf

PHLTONG P H A P D A N H G I A

De giai dup'C h^ phuong trinh bang phuong phap danh gia ta can nim chac

cac bat dSng thuc co ban nhu: Cauchy, Bunhicopxki, cac phep bien doi

trung gian giiia cac bat ding thuc, qua do de danh gia tim ra quan h$ x,y

Ngoai ra ta cung c6 the diing ham so de tim GTLN,GTNN tu do c6 huong

danh gia, so sanh phii hgp

TJty TNHH MTVDVVH KhangVm

V i d y 1: G i a i c a c p h u o n g t r i n h s a u

2 a)

b) Dieu ki^n: x > y^ > 0

Phuong trinh (1) tuong duong: x^ + x(x - y^ J - 2 ^ ( x - y ^ = Q

Dat V x - y 2 = u phuong trinh (1) thanh:

Tu do ta CO cac nghi^m ciia la: Vay h§ c6 nghi^m (x;y) =

i) Ta thay ^y^ +1 - y > y - y S O va f-^/y^ +1 - y A / y ^ +1 + y = 1 nen

phuong trinh (1) ciia h? c6 the viet lai nhu sau: 2x + V 4 x 2 + l = y + 7y^+l

Xet ham so f(t) = t +Vt^+l,f'(t) = l + , * = ^ * J ! I 1 ^ > 0 nen ham so

thiic can chung minh tuong duong vai: -— , — ^ 0 Nhung ( l + ab)(l + a^ l + b^)

dieu nay la hien nhien dung vai a,b ^ 1 Dau bang xay ra khi va chi khi

xay ra khi va chi khi x = 0 => y = 0

Tom lai: c6 mpt cap nghi^m duy nhat (x; y) = (O; O)

Cty TNHHMTVDVVHKhang Vijt

b) Theo bat dSng thuc A M - GM ta c6 :

x^ + y^ + 4x3y3 -3x^y^(x^ + y^) > 0 (2)

Vi y > 0 chia hai ve cho dat t = > 0 bat dMng thiic (2) tro thanh

ki?n J2x + y = 3 o x = y = l l a nghi^m ciia h? • ^ ,

Vi dy 3: Giai cac h? phuong trinh sau

b)

x/Zxy + Sx + S = 4 x y - 5 x - 3

Giai a) Phuang trinh (1) tuong duong:

Thay x = y vao phuong trinh con lai ta c6: x^jlx^+5x + 3 = 4x2 - 5x - 3

De y rang x = 0 khong phai la nghi^m Ta xet x > 0, chia phuong trinh cho

5 3

2 + - + — = 4 -'5 3 ^ Dat t = x^ thi thu duQic:

W^y h^ phuong trinh c6 nghi^m duy nhat (x; y) = (3; 3)

V i d^ 4: Giki cac h ? phuang trinh sau

1- A J

Giai

0 < x < 32

[ y < 4 Cpng hai phuang trinh ve theo ve ta c6:

a) Dieu ki^n:

>^ + V32 - X + ^ + 1/32 - x = y2 - 6y + 21 (*)

Ta c6: y2 - 6y + 21 = (y - 3)^ +12 > 12 Mat khac theo bat dang thuc Bunhiacopxki ta c6:

> y x + V 3 2 - X < ^ ( l + l)( x + 3 2 - x ) =8 + t/32-x < ^(l + l)(V^ + V 3 2 - x ) - 4

Vay Vx + V 3 2 - X + ^ + t/32-x < 12 Tir do suy ra h? c6 nghi^m khi va chi

khi x,y phai thoa man:

Vay h^ phuong trinh c6 1 nghi^m duy nhat (x;y) = (16;3)

Ngoai each dimg bat dang thuc ta cung c6 the diing ham so' do'i voi f(x) = ^ + N / 3 2 - X + ^ + ^ 3 2 - X tren [O;32]

That vay taco: ^'

