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 1 166

• Danh cho hoc sinh Idp 12 chiicrng trinh chuan va nang cao • On tap va nang cao k i nang lam bai • Bien soan theo noi dung va cau true de thi ciia Bo GD8fDT -T H i ; VIEW -TIfy'HBINH -T

5 1 0 7 6

/cirxg boo vo gioi

PHl/ONG TRlNH

X BAT PHl/ONG TRlNH H£ PHl/ONG TRINH

• D^nh cho hQC sinh Idp 12 chiicfng trinh chuan n^ng cao

• On tap nang cao ki nang 1km hki

m Bien so^n theo npi dung vk cau true di thi cua B Q G D & D T

• Danh cho hoc sinh Idp 12 chiicrng trinh chuan va nang cao

• On tap va nang cao k i nang lam bai

• Bien soan theo noi dung va cau true de thi ciia Bo GD8fDT

-T H i ; VIEW -TIfy'HBINH -THUAN'

Ldi noi dHu

Phuong trinh, bat p h u o n g trinh, h f p h u o n g trinh, bat d5ng thuc la m^ng

kien thuc quan trong trong chuong trinh toan pho thong Dac bi^t cac bai toan

ve p h u o n g trinh, bat p h u o n g trinh, h^ p h u o n g trinh, bat dang thuc t h u o n g

xuyen xua't hien trong cac ky thi chQn h p c s i n h gioi, cOng n h u Tuyen sinh dai

hQC va luon gay kho khan cho hoc sinh

NhSm giiip cac e m hpc sinh THPT cung n h u cac e m hpc sinh chuyen Toan

CO mpt tai l i f u mang tinh h f thong de o n l u y f n , nang cao kien thuc k y nang

giai toan de dat ket qua cao nha't trong cac k y thi Hgc sinh gioi, k y thi Tuyen

sinh dai hpc, cung n h u thi vao cac lop chuyen chpn, toi bien soan cuon: "Tai

li^u on thi Dai hoc - sang tao va giai phuong trinh, bai phuong trinh, h$

phuong trinh, bai dang thitc"

N p i d u n g cuon sach dupe chia lam 4 phan:

,Phan 1: Phuong phap giai p h u o n g trinh, bat p h u o n g trinh v 6 t y

Phan 2: Phuong phap giai h^ p h u o n g trinh

Phan 3: Phuang phap ham so'trong cac bai toan chua tham so'

Phan 4: Phuong phap h a m so trong chiing m i n h bat dang thuc va t i m

G T L N , G T N N

Trong m o i phan toi luon c6' gang h^ thong p h u o n g phap, phan tich, djnh

huong each giai, cuoi m o i phan deu c6 bai tap ren luy^n de cac e m hpc sinh

t h u s u c

Toi hy vpng cuo'n sach se la tai li^u bo ich cho cac em hpc sinh hpc tot m o n

Toan va dat ket qua cao trong cac ky t h i

Mac d i i da co gang danh nhieu tam huyet cho vi?c bien soan cuon sach

song thie'u sot la dieu khong the tranh khoi Rat m o n g s u dong gop phe binh

ciia ban dpc de Ian tai ban sau dupe hoan thi^n hon

Cuoi cung toi xin g u i lai cam o n sau sac den ban be, dong n g h i f p, cac dien

dan toan da cung cap m p t so'tai li^u quy gia de hoan thi^n cuon sach

I NHOTNG KI^N THLfC B6 T R O C H O GlAl P H l / O N G TRINH V O i t

1 Giai phuang trinh bac 4:

a) PhuoiTg trinh dang: x^ = ax^ + bx + c

Phuong phap: Ta them bot vao 2 v e m p t lupng: 2mx^ + m^ k h i d o p h u o n g

trinh tro thanh: (x^ + m)^ = (2m + a)x^ + bx + c +

Ta m o n g m u o n ve'phai c6 dang: (Ax + B)^

Phuong trinh tro thanh: y2 + ^/^^^ ^ 5 ^ j y ' " l O y ' - y + 20 = 0

(0 < y < Vs > ^f,,

Xet p h u o n g trinh: y4 - I0y2 - y + 20 = 0 o y" = 10y2 + y - 20

Ta them vao 2 ve'phuong trinh mpt lupng: 2my^ + m^

K h i d o p h u o n g trinh tro thanh:

y'^ + 2my2 + m^ = (10 + 2m)y^ + y + m^ - 20

Tdt lieu ou tin h,u si'iii^ tao vd ^fiat fi, oai fi, ne yrrmn^i -nguyen irung I V I C T I

Ta them vao 2 ve phuang trinh mot luong: 2my^ + m^

Khi do phuang trinh tro thanh:

y^ + 2my^ + m^ = (22 + 2m)y^ + 8y + - 77

Ta CO Ayp = 1 - 4(22 + 2m)(m2 - 77) = 0 o m = - 9

Ta viet lai phuong trinh thanh:

y'* - 18y2 + 81 = 4y^ + 8y + 4 o (y^ - 9 - (2y + 2 ^ = 0

b) Phuang trinh dang: x"* + ax^ = bx^ + cx + d

Ta se tao ra 6 ve phai mot bieu thuc binh phuong dang:

Bang each khai trien bieu thuc:

Ta thay can them vao hai ve mpt luong:

phuong trinh tro thanh:

V i du 2: Giai cac phuang trinh:

a) 2 x 2 - 6 x - l = V4x + 5 b) x^ = (1 - V5^)(2x - sVx + 3)

Giai:

a) Dieu kifn: x > —

4 Binh phuong hai ve'ta thu duoc phuong trinh:

o x ^ - 6 x ^ +8x2 + 2 x - l = 0<»x^ - 6 x ^ =-8x2 - 2 x + l

Ta tao ra vetrai dang: (x^ - 3x + m)^ = - 6x^ + (9 + 2m)x2 - 6mx + m^ Tuc la them vao hai ve mpt lup-ng la:(9 + 2m)x2 - 6 m x + m2 phuong trinh tro thanh: (x^ - 3x + m)^ = (2m + l)x2 - (6m + 2)x + m^ +1

Ta can A\p = ( 3 m + 1 ) - ( 2 m + I)(m2+1) = 0 o m = 0

Phuang trinh tro thanh: (x^ - 3x)2 = (x -1)^

thoa man phuong trinh

b) Dat N/X = y > 0 thi phuong trinh da cho tro thanh:

y^ = (1 - y)(2y2 - 3y + 3) <=> y^ + 2y^ = Sy^ - 6y + 3

Ta tao ra phuang trinh: (y^ + y + mf =(2m + 6)y2 +(2m-6)y + m2 +3

II MOT S 6 DANG PHUONG TRJNH VO THL/ONG GAP

1 Giai phuong tiinh v6 ty bling phuong phap su dung bieu thuc lien hgp:

Dau hi^u:

+ Khi ta gap cac bai toan giai phuong trinh dang: yfu^ + '^g(x) + h(x) = 0

Ma khong the dua ve mpt an, hoac khi dua ve mot an thi tao ra nhihig

phuong trinh bac cao dan den vi^c phan tich hoac giai true tiep kho khan

+ Nham dugc nghi^m cua phuong trinh do: bang thii cong (hoac su dung

may tinh cam tay)

Phuong phap:

• Dat dieu kif n chat cua phuong trinh ( neii c6)

Vi du: Doi phuong trinh: Vx^ +3 + 3 ^fly^+V + 2x

+ Neu binh thuong nhin vao phuong trinh ta tha'y:

Phuong trinh xac dinh voi moi x € R Nhung do chua phai la dieu ki^n

chat De giai quyet triet de phuong trinh nay ta can den dieu ki^n chat do la:

+ Ta viet lai phuong trinh thanh: N / X ^ + 3 - ^ 2 x 2 + 7 = 2 x - 3

Dey rang: ylx^+S-^ll)^ + 7 < 0 do do phuong trinh c6 nghiem khi

3

2 x 3 < 0 o x <

-2

• Neu phuong trinh chi c6 mot nghiem XQ :

Ta se phan tich phuong trinh nhu sau: Viet lai phuong trinh thanh:

# 0 - + - - ^ ^ g O ^ + h(x) - h(xo) = 0

Sau do nhan lien hgp cho tung cap so hang voi chii y:

+ ( ^ - b ) [ ^ + ^ b + ^ + ( ^ / I - b ) ( V ^ + b) = a -b2

= a-b^

f / + Neu h(x) = 0 C O nghiem x = XQ thi ta luon phan tich dugc

h(x) = (x -Xo)g(x) Nhu vay sau buoc phan tich va riit nhan tu chung x - XQ thi phuong trinh

ban dau tro thanh: (x - Xo)A(x) = 0 <=> X - Xg = 0

A(x) = 0 Vi?c con lai la diing ham so', bat dang thuc hoac nhirng danh gia co ban de ke't luan A(x) = 0 v6 nghiem

• Neu phuong trinh c6 2 nghiem x ^ X j theo dinh ly viet dao ta c6 nhan tu

chung se la: x^ - (xj + X2 )x + Xj Xj

Ta thuong lam nhu sau:

+ Muon lam xua't hien nhan tu chung trong ^ ( ( x ) ta tru di mot lugng ax + b

Khi do nhan tu chung se la ke't qua sau khi nhan lien hgp ciia

+ Hoan toan tuong tu cho cac bieu thuc con lai:

Ta xet cac vi du sau:

Vi d\ 1: Giai cac phuong trinh:

a) V 5 x ^ - l + ^ 2 x - l + x - 4 = 0 b) V x ^ + V 4 - x = 2 x 2 - 5 x - 3

Giai:

a) Phan tich: Phuong trinh trong de bai gom nhieu bieu thuc chua can nhung

khong the quy ve 1 an Neu ta liiy thua de tri^t tieu dau \/~, yT thi se tao ra phuong trinh toi thieu la bac 6 Tit do ta nghi den huong giai: Su dung bieu

thuc lien hgp hoac dimg ham so :

Tu djnh huong tren ta c6 loi giai nhu sau:

Dieu ki^n x > ^—

Ta nham dugc nghiem cua phuong trinh la: x = 1

T u d o ta CO loi giai n h u sau:

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

> A C ^ - 1 + 1 - N / 4 ^ = 2 X 2 - 5 X - 3

^ x - 3 ^ X - - 3 ^(x_3)(2x + l)

1 1 ( x - 3 ) - ( 2 x + l) = 0

T u do suy ra: x = 3 la nghiem duy nhat ciia p h u o n g trinh

N h a n xet: De danh gia p h u o n g trinh cuol cung v6 nghiem ta thuong d u n g

cac uoc l u g n g co ban: A + B > A v o i B > 0 tir do suy ra < 1 v o i m o i

< 0 (Bang each thay m p t

gia tri X > 7 2 ta se thay

I > 2 <=> x2 + 3x - 1 > 27x^ - 2 » x^ + 2x^ + 7x2 - 6x + 9 > 0

7 x ^ - 2+ 5

<=> (x^ + x)^ + 6x^ - 6x + 9 > 0 Dieu nay luon dung

T u d o suy ra p h u a n g trinh c6 nghiem duy nhat: x = 3

V i vay p h u o n g trinh c6 nghi?m duy nhat t = 2 <=> x = 8

N h a n xet: Vi^c dat \/x = t trong bai toan de giam so l u g n g da'u can da giiip

d a n gian h i n h thuc bai toan

Ngoai ra k h i tao Hen h o p do (t^ - 4) > 0 nen ta tach no ra khoi bieu thuc de

cac thao tac tinh toan dup-c don gian hon

V i dy 3: Giai cac p h u o t i g trinh:

19

a) Dieu ki^n: - 3 < x < —

3

Ta nham duoc 2 nghiem la x = l , x = - 2 nen ta phan tich de tao ra nhan t u

chung la: x^ + x - 2 De lam dug-c dieu nay ta th^c hien them bot nhan t u

Vay phuong trinh c6 2 nghiem la: x = 3, x = 8 I ' •

N h a n xet: Neu da nham dugc hai nghiem ciia p h u o n g trinh va d u doan dugc p h u o n g trinh chi c6 2 nghiem thi ta c6 the giai theo m p t each khac

n g i n gon hon nhu sau:

Xet ham so f(x) = 4Vx + 3 + V l 9 - 3 x - x^ - 2x - 9 tren

Tren -3; 19 ta C O

f'(x) = ^ • 2 x 2 ; f ' ( x ) =

-' 3 )

^ - 3 x ' ^ ( x + 3 f ^ ( 1 9 - 3 x )

nen f(x) = 0 c6 toi da hai nghiem tren

Mat khac: f(-2) = f(l) = 0 nen phuong trinh c6 d i i n g hai n g h i f m la

Phuong trinh dugc viet lai n h u sau: 5 V 3 X - 8 - 5 N / 5 ^ = 2 X - 1 1

Ta nham duoc 2 nghiem x = 3, x = 8 nen suy ra nhan t u chung la:

x^ - l l x + 24

Ta phan tich v o i nhan t u 5\/3x-8 n h u sau:

+ Tao ra sVSx - 8 - (ax + b) = 0 sao cho p h u o n g trinh nay nhan x = 3,x = 8 la

nghiem

Tuc la a,b can thoa m a n he: 3a + b = 5 j a = : 3

8a + b - 20 ^ b = - 4 + T u o n g t u v o i 5vx + 1 - ( m x + n) = 0 ta thu dugc:

Phuong trinh da cho tro thanh:

Ta chung m i n h : A ( x ) < 0 tuc la:

Cty TNHH MTV DWH Khattg Viet

Vay p h u o n g trinh c6 2 nghiem la: x = 3, x = 8

C h u y :

N h u n g danh gia de ke't luan A (x ) < 0 t h u d n g la nhirng bat dang thuc khong chat nen ta luon dua ve dugc tong cac bieu thuc b i n h p h u o n g

Ngoai ra neu tinh y ta c6 the thay: 5V3x - 8 + 3x - 4 - 9 ( x + 7 + sVx + 1) < 0

5\J3x - 8 + 3x - 4 < 9x + 63 + s V s i x + 81 N h u n g dieu nay la hien nhien d u n g

do: 5 V 3 x - 8 < 5 7 8 ] x + 8 1 ; 3 x - 4 < 9 x + 63 v o i m g i x > - ,,,,

3 Ngoai ra ta cung c6 the giai bai toan theo each d u n g h a m so 6 cau a) c) Dieu kien: x > 0 j ,

Ta nham dugc x = 1; x = 3 nen bie'n doi p h u o n g trinh n h u sau:

r

7 7

X - 4 x + 3 x - 4 x + 3 Vx3 + 3 x + 2 x " 2(x + l )

x ' ' - 4 x + 3 = 0 (1)

V x ^ + 3 x + 2 x - 2 ( x + l ) (2) Giai (1) suy ra x = 1, x = 3

Giai (2) ta c6:

Vx-* + 3 x + 2x = 2(x + 1) c:> Vx^ + 3 x = 2 < o x ^ + 3 x - 4 = : 0 < = > x = l Ke't luan: Phuong trinh c6 nghiem la x = 1; x = 3

N h a n xet: Ta cQng c6 the phan tich p h u o n g trinh n h u cau a,b ' "

d) Ta c6: x^ + 5x^ + 4x + 2 = (x + 3)(x^ + 2x + 3) - 5x - 7 nen bat p h u o n g trinh

t u o n g d u o n g v o i

; = Vx^ + X + 2 « x + 3 - V x ' + 2 x + 3 = 0

x - + 2 x + 3 x ^ + 2 x + 3

Tai lifu 6n thi d^i hQC sang t^o vd giai FT, bat l^l, WPT, bal VI - Nguyen i rung r^ien

Vi du 4: Giai cac phuong trinh:

D i i n g may tinh bo t u i ta thu dugc 2 nghiem la: x^ - - 0 , 6 1 8 , X 2 =1,618

T u d o ta thay x ^ + X 2 = l , X j X 2 = - l nen n g h l den nhan t u chung la

-0,618a + b = 2,61 Ia = - 1 l,618a + b = 0,38 ' ^ ] b = 2

Ta nham dugc t = 1 N e n phan tich p h u o n g trinh n h u sau:

Tai l.cn 6nthUaihocshngtaovagtdtPl,batl'l,hel^l,bai VI -N^yen imngi^ien

De phuang trinh c6 nghiem ta can: 3 t - 2 > 0 o t > - Nham duoc t = 1 nen

' ta viet lai phuong trinh thanh: Vt^ +15 - 4 = - 3 + 3t - 3

nghiem duy nhat t = 1 <=> x = 1

Truong hgp 2: x < 0 Chia hai ve phuong trinh cho x ta thu dugc:

Suy ra phuong trinh v6 nghiem % ' '

Tom lai: Phuang trinh ban dau c6 2 nghiem la x = 0, x = 1 , ' _

Nhan xet: Trong mpt phuong trinh c6 chiia nhieu dau ta nen dat an

phu la mot bieu thuc chua sao cho vif c bieu dien x theo an do la don

gian nhat Vi^c la nay se giiip cau true phuang trinh mod da phuc t^p hon

b) Dieu ki^n x e

-'•'-1

Ta viet lai phuong trinh nhu sau: sfsx + l - Vx + 3 +1 - x = 0

2x-2 + l - x = 0 o ( 2 x - 2 ) ' , - ± V3x + l + V x + 3 ^ W 3 x + l + V x + 3 2 j = 0

phuong trinh ban dau rat nhieu

+ De y rang khi x 1 thi VSx + l = Vx + 3 nen ta se lien hgp tr\rc tiep bieu

thuc: Tsx + l - Vx + 3

= 0

Tm V!r:N TifvHBINH THUA;\

1 7

2 Dat an phy dya vao tinh ding cap ciia phuong trinh:

Ta thuang gap phuang trinh dang nay 6 cac dang bien the nhu:

De giai cac phuong trinh (1), (2)

Phuong phap chung la:

+ Phan tich bieu thuc trong dau V~ thanh tich ciia 2 da thuc P(x),Q(x)

+ Ta bien doi ax^ + bx + c = mP(x) + nQ(x) bang each dong nhat hai ve

Khi do phuong trinh tro thanh: mP(x) + nQ(x) = d^/P(x)JQ(x)

Chia hai ve cho bieu thuc Q(x) > 0 ta thu dup'c phuong trinh:

P(x) , „ ^ fPix) fPW

m — ^ + n = d J — ^ Dat t = > 0 thi thu dugc phuong trinh:

Q (x) VQ(X) • Wi^)

mt^ - dt + n = 0

Mpt each tong quat:

Voi nriQi phuong trinh c6 dang:

aP"(x) + bQ"(x) + C P "-''(X)Q''(X) + d^^?{x).Qi\) = 0 thi ta luon giai dugc

theo each tren

Gia su x^ -3x + 2 = m(x + 2) + n(x^ -2x + 4) Suy ra m,n phai thoaman

-2(x + 2) + 2(x^ - 2x + 4) - 3yJ{\ 2)(x'^ - 2x + 4) = 0 Chia phuong trinh cho

- 2x + 4 > 0 ta thu dugc: -2 x + 2 - 3 (x + 2) + 2 = 0

(x^ - 2 x + 4) Dat t = — — > 0 ta thu duoc phuong trinh: -2t^ - 3t + 2 = 0

X = 3 + N/I 3

x = 3-Vl3

<=> 0 <x<2-S x>2 + S

x^ - 4x + l >0 Binh phuong 2 ve'ciia phuong trinh ta thu dugc:

x^ <^ + 2x + 1 + 2(x + l)7x^ -4x + l + x^ - 4 x +1 = 9x

c^2x- - n x + 2 ^ 2 j ( x 2 + 2x + l)(x^ -4x + l) =0

Gia su 2x^ -11x + 2 = m(x^ + 2x + l) + n(x^ -4x + l)=>

Phuong trinh tro thanh:

m + n = 2 2m - 4n -11 <=>

-1 + 5 x^ x^ - 4 x - 4 x + l + l J J x^ x^ - 4 x - 4 x + l + l = 0 Dat t - 'x^ = 0 Dat t - - 4 x + l '