Ket lu?in: c6 nghi?m d u y nha't x = y = 1 f

N h ^ n xet: Vi?c n h i n ra du(?c quan h? x = y la chia khoa de giai quye't bai

toan Day la ky nang dac bi?t quan trpng k h i giai h? bang p h u a n g phap

danh gia ciing n h u chung m i n h bat dang thuc

3" < 1 suy ra 2 ' * ( 2 ' ' - l ) + 3 ' ' ( 3 ' ' - l ) < O n h u n g 3 4 y - 2 6 x > 0

Vay h^ v6 nghi^m N h u v^y h? c6 nghi^m k h i x , y > 0

Ta viet lai p h u a n g trinh (1) cua h? thanh:

f (x) = 9" + 4" - 3" - 2" - 68x + 26;

r (x) = 9 ' ' + 4 ' ' - 3 ' ' - 2 ' ' - 6 8 ;

r ( x ) = 9 ' ' + 4 ' ' - 3 ' ' - 2 ' ' > 0 L9P bang bien thien cua ham so ta suy ra f(x) = 0 c6 toi da 3 n g h i f m M$t khac ta thay f(0) = f ( l ) = f(2) = 0 suy ra p h u a n g trinh c6 d i i n g 3 nghi^m

x = 0;x = l ; x = 2

V$y h? p h u a n g trinh CO 3c|pnghi?m: ( x ; y ) = ( 0 ; 0 ) , ( l ; l ) , ( 2 ; 4 )

b) T u p h u a n g trinh (1) cua h? ta suy ra x , y , z e [ 0 ; l "

Truoc he't ta chung m i n h : 3' > 21^ + 1 (*) v a i m p i t e [ 0 ; l "

T h a t v a y x e t h a m s o f(t) = 3 ' - 2 t 2 + l tren [ 0 ; l "

Ta CO f'(t) = 3' In 3 - 4t; f ' ( t ) = 3' In^ 3 - 4 ; f "(t) = 3' In^ 3 > 0

N h u vay h a m so f(t) = 0 c6 toi da 3 nghi^m

Mat khac f(0) = £(1) = f(2) = 0 nen f(t) = 0 c6 toi da 3 nghi?m la

GlAl He BANG SLf DUNG T I N H C H A T CUA s6 PHLfC

M u o n giai du(?c cac h? p h u o n g trinh bang p h u o n g phap s u d u n g so phuc,

can n h o n h u n g cong thuc co ban cua so phuc,

Phuong phap chung la nhan m p t p h u o n g trinh ciia h? v o i / sau do cong

hai p h u o n g t r i n h v o i nhau D y a vao cac tinh chat cua so phuc de thiet lap

Day la h? d i n g cap bac ba T u y nhien, neu giai bang p h u o n g phap thong

t h u o n g ta se d i den giai p h u o n g trinh bac ba: \f3t^ + 3t^ - sVst - 1 = 0

Phuong t r i n h nay khong c6 nghi^m dac bi^t

Xet so phuc z = x + y i V i z^ = x^ - 3xy^ + i ( 3 x V - ) nen tir h? da cho ta

da cho ta suy ra z^ = >/3 + i = 2 COS—+ i s m — 71 71

JW\ + • = 3

V i 4: Giai phuong trinh voi nghi^m voi x, y e M : • I 5x + y ;

1 - ^ -1

I 5x + y , Giai:

Tu suy ra x > 0, y > 0

Bai he nay khong c6 ngay dang nhu tren, tuy nhien voi muc dich chuyen

mau ve dang binh phuong modun ciia so phuc, chi can dat u = Vsx, v = ^

Vay nghi^m can tim la (x;y) =

Mot vi du tuong tu cua bai toan tren

Giai phuong trinh (*), ta c6 A' = 6 + 6V3i = 3(73 + i)

Suy racacnghi^m z = 73 + 3 + (73+ 3)i,z = 7 3 - 3 + ( 3 - 7 3 ) i

Vi u , v > 0 nen ta CO u = 73 + 3,V = 73+3, suy ra nghi^m cua h^ la:

giam tren tap xac djnh va h? c6 nghi^m (xi;x2; ;x^) thi X j = X j = = x„

Tai lifu on thi daihoc ^ng tao vd gidi Fl, mx f i , ftp f i , um u j JIIny^ggin 1 1 W A

Tinh chat 2: Neu f,g la cac ham khac tinh dan di§u va h? c6 nghi^m

I r ( x i ; x 2 ; ; x „ ) thi X j = X 2 = = x„ neu n le va

Giai

N2

T a c o 60x2-150x + 125 = 15

5 ^ ^ 5 ^ Tuang ty x > — — , y > — —

4 4 Xet ham so f (t) = 601^ - 150t +125 vol t > ^

5 ^

125 125 + > nen z > — —

Vi f'(t) = 1 2 0 t - 1 5 0 > 0 V t €

5 ^

-:+co , nen ham so f(t) dong bien tren

+) Ne'u x > y thi Sx^ >Qy^ nen f (y) > f ( z ) , hay y > z d o d o f ( z ) > f ( x ) nen

z > x Vi the x > y > z > X (v6 ly)

+) Tuong tu cung xay ra truong hop x < y

V^y X = y va tu do CO x = y = z Phuong trinh thu nhat ciia h# thanh

Sx^ - 60x2+150x-125 = 0 hay (2x - 3 f = 0 nen x = — 5

2 Vay nghif m can tim ciia h^ la (x;y;z) = f 5 5

2 ' 2

5^

' 2 3x3+3x = + 1

V i dy 2 Giai h? phuong trinh: • 3y3 + 3y = 3z2 + 1

3z3+3z = = 3x2 + 1

G i i i

Vi 3y2 +1 = 3x^x2 + i j nen x > 0 Tuang ty y , z > 0 > Tren khoang (0;+oo) cac ham so f (t) = + 3t,g(t) = 31^+1 deu dong bien

neu x > y thi f ( x ) > f ( y ) nen g ( y ) > g ( z ) suy ra y > z K h i do Igi c6

f ( y ) > f ( z ) nen g ( z ) > g ( x ) hay x > z , tuc la x > y > z > x do do phai c6

x = y = z

Phuong trinh thu nhat ciia h? thanh 'ix^ - 3x2 + 3x - 1 = 0 (1) Phuong trinh (1) tuong duong vox 2x^ + (x -1)^ = 0 nen x = — ^ ^

1 + ^ Vay nghiem can tim ciia h^ la (x; y; z) =

, l + ^ ' l + ^ ' l + ^ 2x (y2+7| = y(y2+63)

V i dy 3 Giai hf phuong trinh: • 2y(z2+7j = z(z2+63)

2z (x2+7J = x(x2+63)

Viet lai h? duoi dang:

G i i i

y y2+63) y2+7

z z2+63 z2+7

|x2+63 x2+7

t]'\O r: •

tft2+63) Xet ham so f (t) = '- voi t e K

t2+7 ( t 2 - 2 l f ( t 2 7 f

Ta CO f'(t) >0 voimpi t € l Suy ra ham so dong bien tren M Vi the neu x > y thi f (y) > f (z) nen y > z , suy ra f ( z ) > f ( x ) dodo z > x , t u c l a x > y > z > x

Vay X = y = z Khi do phuong trinh thu nha't aia h? thanh - 49x = 0 nen

X = 0 hoac X = ±7

Nghi^m aia h# can tim la (O; 0; O), (7; 7; 7;), (-7; -7; -7 )

V i 4 Giai h^ phuong trinh:

V z^ + S ^ x ^ - V x ^ Viet h^ da cho ve dang:

Xet ham so f (t) = Vt^+S va g(t) = t^ - vol t - ; o o 3

Ta CO f'(t)= , * >0,g'(t) = 2t 1 ^ 4 t V t 1 1 ^ ^

t€ 3

Suy ra cac ham so deu dong bien, nen de dang suy ra x = y = z

Khi do phuong trinh thii nhat ciia h^ thanh - 1 hay

Phuong trinh (2) tuong duong voi x = 2

Vay nghi^m can tim ciia hf (x; y; z) = (2; 2; 2)

2x3 + 3x2--18 = y3 + y

Vi dv 5 Giai h^ phuong trinh: + 3y2 -18 = z3 + z

2z3 + 3 z 2 --18 = x3 + x Giai

Idi lieu on tht aat hgc songtao va giat fi, vax ri, ne t-i, oai vi -Nguyen irurixj^^^

5

= 13

2x +

-x + y = 1

1) Bien doi h§ phuong trinh thanh: x 2( x - y ) = (x + y ) ' + 7 ( x - y ) + 4 (1)

4 = - 3 x 2 - y 2 + 8{ x - y ) (2) Thi^c hi^n phep the (2) vao (1) ta c6:

2r) ,v- U ••^;r The V = 5 - 3 u vao phuang trinh (*) giai tim dug^c u = l , t u d 6 v = 2

=>x = -3;y = 2

3) FT thu hai aia h^ - ^

<»x + 2y + 27x + 2 y + l = x2+2x + l o ( 7 x + 2 y + l j ^ = ( x + l f <::>^ + 2y =x hoac ^x + 2y = - x - 2

T H I : 7x + 2y = x o 13x2 - l l x - 3 0 = 0 TH2: 7x + 2y = - x - 2 o

x > 0

2 thay vao phuang trinh thu nha't ta dugc

2y = x - X

x + 2<0 , thay vao phuang trinh thu nha't ta 2y = x 2 + x + l ^ ^ ^

duqc bac hai theo x 4) Dieu ki?n: x > 4;y > 0;x2 > y;4x > y;y > 3x Phuong trinh (1)

O x = 5v ^ _ —

f V x 2 - 1 6 + 3 V x - 4 + 1

phuong trinh v6 nghi^m J i

T o m Igi h? CO nghi^m duy nhat: (x;y) = (5; 16) j

V^y nghi^m cua h$ la (x; y ) = (2; 2)

-, phuong trinh (1) cua h? da cho tuong d u o n g v o l :

Ket lu?in: c6 nghi^m duy nhat: ( x ; y ) =

7) T u phuong trinh (1) ta rut ra dugfc:

+ x 9 i - 3 => y 7 t - 4 thi binh phuong hai ve phuong trinh (*) j

(x + 3)(y + 4 ) < 0

- < » y + 4 = -2(x + 3 ) o y = - 2 x - 1 0 ( y ^ 4 ) ^ = 4 ( x + 3)2 ,.„.x,*

Thay vao phuong trinh (2) va riit gpn ta dugc:

4x2 + 28x + 51 + 3^4x +15 = 0

tk 4(x2+8x + 16) + 3^4x + 1 5 - ( 4 x + 13) = 0

1 tij neu On rnt a«i npc san^ luv r*/ r i , TT^TT imt T J I ~ rs^yeti i T T W ^ W * ^

Ta se chung minh phuong trinh nay v6 nghiem nhu sau:

De thay voi mpi x thi 4x'^ + 28x + 51 > 0

9) Tu phuong trinh (2) ta thu duqc: y ^ = 2 - x ^ - y - ^

Thay vao phuong trinh (1) ta c6:

x^y^ + 8xy^ - 6xy - 12y^ - 7y + 8 = 0 13y^ + y + l - 6 x y ^ = 0 • ; 0-Lay (1) + (2) ta c6 dugc phan tich sau: ,

x^y^ + 2xy2 + y^ - 6xy - 6y + 9 = 0 o [y(x + l)f - 6y(x +1) + 9 = 0 j

Tadugc y(x + l ) = 3<»19y^-17y + l = 0

49-3N/213 17 + 7213 ) f 49 + 37213 17 - 72131

(x;y) =

38 11) Dieu kien: y ;^ 0

Voi y ^ 0 ta bien doi he phuong trinh thanh

Cpng (3) va (4) theo ve va thu gpn ta dugc

a - a - 2 = 0<=> a = - l

a = 2

T H l : a = - 1 => b^ + 2b + 4 = 0 ( VN)