^x^ + 2x + l , i x^ + 2x + l^ • \ ^x2+2x + l ^ > 0 ta CO

Ket luan: Phuong ^rinh c6 2 nghi^m -x = - , x = 4

Nhan xet: Trong lai giai ta da bien doi:

Xet phuong trinh: 2x^ - 2 x - 2 -3xVx +1 = 0 <=> 2x2 - 3xVx +1 - 2 ( x +1) = 0

De thay x = -1 khong phai la nghi^m

Xet X > -1 ta chia cho x +1 thi thu dugc phuong trinh:

-2t2 - 3 t + 2 = 0 < »

J— - — = 0 Dat t = /—-— > 0 phuong trinh moi 2 x - 1 2 x - l

t = -2 = 1

2

Voi t = i taco: F ^ = i c : > x 2 - 8 x + 4 = 0 o x = 4 + 2 V 3

x = 4-2>/3 Nhan xet:

+ D61 voi phuong trinh 2x2 _ ^ 2 - 3 x V 2 x - l =0 ta c6 the khong can dua

X vao trong dau V~ khi do ta phan tich: 2x2 _ 4^ + 2 = mx2 + n(2x -1) va

chia nhu tren thi bai toan van dugc giai quyet Vif c dua vao la giiip cac

em hgc sinh nhin ro hon ban chat bai toan

+ Ngoai ra can iuu y rang: Khi dua mgt bieu thuc P(x) vao trong dau thi dieu kien la P(x) > 0 Day la mgt sai lam hgc sinh thuong mac phai khi

giai toan

Tni lieu on thi iliii hoc •.mix tao vii giai PT, hat PT, he PT, boT DT- Nguyen TrungKien

b) D i e u ki^n:

- 3 x - 1 8 > 0

x > 0 < = > x > 6 5x^ + 4 x > 0

P h u o n g t r i n h da cho dvtqc viet lai thanh: Vsx^ + 4x = Vx^ - 3x - 1 8 + 5^/x

Cty TNHH MTV DWH Khang Viet

T o m lai: P h u o n g t r i n h c6 2 n g h i f m la: x = ^ va x = 9 c) D i e u ki?n x > 5

C h u y e n ve b i n h p h u o n g ta dugc: 2x2 - 5x + 2 = S^^x^ - x - 20J(x +1) Gia sir: 2x2 _ 5 x + 2 = m|x2 _ x - 2 0 J + n ( x +1)

^ ~ 4

V i du 3: Giai cac phuong trinh:

a) 7x2 + 2 x + 7 2 x - l = 7 3 x 2 + 4 ^ + 1 b) x ^- 3 x 2 + 2 ^ / ( x + 2 ) ^ - 6 x = 0

(x^ + 2x) - (2x -1) - ^(x^ + 2x)(2x -1) = 0 - J ( 2 x - l 1

,x^ +2xJ i + 1 = 0 Dat t = 2 x - l

Ix^ +2x > 0 = > - t ^ - t + l = O o t =

Ve CO ban den day ta hoan toan tim dugc x Nhung voi gia tri t nhu vay

viec tinh toan se gap kho khan

De khac phuc ta c6 the xu ly theo huong khac nhu sau:

Ta viet lai: ^(x^ + 2 x ) ( 2 x - l ) = 7(x + 2)(2x^ - x ) liic nay bang each phan tich

nhu tren ta thu dugc phuong trinh:

Neu ta dat y = Vx + 2 thi phuong trinh tro thanh: x^ - 3xy^ + 2y'^ = 0 Day

la mot phuong trinh ding cap bac 3 Tu djnh huong tren ta c6 loi giai cho

bai toan nhu sau:

+ Xet truong hop: x = 0 khong thoa man phuong trinh:

+ Xet X 0 Ta chia phuong trinh cho x'^ thi thu dugc:

l _ 3 ( ^ 2 i ^ = 0

Dat t = ^ ^ ^ ^ ta CO phuong trinh: 2t^ - 31^ +1 = 0 » t = - i 2

t = l Truong hgp 1: * =

Jx + 2 1 „ I r

< r > = — <=> 2\lx + 2 = -x <=>

x < 0 x^ - 4 x - 8 = 0 <=>x = 2-2^3 Truong hgp 1: t = 1

Vx + 2 ^ I

-<=> = !<:=> \/x + 2= x-<=>

x > 0 x^ - x - 2 = 0 «>x = 2 Ke't luan: Phuong trinh c6 2 nghiem: x = 2;x = 2 - 2^3

Vi du 4: Giai cat phuong trinh:

ml im on tm a^t H^J sang t4d tagtat m. bat l^l, Vi, uai trr^ Nguyen TnmgKien

2(x2 - 2x + 4) + 2(x2 + 2x) - 5^{x^ - 2x + 4)(x2 + 2x) = 0 Chia hai ve cho

Ket luan: Phuong t r i n h c6 hai nghi^m la:

N h a n xet: Ta c6 the phan tich:

Chu y r5ng: Trong m p t so p h u o n g trinh: Ta can dua vao tinh ddng cap cua

timg nhom so'hang detit do phan tich tao thanh nhan tu chung

<=> (a - 2b)(a - b)(a + b) - (a - b)(a - 2b) = 0 c:> (a - 2b)(a - b)(a + b - 1 ) = 0

N/2X + 3 - Vx + 1 = 0 2N/2X + 3 -N/X + 1 = 0

Ta thay rang ne'u b i n h p h u o n g true tie'p se dan den p h u o n g trinh bac 5

De khSc phuc ta se t i m each tach x2 + 4 ra khoi V2x + 4 tj •

T u do ta viet lai p h u o n g trinh n h u sau: (x2 + 4)\/2x + 4 + x2 + 4 = 4x2 ^

f m = l Gia sir x2 - 6 x + l l = m(x2 - x + l) + n ( x - 2 ) - m + n = - 6 <=> m = 1, n = -5

m - 2 n = l l

Tai lieu oil thi dai hoc sang tao va giai PT, bat PT, hf PT, boT DT - NguyettTrung Kien

2 Giai phuong trinh v6 ty bang phuong phap dat an phy khong hoan toan

+ Dat an phu khong hoan toan la phuong phap chpn mpt so hang trong

phuong trinh de dat lam an sau do ta quy phuong trinh ban dau ve dang

mpt phuong trinh bac 2: mt^ +g(x)t + h(x) = 0 (phuong trinh nay van con

an x)

+ Van de ciia bai toan la phai chpn gia trj m bang bao nhieu de phuong trinh

bac 2 theo an t c6 gia trj A chan A = A(x) nhu the viec tinh t theo x

se dupe de dang

+ Thong thuong khi gap cac phuong trinh dang:

ax^ + bx + c + (dx + e)^jp\^ + qx + r = 0 hay

ax^ + bx + c + (dx + e)^px + q = 0 thi phuong phap dat an phu khong hoan

toan to ra rat hi^u qua:

+ De giai cac phuong trinh dang nay ta thuong lam theo each:

- Dat 7f(x) = t => t^ = f(x)

- Ta t^o ra phuong trinh: mt^+g(x)t + h(x) = 0

Ta CO A = [g(x)]^-4m.h(x) = fj(m)x^+gj(m)x + h j ( m ) De A eo dang A(x)]^ thi dieu ki§n can va dii la A^j, = [gi(m)]^ -4£j(m).gj(m) = 01=> m

Ta xet cac vi du sau:

Vi 1: Giai cac phuong trinh:

a) x ^ + l - ( x + l)\/x^-2x + 3 = 0 b) 2V2x + 4 + 4 ^ 2 ^ = 79x^+16

Giai:

a) Dat t = Vx^-2x + 3 > 0 = > t ^ - x 2 -2x + 3 Phuong trinh da cho tro thanh: x^ +1 - (x + l)t = 0 ' '

Ta se tgo ra phuong trinh: mt^ - (x + l)t + x^ +1 - m(x^ - 2x + 3) = 0 (Ta da them vao mt^ nen phai bot di mpt lupng mt^ = m(x^ - 2x + 3)) Phuong trinh dupe vie't lai nhu sau:

^ , f x > i + Truong hpp 2: t = x - 1 o Vx -2x + 3= x - l<t:> -2x + 3 = x^-2x + l

Phuong trinh v6 nghiem

Tom lai: Phuong trinh c6 2 nghiem la: x = 1 ± %/2

8 + (2x + 8) x ,

t = - = 4 -4 2

T o m lai p h u o n g trinh c6 nghiem duy nha't x = 4V2

Vi d\ 2: Giai cac phucmg trinh:

-3

Ket luan: Phuong trinh c6 2 nghiem x = 0, x = —

Chii y : O buoc cuoi cung khi giai ra nghiem ta phai t h u lai v i phep binh

phuong liic dau khi ta giai la khong tuong duong

T u do ta can luu y ; Khi gidi mot phuong trinh md cdc phep dat dieu kien

phuc tap ta c6 the bo qua buoc nay nhung khi gidi xong phuang trinh ta phdi thu lai vdo phuong trinh ban ddu detim nghiem chinh xdc

b) Dieu ki?n: x > - 2 ^ Dat t = 2 ^ x 7 2 - V2x + 5 thi t2 = 6 x + 13 - 4^2x2 + 7x +10

Phuong trinh da cho tro thanh: t2 - (3x + 2)t + 2x2 + x - 3 = 0

T a c o A = 9x2 + 12x + 4 - 8 x 2 - 4 x +12 = (x + 4)2 t = 2 x + 3

t = x - l

Truong Rgp 1: t = 2x + 3 <=> Isjx + l - sJlx + B = 2x + 3

Nhan thay tren moi khoang (-2;!) va (l;+oo)

ham so f(x) = 2\/x + 2 + -Jlx + S _ ^ x - 1 lien tyc va c6

f ( x ) = 1 1 • + — = = = + - > 0 nen ham so' dong bien Suy ra tren moi

s/x + 2 yJ2x + 5 {x-lf

khoang d o p h u o n g trinh c6 nhieu nhat mot nghi?m

Xet tren khoang (-2;!) ta c6 f(-2) > 0 nen phuong trinh v 6 n g h i f m

Xet tren khoang (l;+oo) c6 f(2) = 0nen p h u o n g trinh c6 n g h i ^ m d u y nhat

Truong h o p 2: t = 2x <» ^j2x^ 3x + l =2x<=>i'^^^ <=>x =

-[-3x + l = 0 3

3 3 1 Ke't luan: Phuong trinh c6 3 nghif m: ' ' ^ ^ ' ' ^ ^ y ' ' ^ " ^

b) Dieu kien: x > 1 Dat t = 7 x ^ + 3 > 0 <=> x^ = t2 - 3 Do h? so cua x^ trong p h u o n g trinh la: 1 Phuong trinh da cho tro thanh: t2 - (5x - l ) t + 6x2 - 2x == 0

Ta coi day la phuong trinh bac 2 cua 5 ta c6:

Doi chieu voi dieu ki^n ta c6 4 nghi^m deu thoa man phuong trinh

b) Ta viet 1 ^ phuang trinh thanh: 3 - (2x2 _^ + x + x'* = 0

Ta coi day la phuang trinh bac 2 cua Vs ta c6:

A = ( 2 x 2 + 1 ) 2 - 4 ( x + x^) = 4 x 2 - 4 x + l = (2x + l)2 \ «

> / 3 = - ( 2 x 2 + l + 2 x - l ) = x2 + x Tir do suy ra

V 3 = - i ( 2 x 2 + l - 2 x + l ) = x 2 - x + l

x2 + X - Vs = 0 x2 - x + l - V s = 0 Giai 2 phuang trinh tr§n ta thu dugc cac nghi?m ciia phuong trinh da cho

- l ± V l + 4V3 „ - 1 ± V 4 N / 3 - 3 la: x = ^ hoac x = ^

2 2 c) Dieu ki?n x > - 4

Ta viet lai phuang trinh thanh: x + 4 + (4x2 ^^_2|7x + 4 + 8x2 + 2 x - 8 = 0 Coi day la phuang trinh bac 2 an Vx + 4 thi

1 b) Dieu k i ^ n : x >

Ta viet lai phuong trinh thanh: N / X2+ 3 X + 6 = 3 X + 1 - \ / 2 X 2 - 1

Binh phuong 2 ve'va thu gon ta dugc phuong trinh m o i :

+ Bai toan phuong trinh giai bang phuong phap ham so' thuong c6 dac diem

la luon dua ve dugc dang: f u(x) = f v(x) trong do ham so' dac trung

thuong la ham don dieu tang hoac don d i ^ u giam tren mien xac dinh D + Dac diem noi bat nhat ta c6 the de phat hif n la: Trong phuong trinh c6 nhieu bieu thuc chua can, hoac da thuc bac cao ma ta khong the quy ve mpt an

Ta thuong giai cac phuong trinh dang nay theo each:

nghiem do la duy nhat

* Ta xet cac vi dvi sau:

V i dy 1: Giai cac p h u c m g trinh sau:

Xet h a m so f ( x ) = V x - 1 + x^ - 7 + ^ x + 6 = 0 tren (1; +oo) ta c6

f '(x) = — / + 2x + — , ^ > 0 V i v|iy h a m so dong bien tren (1; +oo)

2 V x - l 3^(x + 6)2

M a t khac ta c6: f(2) = 0 => x = 2 la nghi^m d u y nhat cua p h u a n g trinh

b) D i e u ki?n: x > 8 ' ' -i"'

De d a n gian h i n h thuc ciia p h u a n g trinh ta dat ^ x - 1 = t t > ^

P h u a n g t r i n h da cho tro thanh: t^ - 2t - (t^ - A)yjt^-7 - 3t^ + 28 = 0

<=> 3t^ -1^ + 2t - 28+(t^ - 4)7t^-7 = 0 Nhan thay t = y/? khong phai la nghi^m

Xet h a m so f (t) = 3t^ -1^ + 2t - 28 + (t^ - 4)ylt^-7 tren + o o )

T a c o f('t) = 9 t ^ - 2 t + 2 + 3 t ^ ^ t ^ - 7 + "^^'^^ V i 9 t ^ - 2 t + 2 > 0 nen suy

W-7f

ra f ' ( t ) > 0 V t e ( \ / 7 ; + o o ) N h u vay h a m so f ( t ) l u o n d o n g bien tren

^\/7; +00j Ta c6 f(2) = 0 t = 2 la n g h i f m d u y nhat cua p h u a n g trinh:

T u d o suy ra p h u o n g t r i n h ban dau c6 n g h i f m d u y nhat: x = 9

V i 2: G i i i cac p h u a n g t r i n h s a u :

a) 2x'' - 3x^ - 14x + 16 = (28 - i\^)\l2x^ -15

b) 2\/4x^-x + l + 2 x = 3 ^ 2 x 2 - x ^ + V 9 x ^ - 4 x + 4

G i a i : a) D i e u k i f n : 2x^ - 1 5 > 0 o x > 3/^

Ta V i e t lai p h u o n g t r i n h thanh: 2x(x^ - 7) - 3(x^ - 7) - 5 = 4(7 - x^ )^j2{x^-7)

D|t t = ( x ^ - 7 ) = > x = ^/t + 7 v o i t > i

P h u a n g t r i n h da cho tro thanh: 2 t ^ t + 7 - 3t - 5 + 4 t 7 2 t - l = 0

De thay t = ^ k h o n g phai la nghi^m cua p h u o n g trinh, nen ta chi giai

De y rang: V o i t e 1 thi 2\/t + 7 > 23/^ > 3 n e n suy ra

f ( t ) > O V t e - ; + o o 1 , n h u v^y h a m so' f ( t ) d o n g bien tren - ; + o o 1 Lai CO

f ( l ) = 0 => t = 1 la n g h i f m d u y nha't cua p h u a n g trinh Tu d o ta c6 : P h u a n g

trinh ban dau c6 n g h i ^ m d u y nhat x = 8 b) Ta thay: x = 0 thoa man p h u a n g trinh da cho

Xet t r u a n g hg-p x > 0 Chia hai ve p h u a n g t r i n h cho x va dat t = — ta

du(?c: 2Vt^ - 1 + 4 + 2 = 3^2t - 1 + ^41^ - 4t + 9 Dat a = ^ 2 t - l ta c6 p h u o n g trinh m a i : 3a + yja^ +8 - Va^ +15 - 2 = 0

Xet ham so: f(a) = 3a + Va^ + 8 - Va^+15 - 1 De thay a < 0 t h i f(a) = 0 v 6

n g h i ^ m

Ta can xet k h i a > 0 c6: f (a) = 3 + • 3a= 3a^ > 0

M a t khac: f ( l ) = 0 a = 1 la nghi^m d u y nhat x = 1

* Xet t r u o n g h^p x < 0 Chia hai ve cho x va dat t = — ta duQc:

-2\/t^ - t + 4 + 2 = 3 ^ 2 t - l - V 4 t 2 - 4 t + 9 Dat a = v 2 t ^ 1 => a < - 1 ta c6 p h u a n g trinh m a i :

V a ^ + 8 - 3 a - V a ^ + 1 5 + 2 = 0 Xet ham so f(a) = Va^ + 8 - 3a - Va^ +15 + 2 v a i a < - l

1 1 Taco: f'(a) = - 3 + 3a'

N h u vay ham so' dong bien tren 7 Mat khac f (4) = 0

nen p h u a n g trinh f(x) = 0 c6 nghi^m duy nha't x = 4

b) Dieu k i ^ n x < 4 x < 4

2x^+3x2 +6x + 1 6 > 0 (x + 2)(2x2-x + 8 ) > 0 <=> - 2 < X < 4

Ta thay x = -2;x = 4 khong phai la nghiem nen ta chi xet tren (-2;4)

Xet ham so f(x) = V 2 x ^ + 3 x 2 + 6 x + 16 _ ^J^Z^ _ 2^ j^en (-2;4)

T ' f / X +6X + 6 1 „ r r>

Ta CO f (x) = + — > 0 vol mpi x e (-2; 4)

V2x^+3x2+<ix + 16 2 V 4 - X Suy ra ham so dong bien tren (-2; 4) Mat khac f(l) = 0 nen x = 1 la nghiem

duy nha't ciia phuong trinh

Cac V I du sau se tap trung vao lop cac bai toan s u dung phuang phap ham

so bling each 2:

+ Bien doi phuong trinh ve dang: f u(x) =f v(y) thong qua he tarn

+ Tu do dya vao tinh chat Neu f(t) dan di|u tang hoac dan di^u giam tren

D ma f ru(x)l = f rv(y)1 <=> u(x) = v(y)

V i dvi 4: Giai cac phuang trinh sau:

a) 8x^ - 36x2 + 53x - 25 = ^ 3 x - 5 b) 8x^ - 13x^ + 7x = 2 \ / x 2 + 3 x - 3

c) ^ 2 4 x - l l - 1 6 x V 2 x - l - 1 = 0 c) x^ - ^ x + 2 1 n x - | l n ( x + 21nx) = 0

G i i i :

Nhu'ng p h u a n g trinh c6 dang: ax"' + bx^ + cx + d = (ex + h ) ^px + q (1)

Hoac: ax^ + bx^ + cx + d = e^px'^ H-qx^ + r x + h (2)

ta thuong giai theo each:

Doi v o i (1): Dat ^ p x + q = y khi d o x = thay vao p h u a n g trinh ta dua ve dang: ax"^ + bx^ + cx + d = A y ' ' + By Sau do bien doi p h u a n g trinh thanh: A [ u ( x ) ] ^ + B.u(x) = A y ^ + By

Doi voi (2): Dat gyjpx^ +qx^ + r x + h = y sau do tao ra h ^ tam:

3 cpng hai p h u a n g trinh ta thu dup-c:

ax'^ + bx^ +cx + d = s.y

g ^px + qx + rx + h I = y^

Ax'' + Bx^ + Cx + D = s.y + y^ sau do dua phuang trinh ve dang:

u(x) l3 + S u(x) = y +s.y

Ta xet cac v i d u sau:

X r^^ 3/; 7 , f 8 x ^ - 3 6 x 2 + 5 3 x - 2 5 = y

3 x - 5= : y ^ Cpng hai p h u a n g trinh ciia he voi nhau ta thu dupe:

8x^ - 36x2 + - 30 = y^ + y (*) Ta nghi den v i f c bien doi ve'trai thanh:

A(x)]'' + A ( x ) de p h u a n g trinh c6 d^ng: [ A ( X ) ] ^ + A ( x ) = y^ + y

G i a s u : 8 x ^ - 3 6 x 2 + 5 6 x - 3 0 = (2x + a)^+(2x + a) ^ •' Dong nha't h f so ciia x2 => a = - 3

N h u vay p h u a n g trinh (*) c6 dang: (2x - 3)'' + (2x - 3) = y'' + y (1) Xet ham so f(t) = t^ + 1 ta c6 f'(t) = 3t2 + 1 > 0 f(t) dong bien tren R

Tai lieu on thidai UQC sang tao va gidi PT, bat PT, FT, barUT-Nguyen ThmglOen

Qua vi dy tren ta thay vi^c chuyen qua h^ tarn (I) giup ta hinh dung bai

toan du<?c de dang hon Khi da thanh thao quy tac bien doi ta c6 the thu

du<?c cac loi giai ngin gpn hon nhu sau:

b) Dat ^x^ + 3x - 3 = y ta thu duQ-c h^ phuong trinh sau:

Cpng hai phuong trinh ciia h? ta thu du^c:

^123^ +1 - 8 a ^ - 8 a - l = 0 o 8 a ^ + 8 a + l = \/l2a2+l

Dat \^12a^n = y ta thu dugc h$ sau: ^ y Cgng hai phuong

\ld} + l = y^

trinh ciia h? voi nhau ta thu dugc: (2a +1)^ + (2a +1) = y^ + y (*)

Xet ham so f(t) = t^ +1 ta c6 f'(t) = 31^ +1 > 0 => f(t) dong bien tren R

Theo (*) ta c6

f(2a +1) = f(y) <=>y = 2a + l=>8a^+8a + l = 2a + l o a = 0=>x = ^

Ke't luan: x = i la nghif m duy nha't ciia phuong trinh:

d) Dieu ki^n: x > 0

x + 21nx>0 Dat y = \/x + 21nx > 0 ta thu dugc h§ sau:

x - y - 2 1 n y = 0 _ = y + 21ny _ _ 3 _ , ^,

y =x + 21nx y^ = x + 21nx x'^ + x + 21nx = y'^ + y + 21ny

Xethamso f(t) = t^ + t + 21nt tren ( 0 ; + o o ) ta c6 f'(t) = 3 t 2 + l + - > 0

Cty TNHH MTV DWH Khang V i g /

Nen ham so' f(t) dong bien tren ( 0 ; + o o )

Tu do suy ra f(x) = f(y) <» x = y Ta quy bai toan ban dau ve giai phuong trinh: x'^ - x - 2 1 n x = 0 Xet ham so f(x) = x ' ' - x - 2 1 n x tren (0;+oo).Tac6 f(x) = 3 x ^ - 1 =

Tu do suy phuong trinh c6 nghi^m duy nha't x = 1

V i dv 5: Giai cac phuong trinh sau:

fdi lifu on thidai hoc sang tao va giai PT, bat PT, h? PT, baT DT- Nguyin Trung KietT

b) Nh|n thay x = 0 khong phai la nghi#m cua phuong trinh nen ta chia hai ve

phuang trinh cho x thi thu duQC phuang trinh tuang duang

la: 3 x 2 + 4 x - - ! - = 3 x 3 + 2 + i

Dat y = + 2 + — ta c6 h§ sau: 3x^+4x- —= y ^ Cpng hai phuong trinh

, ^ + 2 - f i = y^

cua he ta c6: (x +1)^+(x +1) = y^ + y Xet ham so f ( t ) - t ^ + t ta c6

f'(t) = 3t^ +1 > 0 f(t) dong bieh tren R

Tu phuong trinh ta suy ra

Phuong trinh da cho tuong duang vai

dong bieh tren [0;4

Phuong da cho CO dang: f x - x + 2 -f(x2+x)-(x2-ij=on

Ta xet hai truong hg-p:

T H 1: Neu: 1 < x < - I + N/T?

< » x ^ - x + 2 < x ^ + x = > f ( x ^ - x + 2J<f^x^ + x) dong thoi x ^ - l > 0 khi do f^x^ - x + 2 J - f ^x^ + xj-^x^ - l j > 0 Nen phuong trinh v6 nghi^m

T H 2: Neu - 1 < x < 1 khi do x^ - x + 2 < x^ + x nen suy ra

f ( x 2 - x + 2 ) > f ( x 2 + x ) , mat khac x^ - 1 < 0 nen phuong trinh v6 nghiem

Thu true tiep ta thay x = 1 thoa man (*)

Vay phuang trinh da cho CO nghiem duy nhat la x=l j :;

d) De y rang x-" +1 - X X + 1 + X = 1 nen phuang trinh da cho tuong duang voi: 2014"

Lay logarit theo co so e ca hai ve ta thu dugc phuang trinh:

Xet ham so' g(x) = x - In x + V ? + l ta CO gXx) = l

-Vx^+1 Vx^+l

g'(x) = O o x = 0

K h i X > 0 thi g'(x) > 0, k h i x < 0 thi g'(x) < 0 tir do suy ra f (x) doi dau tir

(-) qua (+) k h i d i qua x = 0 Lap bang bien thien ta suy ra f(x) > f(0) = 0

Do do p h u o n g trinh da cho c6 nghif m duy nhat x = 0

f (x) > 0 N h u vay ham so f(x) dong bien tren R

Mat khac ta c6: f(0) = 0 suy ra phuong trinh c6 nghi^m d u y nhat x = 0

c) Ta Viet l^i phuong trinh thanh: ( x ^ - x ) ^ - 1 6 x = 4\^4x^Tl2x

D^t y = V 4 x ^ 12x ta c6 h ^ t?m sau: ^ 3 - x - 1 6 x = 4y

4x^+12x,= y^

Cong hai ve h? p h u o n g trinh ta thu du<7c: {x^ " ''j + '^{^^ - xj = y^ + 4y

Xet ham so f(t) = t^ + 4t ta c6 f'(t) = St^ + 4 > 0 nen ham so f(t) dong bien

Ta c6: f (x^ - x) = f ( y ) » x^ - x = y <» (x^ - x)^ = 4x^ + 12x

o X x ^ ^ x ^ - 1 ) ^ - 4 x ^ - 1 2 = 0 o x = 0

x 2= 3 Vgy p h u o n g trinh da cho c6 3 nghif m la: x = 0; x =

V i d v 7: G i a i cac p h u o n g t r i n h sau:

a) ^6x +1 = 8x^ - 4x - 1 (2x + l ) 2 + V4x2+4x + 4 +3x 2 + 79x^+3 = 0

a) Phuong trinh tuong d u o n g voi 6x + l + \/6x + l = (2x)"' + 2x ' ' '

Xet f(t) = t^ +1 ta CO f'(t) = 3t2 + 1 suy ra ham so f(t) dong bien Theo de bai

taco f ( ^ 6 x + l ) = f(2x) <=>2x = ^6x + l o 8 x 3 - 6 x - l = 0 o 2 x ( 4 x 2 - 3 ) = 0 Neu x > l = > V T > 2 N e u x < - l thi V T < - 2

Suy ra m p i n g h i ^ m phuong trinh deu thupc [ - 1 ; ! ] Dat t = cos X, X e 0; t

Phuong trinh tro thanh cos3t = = cos - « 3t = ± - + k27t V i x e [O; 7t] nen

2 3 3

ta t i m dugc 3 gia trj t tuong u n g thoa man dieu k i f n 1^: t = - ; t = — ; t = —

9 9 9 V^y p h u o n g trinh c6 3 nghi?m x = cos - ; x = c o s — ; x = c o s —

9 9 9 b) Phuong trinh da cho c6 dang: ^

(2x + l ) f 2 + ^(2x + l f + 3 ] = ( - 3 x ) f 2 + yj{-3xf+3

Xet ham so f { t ) = t f 2 + V t ^ ] taco i'{t) = 2 + ^t^+2 + > 0 , l a ham

dong bien tren R,

Ta CO £(2x +1) = £ ( - 3 x ) » x = -1 t h u lai ta thay thoa m a n dieu ki$n

lai lieu on f«i iTiu nor san^inv r n g m i i i, vui i i, nc i i, vui

Xet ham so f (t) = logj t - 2t +1^, Vt > 0 Ta c6:

f'ft) = — ^ + 2 t - 2 > 2 J — ^ 2 t - 2 = 2 J — - 2 > 0

t.ln2 Vt.ln2 Vln2

Vay ham so f (t) dong bien tren khoang (0;+oo), do do:

I ( i ) o f ( V ^ ) = f <=> Vx + 2 = 2 + - (2)

Voi dieu ki^n x € - 2 ; - l

V 2 , u ( 0 ; + o o ) , binh phuong hai véphuor\ trinh (2)

-3 + ^/l-3

X =

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

Nhung ky thuat qua trong de giai phuong trinh giai bang phuong phap

danh gia ta thuong sir dung la:

+ Dung hang d^ng thuc: Â^ + + Ấ^ = 0 o A, = = = = 0

+ Diing cac bat dang thuc c6 dien Co si, Bunhiacopxki, Bat dang thuc hinh

hpc

+ Dung phuong phap khao sat ham so de tim GTLN,GTNN :

Vi du 1: Giai cac phuong trinh sau:

a) lex"* + 5 = 6\/4x^ + x

b) 4x'' + x2+3x + 4 = 3^16x^+12x

c) 96x^ - 20x + 2 + xVSx - 1 - ^4x(8x +1) = 0

Giai:

a) Vi 16x'*+5>0 nen phuong trinh da cho c6 nghiem khi

4x^ + X > 0 <=> x(4x^ + 1) > 0 <=> X > 0 De y rang khi x = thi VT = VP nen

ta nghi den sir dung bat dMng thuc Co si sao cho dáu bang xay ra khi = •

Tir nhiing co so tren ta c6 161 giai nhu sau:

4x = (4x^+1) = 2

1 Tom lai: Phuong trinh c6 nghỉm duy nhát x = — b) Vi 4x^ + x^ + 3x + 4 > 0 nen phuong trinh da cho co nghiem khi

16x^ + 12x > 0 <=> 4x(4x2 + 3) > 0 « X > 0 De y rang khi x = i thi VT = VP

nen ta nghi den sir dung bát dSng thuc Co si sao cho dáu bang xay ra khi

J X = Tir nhiing co so tren ta co I6i giai nhu sau:

Theo bat dang thuc Co si dang 3\/abc < a + b + c ta co

Vi 2: Giai cac phuang trinh sau:

x^ > 2 x - l o ( x - l ) ^ > 0 Dieu nay la hien nhien dung

Dau bang xay ra khi va chi khi x = 1

Ta chung minh:

^ 4 x - 3 > 1 That vay bat dang thuc tuong duong voi

x^ > 4x - 3 « x^ - 4x + 3 > 0 (x^ - 2x + l){x^ + 2x + 3) > 0

o ( x - l ) 2 ( x 2 + 2 x + 3)>0

Dieu nay la hien nhien diing Dau bang xay ra khi va chi khi x = 1

Tir do suy ra VT > 2 Dau bang xay ra khi va chi khi x = 1

Tuong ty ta cung co: + > , :Mnm ^lituti-i

c) Ap dung bat dang thuc Bunhiacopxki dang:

Dau bang xay ra khi va chi khi x = 1

Vay phuang trinh c6 nghiem duy nhat x = 1

Vi dy 3: Giai cat phuang trinh sau:

a) 2x^sinx + x.cosx + V2x + l =x^-x^+x + l \'''

b) V-x^ + 4x + 21 - V-x^ + 3x +10 = V2

G i i i : 3) Ta tha'y phuong trinh khong c6 nghiem x = nen ta chi xet x 5^

Xet ham so f(x) = 2x^.sinx + x.cosx + \/2x + l +x^ -x'^ - x - l , x —^

- + 5 x ' ' - 3 x ^ - 1

Taco f (x) = 3x.sinx + (2x^+l]cosx + —, ^

Ta se chung minh: Bx.sin x + (2x2 +1)cos x + _ _ 1 > 0, Vx e j^^*^

Ta thay bieu thuc nay khong thay doi khi thay x bai - x nen ta chi can xet

x > 0 Ta can chiing minh bat dSng thuc quen thupc sau:

x3 , , x^

sinx > X ,cosx > 1 , Vx > 0 Xet ham so g(x) = cosx - 1 + — , x > 0,

6 2 2 Tac6g'(x) = - s i n x + x;g"(x) = -cosx + l > 0 = > g ' ( x ) = - s i n x + x >g'(0) = 0

Do do g(x) la ham dong bien tren 0; +oo), suy ra

x2 _ x2

g(x) > g(O) = 0 => cosx 1 + — > 0 => cosX > 1

-Tuong t u ta cung c6 sin x > x

Do do (*) diing hay f'(x) > 0, Vx G R Suy ra f(x) la ham dong bien tren R,

Mat khac ta c6 f(0) = 0 => x = 0 la nghi^m duy nhat cua phuang trinh:

b) Taco VT = J(x + 3 ) ( 7 - x ) - J ( x + 2 ) ( 5 - x ) = , ^'^^] •

V(x + 3){7-x)+V(x + 2)(5-x)

Dieu ki^n xac dinh la -2 < x < 5

Theo bat dang thiic Cauchy ta c6:

Vi dy 4: Giai cac phuong trinh sau: