Phương Trình, Bất Phương Trình Hữu Tỉ Vô Tỉ Mũ Logarit
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512.9
PH561T
•4 (CB) • ThS. PHAN VIET BAC
MHAN - CN. LE PHUC LUf J f
HAT l»HI]liSS •nrnCo
^ D U N G CHO HS GIOI
THI TRirOfNG C H U Y E N
^ O N THI THPT QUOC GIA
(2trong1)
I^NrtA'XUAT BAN
•
C QUOC GIA HA NOI
5^1. g
TS. LE XUAN S d N (CB) • ThS. PHAN V I E T BAC
ThS. TRAN NHAN - C N . LE PHUC L Q
HUUTI
VO Tl
MO
LOGARIT
^ D U N G CHO HS GIOI,
THI TRI/flfNG CHUYEN
y ^ O N THI THPT QUOC GIA
DTR:
N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI
NHA XUAT BAN DAI HOC QUOC GIA HA NQI
16 Hang Chuoi - Hai Ba Triing - Ha Npi
Dien thoai: Bien t$p - Che bkn: (04) 39714896;
Quan ly xuat ban: (04) 39728806; long bien tSp: (04) 39715011
Fax: (04) 39729436
*
* *
Chiu trdch
nhi^m xuat ban:
Gidm doc - Tong bien tdp: T S . P H A M T H I T R A M
NHA SACH HONG A N
Trinh bay bia:
NHA SACH HONG AN
ban:
Che
tdp:
Bi^n
VAN ANH - PHUONG ANH
^'
Doi tdc lien ket xuat ban:
NHA SACH HONG A N
SACH L I E N K E T
PHaONG TRINH - BAT PHUONG TRINH HLJfU TJ, VO TJ, MU, LOGARIT
Ma s5: 1L - 99OH2015
In 2.000 cudn, kh6 17 x 24cm tgi COng ti Cd phin VSn h6a VSn Lang.
Dia chl: S6' 6 Nguy§n Trung Tri;c - P5 - Q. Binh Thanh - TP. Ho Chi Minh
So xua't bSn: 351 - 2015/CXB/4 - 74/DHQGHN, ngay 09/02/2015.
Quyg't djnh xuS't ban s6: 120LK-TN/QD - NXBOHQGHN.
In xong va nOp liAi chieu quy II n3m 2015.
Cac em hoc sinh than men!
Phuomg trinh, bat phuang trinh la nhOng noi dung can ban trong chuong
trinh toan ph6 thong. Co dugc ky nang t6t trong viec giai phuang trinh, bSt
phuang trinh se khong nhung gop phan quan trong dS hinh thanh va phat trign
nang lire giai quyet van de ciia hoc sinh ma con giup cac em dat k6t qua t6t
trong nhtjng ky thi quan trong nhu: thi vao truang Chuyen, thi dai hoc, thi hoc
sinhgioi cac d p .
Vai muc dich ay, chung toi bien soan cuon sach nay nham cung cap cho ban
doc mot he thdng bai tap phong phu, da dang vai nhi§u bai mai la va cac
phuang phap giai hieu qua ve phuang trinh, bdt phuang trinh.
Noi dung ciia cu6n sach dugc trinh bay thanh ba chuong:
Chmmg 1 de cap den phuang trinh bat phuang trinh dang da thuc va huu ty;
ChiroTig 2 de cap den phuang trinh, bat phuang trinh v6 ty;
ChiroTig 3 va Chirong 4 theo thu tu de cap den phuang trinh, bat phuang trinh
mu va logarit.
Trong tung muc, tung phuang phap deu c6 cac vi du minh hoa tieu bieu; c6
phan bai tap de ban doc ren luyen; c6 phin huong din giai bai tap ngay sau do
de ban doc tham khao, so sanh vai lai giai cua minh. Sau moi chuong deu c6
phan bai tap tong hgp, phan Ian la bai tap hay va kho.
Chung toi hy vgng rang cuon sach "Phuang trinh, bat phiromg trinh va
phuang phdp gidi" se thuc su huu ich cho cac em hoc sinh cung nhu cac thay,
CO day Toan a truang pho thong.
Du da hit sue c6 ging trong qua trinh bien soan, nhung bg sach kho tranh
khoi nhung thieu sot nhat dinh. Cac tac gia chan thanh cam an y kien dong gop
cua cac thay giao, c6 giao va cac em hoc sinh gan xa de Ian tai ban bg sach se
dugc hoan thien hon.
Mgi y kien dong gop cho tac gia xin quy ban dgc gai ve:
CAC TAC GIA
1
ChUctng 1.
PHLfdNG TRINH, BAT PHtTdNG TRINH HlTU TI
§1. TAM THlTC, PHirONG TRINH, BAT PHlTOfNG TRINH BAG HAI
1) DIU cua tarn thipc bac hai
Tom tit ly thuyet
7.7. Dinh ly ve ddu cua tarn thuc
Cho tarn thuc bac hai f{x)^ax^+bx + c,a^Q. Dat A = b^ - 4ac. Khi do:
NSu A<0 thi af (x)>0 vai moi xeR;
NSu A = 0 thi af (x)>0 voi moi x^2a
Ngu A>0 thi af{x)>0 vai moi x G ( - O O ; X , ) U ( X 2 ; + C O ) va af(x)<0 vai
moi xe{x^;x2), trongdo x,
Cho tam thuc bac hai f{x) = ax^+bx-\-c,a* 0. Dat A = b^ -4ac. Khi do
2./(x)<0,VxeR« \a<0
l./(x)>0,VxeM<» a>0
A<0
\A
3./(x)>0,VxeM<:> a>0
A<0
A<0
1.3. Gid tri Ian nhdt, gid tri nhd nhdt cua tam thuc
Cho tam thuc bac hai f(x) = ax'^+bx + c,a^ 0. Ta c6
2
b^^
fix) = a x + — + c, nen
I 2aj V 4a
b_
Vai a > 0, f{x) c6 gia tri nhoJ nhat la c , dat dugc khi x = - la
4a
5
Vdfi a < 0 , / ( x ) CO gia t r i Ion nhdt la
V i d u 1 . Cho cac so thirc x,y,z
c-
4a
, dat dugc k h i X = -
2a
thoa man x + y + z^\. T i m gia t r i lom nhdt
c u a b i d u t h u c A = 9xy + I0yz + 1 I z x .
LcigiaL
Thay z = \-x-y
vao ^ t a c 6
A = 9xy + \(iyz + nzx^9xy
+ z(\Qy + \\x)^9xy
+ (\-x-y){\Qy
+ \\x).
K h a i trien va n i t gon ta c6
^ = - l l x ' - 1 0 / + l l x + 10>;-12xv
•
D o d o \ +{\2y-n)x
-
---^
=-llx'+(ll-12>;)x-I0/+10>;.
+ \0y^ -\0y + A = 0.
De CO gia t r i Ion nhat ciia A t h i phuong trinh c6 nghi?m. T a c6
A > 0 < » - 2 9 6 / + 1 7 6 ; / + 121-44^>0.
l l V 495 495
74
2
22
121
74 '
D o do ^ <
y- y + — V
+
<
27
11
11
37
296
148
148
27
25
11
Dku dang thuc xay ra k h i x = — ; v = — : z = •
^
^
74
37
74
495
0V a y gia t r i I o n n h i t ciia A la
148
V I d u 2. Cho hai s6 thuc x,y thay d6i thoa man x^ +y^ = \. T i m gia t r i Ion
nhat va nho nhat cua bieu thuc P =
2 •
1 + 2xy + 2y
(Di thi dgi hoc khoi B 2008)
L&igidL
Voi
= 0 ta CO
= 1 nen P = 2 .
X
2fx^+6xv)
Vai>;^0,datr = - t a c 6
y
Dodo
P[t'+2t
2(x^+6xy)
P^-^
- ^ =- ^
^ =
x'+2xv + 3 /
l + 2xy + 2 /
+ 3) = 2t^+\2tc>{P-2y+2{P-6)t
2/^+12^
7
•
t^+2t + 3
+ 3P = 0.
, _
^.l.
V a i P = 2, phuong trinh c6 nghiem t = — .
4
Voi P^2, phuomg trinh c6 nghiem khi vachi khi
A' = - 2 P ' - 6 P + 3 6 > 0 o - 6 < P < 3 .
P = ?> )&h\ = —F=,y
= ^ =
hoac x = — = , j j ; = -
VTo •
3
2
3
P = -6 khi x = — v = — p = hoac x = — i = , y
VTI
713
•
Vl3
=
2
— 1 = .
Vn
Ket hop lai ta c6 gia tri nho nhdt cua P la - 6 , gia tri Ion nhdt cua P la 3.
Vi du 3. Cho a,b^O. Tim gia tri nho nhdt cua bi^u thuc
1 b
P = a' +b' +^ + - .
- .
a
a
L&igidL Xem P nhu la mot tam thuc bac 2 doi voi bien b.
Taco P = b' +2b— + -K +
+a = b+
2a
la 4a' 4a'
+ •4a'
4a'
D4U bang xay ra khi
+a
+ a'>2.
-.a'
\4a-
b= 2a
a' =
4a'
2
Vi du 4. Cho cac so duong a, b, c thoa man a + b + c = 3. Chung minh rang
9
a + ab + 2abc< — .
2
.
Laigiau Tugiathiet 6 = 3 - a - c.Tac6
9
9
a + ab + labc < — • o a + a ( 3 - a - c ) + 2ac(3 - a - c) < —.
Dat / ( a ) = (2c + l)a^ +(2c^ - 5 c - 4 ) a + ^ > 0 . Tachung minh / ( a ) > 0 .
Ta CO / ( a ) la mot tam thuc bac hai c6 he so ciia a' la 2c +1 > 0, va lai c6
A = ( 2 c ' - 5 c - 4 ) ' - 1 8 ( 2 c + l ) = ( 2 c - l ) ' ( c ' - 4 c - 2 ) < 0 do 0 < c < 3 .
3
1
Tir d o / ( a ) > 0. Dau bang xay ra khi a = — ;6 = l:c = —.
2
2
Vi du 5. Cho 4 s6 thuc a,6,c,£/thoa man: + 6^ = 1; c - Chung minh rSng F = ac + bd-cd< ^^^^
4
. (jjsG Nghe An 2005)
2^
L&igidL Tu c-d = 3 taco +
b
+
3)
{a
fa
+
b
+
3^
(
a
+
b
+
3\
-36
c + = ac + (b - c){c
F ^ ac + bd - cd = ac + {b - c)d
[ 2 J J + K- 3)2= -c^J + (a + b + 3)c - 3b
a + 6 + 3V fl + 6 + 3 Y ^ ^ ^ ( a + 6 + 3) 126
4
c-
+
2
Su dung 1 = + 6^ = (a - 6) + 2a6 nen 2(36 = 1 - (a - 6) , thay vao ta c6
(a + 6 + 3)'-126_(fl + 6)'+6(a + 6) + 9-126 - ( a - 6 ) ' + 6 ( a - 6 ) + ll
4
~
4
~
4
Vi a^+6^=l nen
-y[i
Do do F <
.
4
Dau dang thuc xay ra khi d = —, c-- — , a = — , 6 =
2
2
2
Vi dy 6. Cho 3 so thuc a,6,c thoa man di^u kien
2
.
> 36;abc = 1,
Chung minh rang— +6 +c >ab + bc + ca.
L&igidL— Ta+ c6
b^+c^>ab
+ bc + ca<^(b + cY-a(b
>0
+ c) + -
8
<;=>(Z) + c) -2-{b
+ c) + — +
>0
b+c
+ • 12a > 0, luon dung vi > 36.
Vi du 7. Cho hai s6 x,y thoa man
-2xy-2x + 4y-7 = 0. Tim gia tri
cua x \ihi y dat gia tri Ion nhit.
(Di thi tuyin sinh THPT Chuyen Qudng Ngai 2013)
LcfigidL TsiCO x^+y^-2xy-2x
+Ay-1 = Q
ti.
<::>x^ - 2{y + \)x + y^ +Ay-1 = 0
De ton tai gia tri cua x thi phuomg trinh tren phai c6 nghiem, do do
A' = (>; + 1)^ - - 4;; + 7 > 0 O 2>; < 8 >; < 4 .
Khi y = A thay vao phuong trinh ta c6 x = 5.
Vay X = 5 thi dat gia tri Ion nhdt.
Bai tap phan 1.1
+z =\
loa man
1. Cho x,y,z thoa
man <
[2x'+3/+4z'=3
Tim gia tri Ion nhdt, gia tri nho nhat cua y.
2. Cho 3 so x,y,z thoa man 1'xxy++y_yz+ z+=zx5= 8 .Chungminhrang \
—.
3
x^ - xy + y
,.
X +xy + y
4. Cho cac so thuc a,b,c,d sao cho \ <2 \k a + b + c + d-6.
Tim gia tri lom nhat cua P = a^ + b^ +c^ +d^.
(Di thi tuyin sinh THPT Chuyen Long An 201 A)
5. Cho a,b la cac s6 thuc thoa man a + b = a^ -ab + b^.
Tim gia tri Ion nhat ciia +b^.
{De thi tuyin sinh THPT Chuyen Bdc Giang 2007)9
3. Tim gia tri Ion nhat, gia tri nho nhat cua P=
6. Tim gia tri nho nhat cua A =
.
«
X
7. Tim gia tri ion nh4t cua ^ = -4(x^ - x +1) + 3|2x - 1 | , voi - 1 < x < 1.
8. Chobaso x,y,z
thoaman • ' ' ^ ' - ^ ' ^ ^ l - ^'^-1
x+y+z^3
Chung minhrSng x^ + y^ + z^
H i r o n g d a n giai b a i tap p h a n 1.1
1. Ta
CO
z=1-
X-
+ 3y^ + 4z^ = 3
, do do
2x^ + 3>;^ + 4 (1 -
X-
=3
yf
< = > 6 x ^ + 7 / - 8 x - 8 > ; + 8xv + l = 0
,.
^ . ^ 4>
<=>6x^+8(>;-l)x + 7 / - 8 > ; + l = 0 .
,
i^-xviv
D I phuong trinh c6 nghiem ta c6
A ' = 1 6 ( : i ; - 1 ) ' - 6 ( 7 / - 8 3 ; +1) = - 2 6 / + 1 6 > ; +10 > 0
< >^ < 1.
Voi y = l khi do ta CO x = z = 0 . Vay gia tri lom nhdt cua 3^ la 1.
,,
Voi y = —— khi do x = — , z = — . Vay gia tri nho nhat cua >> la - ^
13
13
13
13
2.Tir he ta c6
Do do
[jj;z = x ^ - 5 x + 8 '
>'z = 8 - x ( > ; + z )
J>' + z = 5 - x
j +z =5- x
xjr-:
,«f
c or
- (5 - x ) / + x^ - 5x + 8 = 0.
z la hai nghiem cua phuong trinh
Phuong trinh c6 nghiem
A' > 0 <=> (5 - x)^ - 4(x^ - 5x + 8) > 0
"
' ^
<::>-3x^+10x-7>0<»l
Tuong t u voi y,z ta c6 dieu phai chung minh.
_
3. V o i >; = 0 , t a c 6 x ; t O , P = l .
'T'
' '
^^r^^^i
^il^
ml
10
r ^^
X
Vai
;;7tO,tac6
P
=
/
\
y
r-/ +l
y
~
= ^
X
,
+ - +1
-,vai
X
t= -.
^ +^ + 1
J'
y
Taco
y.rs
t — f +1
^=^r^y^<^/'(r'+/ + i) = ^'-^
+ i < » ( p - i ) ? ' + ( p + i ) r + ( p - i ) = o.
De ton tai gia t r i lom nhSt, gia t r i nho nho n h i t cua P, phuang t r i n h n o i tren
phai CO nghiem, k h i do
A'= (P + 1 ) ' - 4(P - 1 ) ' > 0
V a i P = ^ xay ra k h i r = 1 hay
-3P^ + lOP - 3 > 0 <=> ^ < P < 3 .
=
X
0.
^ 'v;
u
V a i P = 3 xay ra k h i ^ = - 1 hay x = ->» ^ 0 .
Vay, gia tri Ian nhSt cua P la 3, gia tri nho nhdt cua P la ^ .
4. T a c o \
Tudotaco
suyra ( a - l ) ( a - 2 ) < 0 < : : > a ' < 3 a - 2 .
b,c,d.
P-a^
+6^+c^ +
<3(a + 6 + c + ^)-8
= 10.
V a i a = 2,6 = 2,c = 1,^/ = 1 hoac cac hoan v i t h i P = 10.
Vay gia tri Ian nhdt ciia P la 10.
,2
5. Tir a + b = a^-ab
Dodo
{a + bf-4(a
Taco
DSu
+b'
=(a
_ L
.
L2
+ b" =(a
+ b)^-3ab>(a
+ b)<0<^0
+
+ b^)^{a
+ bf
-^{a
+ bf
1
=-^(a
+
bf.
b<4.
+ bf
<16.
bang xay ra k h i a = 6 = 2 . Vay gia tri Ian nh4t cua ct^ + 6 ' la 16.
6. Ta CO A =
x'-2oi6x+2
_riV
= 2
-2016.
rr
+ 1.
1
Dat ^ = - , ta CO ^ = 2 r - 2016r +1 - 2(r - 504) - 508031 > -508031.
11
Do do gia tri nho nhk cua A la -508031, dat dugc khi t = 504 hay x =
504
V.Taco ^ = - 4 ( x ' - ; c + l) + 3 | 2 x - l | = - ( 2 x - l ) ' + 3 | 2 x - l | - 3
= -2x-lp+3|2x-l|-3.
Dat/ = |2x-l| v i - l < x < l nen ^e(l;3).Khid6
3
1
3^' 3 3 '
A = -t +3t-3 = - t - - — < — . Dau ' =' xay ra khi t = — hay x- —,
4 4
^
2 ^ 4
vi -1 < X < 1. Vay gia tri Ion nh^t ciia A la — .
4
8. Dat x = a + l,Z) = j + l,c = z + l. Taco a,b,ce[-2;2], a + b + c = 0.
Ta can chung minh + 6^ + < 8. Vi trong 2 s6 3 s6 a,b,c c6 hai s6 cung
d4u, giasu 6,c>0. Khi do +b^
Dau " = " xay ra khi \a = -2;6 = 0,c = 2) va cac hoan vi hay
a = -2
x = -l;>' = - l ; z = 3 va cac hoan vi. TIT do ta CO dieu phai chung minh.
2) Bit phipcng trinh bac hai
Tom tat ly thuyet.
De xet nghiem cua mot bat phuong trinh bac hai ta dua vao dinh ly vS dSu cua
tarn thuc, no phu thuoc dau cua he so a va ddu cua A.
Gia su /(x) = ay?' + 6x + c, a 0, ls. = b^ - 4ac, ki hieu Xj, X2 (xj < X2 ) la hai
nghiem cua / ( x ) . Khi do
• Tap nghiem S cua bat phuong trinh / ( x ) > 0 dugc xac dinh boi bang sau
a \
—
0
-
+
+
5 = (x,;x2)
S =R
S = R\l
1
* - 5 = ( - c o ; x , ) u ( x 2 ; + co)
2al
12
• Tap nghiem S cua bat phuong trinh / ( x ) > 0 ducrc xac dinh boi bang sau
a
\
-
—
0
+
S =0
s = -'
S = x^;x2
2a
+
S =R
S - (-oo;xi]u[jC2; + co)
• Tap ngliiem S cua bat phuong trinh f(x)<0
a
—
\
•
+
0
-
5 = (-oo;x,)u(x2; + oo)
S = R\\ •
+
dugc xac dinh boi bang sau
5= 0
S =0
5 = (x,;x2)
Tap nghiem S cua hk phuong trinh / ( x ) < 0 dugc xac dinh boi bang sau
—
0
+
-
S =R
5=M
5 = (-oo;xi]u[x2; + oo)
+
5= 0
s = -'
a
\
5 = [x,;x2]
2a
Vi du 1. Tim a dk hk phuong trinh sau c6 it nh4t mot nghiem nguyen
x^-x
+ a{l-a)<0.
(1)
L&igidL Xet A = 1 - 4a(l - a) =4a^ - 4a + 1 = (2a - 1)2 > 0 Va
Vay (1) CO hai nghiem Va. x = x,; x = ^2 thoa man x, + X j = 1.
X
X
Khi do (l) <=> X, < X < X2. Do ' ^ ^ =
chi khi
X,
1
nen (1) c6 nghiem nguyen khi va
< 0 < 1 < X2.
13
Lai do X, + ^2 = 1 nen dieu nay tuomg duong vai
<0<:>a(l-a)<0<=>
a<0
a>\'
Vay a < 0 hoac a>l.
Vi dy 2. Tim p de bat phuong trinh sau c6 dung mot nghiem nguyen
(2x + 1 4 p ' - 7 ) ( 4 x - 4 p ' - 1 5 ) < 0 .
L&igidL Taco 2x + \4p^-7
Vi -7/?^ +
<
+^
= 0^x
= -lp^
nen bat phuong trinh tuong duong voi
-lp'+-
'
^
•• '
Gia su a la mot nghiem nguyen cua bSt phuong trinh.
+ — > 3, liic nay bit phuong trinh c6
Neu a < 2, ta CO -Ip^ + — <2 va
2
4
them nghiem nguyen x = 3, trai voi yeu cm bai toan.
15
1
7
7, 7,
Neu a > 4, ta CO
+ — > 4 nen p^ >— do do -1 p^ + — < —^++ —
ao
ao
-/p+
S
—
- < 2 nen
4
^ 4
2
4 2
bat phuong trinh c6 them nghiem la 2; 3, trai voi yeu cku bai toan.
Neu a = 3, de phuong trinh c6 dung mot nghiem nguyen tuong duong voi
2 < - 7 / 7 ' + - < 3 < p ' + —<4,nentaco—2
4
14
14
Vi dy 3. Chung minh rang voi moi x > 1, ta luon c6
<2
L&igidL Ta c6 3
<2
-
•
14
<=>3 X
x + - <2
X—
x' +1 + -^
^x'-l^'
N2 ^
1 <2
1 -1
o3
V
^ y
x+—
x+—
Xy
1^ <2 Xy 1 \2
x^-1
- 1 (vi X > 1 nen
o3
>0)
X+—
x+ Datr = x + - > 2 J x . - = 2 , d a u ' = 'xayrakhi x = l,do x > l nen t>2.
X V X
Bat dang thiic can chung minh tra thanh
3r<2(^'-l)<:>2r-3/-2>0<^(2? + l)(f-2)>0,diingvi t>2.
Vi dy 4. Giai bit phuong trinh (x - 6)''+ (x - S)"* < 16.
LoigidL Taco (x-6)V(x-8)^ <16
•
«((x-7) + l ) V ( ( x - 7 ) - l ) ' < 1 6
«2(x-7)'+12(x-7)'+2<16«(x-7)'+7(x-7)'-7<0
o - 7 < ( x - 7 ) ^ < l < ^ ( x - 7 ) ^
Vi du 5. Tim m sao cho bat phuomg trinh dung vai moi x e ^
.. ^
2x^ +/nx-4
>-6.
-^^' ^
x'-x +l
Lof gidi. Ta CO x^ - x +1 > 0, Vx e R nen bat phuomg trinh tucmg duomg voi
2 x ^ + m x - 4 > - 6 ( x ^ - x + l), V X G R
<::>8x^ + ( w - 6 ) x + 2>0, VxeR
o A = (m - 6)^ - 64 < 0 -2 < w < 14.
Vay-2
r 1
Bai tap phan 1.2
1. Tim m de he bat phuong trinh sau c6 nghiem
-{m + 2)x + 2m<0
x^ +{m + 7)x + 7m<0
2. Tim tat ca cac gia tri cua a de bat phuong trinh sau dung voi moi x
x+\
—z
<1.
ax -4x + a-3
3. Cho f[x) = x^+ax + l voi 3
4. Tim m de bat phuong trinh sau dung voi Vx e R
(m-l)x^
+{m + \)x +
m-\>0.
5. Tim m de bat phuong trinh sau c6 tap nghiem la M
.^^'^
^ 3x^ -mx + 5 ^
1<—
<6.
lx^-x + \
Huang dan giai bai tap phan 1.2
x ' - ( w + 2)x + 2m<0 (1)
1. Ki hieu
+ ( 7 w + 7)x + 7m<0 (2)
Taco A, = ( w - 2 ) ^ > 0 ; A 2 = ( m - 7 ) ^ > 0 .
Ta CO phuong trinh
Phuong trinh x^ +
- (m + 2) x + 2w = 0 luon c6 nghiem x = m hoac x = 2.
+ 7) x + 7m = 0 luon c6 nghiem x = -m hoac x = - 7 .
Voi m = 2 hoac voi m = 7 thi he v6 nghiem.
Voi m^2\m^l,
m>0 nghiem cua ( l ) la 5, =(w;2) hoac
S^-[2•,m)•,
'nghiem cua (2) la ^2 = ( - m ; - 7 ) hoac ^3 = ( - 7 ; - w ) . Ro rang 5, f l ^ j = 0
nen he v6 nghiem.
Voi m<0 nghiem cua ( l ) la 5, = ( w ; 2 ) , nghiem cua (2) la
-{-l;-m).
16
Ta CO iS", n
Taco ca^-4x +
a-3^0yxsR^
A' = 4 - a ( a - 3 ) < 0
<=>
a<-l
a>4
Neu a<-l thi ox^ - 4 x + a - 3 < 0 , V x e M . Tirdo
x+1
<=>x + l>ax^ - 4x + a - 3 , V x e M
<=>ax^-5x + a - 4 < 0 , V x e M
<=>A = 2 5 - 4 a ( a - 4 ) < 0 (vi a<-l)
\a<-l
- V 4-V4T
<=>S ,
<=>a<
[4a'-16a-25>0
2
N6u a > 4 thi ox^ - 4x + a - 3 > 0, Vx e M.
Tirdo
:0
x+1
•
o
X +1
< ax^ - 4x + a - 3, Vx e M
oox^ - 5x + a - 4 > 0 , V x e M
O A = 2 5 - 4 a ( a - 4 ) < 0 ( v i a>4)
<=>
4 + V4T
a>4
4a'-16fl-25>0
Ket hop lai a e
-Qo;
4-741
u
4+V4T
-; + oo
3. Taco / ( / ( x ) ) >x<=>/(/(x))-x>0
Taco
/(/W)-^ = /(/W)-/W+/W-^
i / ( x ) + 1 - (xL+.flx+-i-W-/fx)-x-~':":7:^7,'i
'/'{x)-x']
^(x^ +(a-\)x
+ a[f{x)-x]
=[f{x)-x][f{x)
+ x + a]
el
+ \)[x' +{a + \)x + a + 2).
Bat g{x) = x'+{a-l)x + lc6
A^=a'-2a-3
h(x) = x^+{a + \)x + a + 2 CO
A^=a^-2a-7.
Do 3 < a < | nen
tudo
^ ^ , d o d 6 h{x) = x^ +{a+ \)x + a +
2>0,\fxeR,
f(f{x))-x>0
<=>g(x)>0<=>x^ + ( a - l ) x + l > O o
l-a-Va'-2a-3
2
x> l-a + V a ' - 2 a - 3
fm-l>0
4.Tac6 ( w - l ) x ^ +(m + l)x +
m-l>0,\/xGR<^
A<0
m-l>0
/w-l>0
/
w
l
>
0
m<•ow>3.
3
(m + 1)2 - 4 ( w - l ) <0
[-3m'+10/n-3<0
m>3
5. Ta CO 2x^ - X +1 > 0, Vx G M. Do do bat phirong trinh tuong duong voi
3x^ - mx + 5
-,Vxe:
1<
x^ + ( l - m ) x + 4>0,VxGlS
2x'-x + l
9x'+(m-6)x + l>0,Vxe
3x -tnx + 5 < 6 , V x e :
[ 2x'-x4-l
A,=(l-m)'-16<0
{-3
[0
18
3) Mot s6 dang phu'O'ng trinh hCeu t i dipa v§ b^ic hai
Dang t6ng quat: au^ (x) + bu{x) + c = 0.
3.1. Dang trung phuong ox^ + + c = 0.
Cdch gidi: Dat = ^ > 0 dk dua wh phuong trinh bac hai theo t.
3.2. Dang (x + a)"^ + (x + Z?)'* = c.
Cdch gidi: Dat x +
= t se thu dugc phuong trinh trung phuong theo /.
3.3. Dangnghich dao ox'* + bx^ + cx^ ±bx + a ^ 0, a ^0.
Cdch gidi: Chia ca hai ve cua phuong trinh cho
n x^ 9^ 0 thu dugc phuong trinh
+ b x±
a
1 = / (hoac x- \ — -t), bieu dien x^ + theo t, thay vao phuong
Dat X + —
V
X
X
X
trinh tren se thu dugc phuong trinh bac hai theo t.
3.4. Dang h6i quy ax"^ + bx^ + cx'^ + dx + e = 0, voi — =
Kb
a
Cdch gidi: Gia su — = k^0. Chia ca hai vg cua phuong trinh cho x^ ^0 thu
b
dugc phuong trinh aa x ^ + ^ + 6 x + — + c = 0.
I xj
k
~
-)
k~
Dat x + —= bieu dien x + ^ theo t, thay vao phuong trinh tren se thu
X
x^
dugc phuong trinh bac hai theo t.
3.5. Dang (x + a)(x +fe)(x+ c)(x + c/) = e, voi a + CflcA ^ifl/; Viet phuong trinh duoi dang
((x + a)(x + ^))((x + 6)(x + c)) = e
«> ^x^ +{a + d)x + ad^{x^ +{b + c)x + bc^ = e
D?lt x^ +(a + d)x = r thu dugc phuong trinh bac hai theo /.
19
3.6. Dang (x + a)(x + b)(x + c)(x + d)-ex
,\(n ad = be
Cdch giai: Viet phuong trinh duai dang
((x + aXx + cl)){(x + b){x + c)) = ex^
<=>(x^ + ( a + (i)x + fl
a + b-\-c + d
x + ad-t
thu dugc phuong trinh bac hai theo t.
Cdch 2. Xet X = 0, thu true tiSp.
Xet X ?t 0, chia ca hai ve cho x^ ta c6
ad
x + {a + d) + —
X+
be
(6 + c ) +
-e.
ad
Dat / = X + — , dua ve phuong trinh bac 2 doi voi t.
\2
ax
= b,
a^O,x^-a.
X
3J. Dang x^ +
yx + aj
Cdch giai: Viet phuong trinh duai dang
X-
y
ax
+ 2a
=
b^
x+a
x+a
- + 2a
(x + aY
x+a
X
Dat
x+a
= ?, thu dugc phuong trinh bac hai theo t.
ax
3.8. Dang
-^
+
X +inx + k
bx
^
= e,abc ^ 0.
X +nx + k
Cdch giai: Xet x = 0. Khong phai la nghiem ciia phuong trinh.
Xet X ^ 0. Vidt phuong trinh dudi dang
—— +
—— = c.
x+m +— x+n+—
X
X
Dat x + — = t, phuong trinh tra thanh — ^ + — ^ = c, tu do dua
X
t+m t+n
phuong
trinh bac hai theo
3.9. Dang dang cap bac hai aw^(x) + ZJV^(x) + CM(X)V(X) = 0 .
Cdch gidi: Gia sii v(x) ^ 0. Chia ca hai vg cho v^(x) ^ 0, r6i dat
= f d6
v(x)
dugc phucmg trinh bac hai theo t.
3.10. Dang au^(x)v^(x) + b{u{x) + v{x)f +c = 0, vai u(x)-v(x) = k.
Cdch gidi: Viet phuong trinh da cho ve dang
au^ (x)v^ (x) + b {u{x) - v(x))^ + 4 6 M ( X ) V ( X ) + C = 0
aw^ (x)v^ (x) + 46w(x)v(x) + c + 6yt^ = 0.
Dat u{x)v(x) -1, thu dugc phuong trinh bac hai theo t.
3.11. Dang x'=ax^+bx + c.
Cdch gidi: Ggi a la s6 thuc thoa man 6^ = 4(a + 2a)(c + «^).
(Day la mot phucmg trinh bac ba d6i vai a nen luon ton tai a).
Khido: x'=ax'+Z>x + c ^x' +2ax'
={a + 2a)x^ +bx + (c + a')
j
<^(^x^ +af ={a + 2a)x^ +bx + (^c + a^y
NSu a + 2a ;t 0, khi do vl phai (a + 2a)x^ + 6x + (c +
) la mot tam thuc bac
hai CO A = b^-4{a + 2a)[c + a^) = Q.
N6u fl + 2a > 0, ta CO (JC^ + a)^ = |Va + 2a x + Vc + o^j
x^ +a = |Va + 2a x + Vc + o^ j
x^ + a
=-{^4a + 2a X +
4c'+a^
NIU <3 + 2a < 0 thi FP < 0, > 0, do do ding thuc khong xay ra.
NSu fl + 2a = 0=>6 = 0.Tac6: (x^ + a)^ =(c + a^), tu do tim dugc x.
12x
3x
Vi du 1. Giai phucmg trinh—^
z
= 1.
^
^
x'+4x + 2 x'+2x + 2
{Dk thi tuyin sink THPT Chuyen DH Vinh 2010)
21
LoigidL
DiSukien:
Ro rang x = 0
+Ax + 2^Q,
+lx
+ 2^Q.
khong phai la nghiem cua phuang trinh. D o do p h u o n g trinh
tuomg duong v a i
12
X + -
+
4
3
X + -
X
+
-1.
2
X
2
D5t X + - = ? phuang trinh t r a thanh
X
12
3
t+A
=1
t+2
<:>9/ + 12 = (/ + 4)(r + 2)
o r - 3 / - 4 = 0<»
t = -\
t =4
V a i f = - l t a c 6 x + — = - l < = > x ^ + x + 2 = 0, v 6 n g h i e m .
X
Vai t = 4 t a c 6 x + - = 4<::>x^-4x + 2 = 0 < ^ x =
2±V2.
X
Vay nghiem ciia phuang trinh l a x = 2 ± V2 .
V i d u 2. Giai phuang trinh x " - 3x^ - 2x^ + 6x + 4 = 0.
(De thi tuyin sink THPT
LM
Chuyen DH Vinh 2009)
gidL V i X = 0 khong la nghiem cua phuang trinh nen phuang trinh tuang
duong v a i
x'+4-3x + - - 2 = 0 ^
X
X
X
2^
2
-3
-—
x)
V
X
-
2' + 2 = 0
V
Dat ? = X - - , k h i do phuang trinh c6 dang
- 3t + 2 = 0<^
X
Vai/ = 2<»x —
2
X
=2«x^-2x-2 =0 « x =
t =2
t = \'
l±V3.
X
Wai t = l<^x--
2
= l<i:>x^ -x-2
=
x =- l
0^
x =2
Vay nghiem cua phuang t r i n h la x = 1 ± V3 , x = - l ; x = 2 .
• - {
V i d u 3. Giai phuang trinh (x^ + x + l)(x^ + x + 2) = 12.
(Di
thi tuyin sink THPT
Chuyen DH Vinh 2008)
22
L&igidL Qat +x + l = t,(t >0) phuang trinh da cho tra thanh: /(r +1) = 12
» r^+^-12 = 0»r = 3 , d o r > 0 .
Vol/ = 3 taco +x + l = 3 <=> + x - 2 = 0 <=>''x = l
x^-2
Vay nghiem cua phuang trinh la x = -2; x = 1,
Vi du 4. Giai phuang trinh jc" + - 8JC -12 = 0.
(Di thi tuyin sinh THPT Chuyen DH Vinh 2006)
LoigiaL Ta c6
/ + 4x' - 8JC -12 = 0 <=> / + + 4x' - 4JC' - 8x -12 = 0
o (x' + 2x)' -4(x' + 2x)-12 = 0 «-4r-12 = 0, vai f = x' + 2x > -1
t^-2 <^t = 6(t>-\)
t=6
<;:^x'+2x-6 = 0«x = - l + V7.
Vay nghiem ciia phuang trinh la x = -1 ± V? .
Vi du 5. Giai phuang trinh x^ + 25x^ = 11.
(x + 5)^
(De tuyin sinh THPT Chuyen Lam Son Thanh Hoa 2013)
L&igidL *) Dieu kien x^-5.
Phuang trinh da cho tuomg duang vai
5x + -lOx'
X
x + 5 = 11<:> x + 5^ + 10-x + 5 -11 = 0
2
V
Dat t =
x+5
,
/
x+s;
,tac6 r + 1 0 / - l l = 0 o
2
N2
t^l
t^-n
x^ - X - 5 = 0
x^ +1 Ix + 55 = 0(v6 nghiem )
x+5
=1
= -11
x+5
I + V2T
ox =
m
1±V2T
Vay phucmg trinh c6 hai nghiem x =
2
Vidu 6. Giai phucmg trinh (x^+3x +3)\ +(x^+3x + 5) =82.
(Di tuyin sink THPT Chuyen Lam San Thanh H6a 2013)
LM giaL Dat ? =
(^-1)'+
+ 3x + 4 . Ta c6
=82<»r'-4f'+6/'-4/ + l + f ' + 4 / ' + 6 r ' + 4 ^ + 1 = 82.
t = -2
r = 4 <=>
+ - 40 = 0 o
t =2
r =-10
Vai t = -2 taco +3x + 4 = -2<»x'+3x + 6 = 0 (v6nghi?m)
Vcfi / = 2 taco x^ +3x + 4 = 2<»x^ +3x + 2 = 0 o x = - l
x = -2'
Vay phucmg trinh c6 hai nghiem x = - 1 ; x = - 2 .
Vi du 7. Bilt phucmg trinh x" + ox^ + 6x^ + ax +1 = 0 c6 nghiem. Chung minh
4
rang a 7 +b 7 >-.
L&igidL
(Di thi tuyin sinh THPT Chuyen Ha Tinh 2014)
Gia su x^ la nghiem ciia phucmg trinh da cho, ta c6
Ro rang Xg ^ 0. Ta c6
XQ + OXQ + 6X0 + OXQ
+ 1 = 0.
Xn + OXn + ZJXQ + OXQ + 1 = 0 <=> XQ + OXQ + 6 + — + ^ = 0
^0 ^0
\2
Xo+-
+a
Xo+-
+ 6 - 2 = 0.
1
Dat r = Xn + — ta CO t\>2.
Tu do phucmg trinh tucmg duong t^+at + b-2 = 0<:^at + b = 2-t^.
24
Do do ( 2 - f = (at + bf <{a'+b'){t'+!)<:>
—
Tachimg minh —
—
a'+b'>
+4
4
> - <=> 5t^ - 24t^ + \6 > 0 <^ [St^ - 4 ) ( / ^ - 4 ) > 0 ,
luon dung v i \t\>2.
Vay ta c6 di6u phai chung minh.
V i du 8. Chung minh rang phuomg trinh sau khong c6 nghi?m nguyen
(x-y)(x-2y)(x-3y)(x-4y)
+ /+2
= z\
(De thi tuyin sink THPT Chuyen Phan Boi Chdu Nghe An 20\4)
L&igiaL Ta c6
(x-y)(x-2y){x-3y)(x-4y)
<=> ((X -y){x^[x^-5yx
Bat t = x^-5xy
+
4y)){{x
+ y' +2 = z^
- 2y){x - 3y)) + /+2
+ 4y^)[x^-5yx
+ 6y^) +
= z'
y*+2^z\
5y\Tac6
(t'-/)(t'+/)
+ /+2
= z'<^t'+2
= z'<=>(z-t')(z
Khong m4t tinh tong quat gia su z > 0, khi do ta c6 • ^ ' ^ ^
+ t') = 2.
^,
z-e=\
suy ra z = —. V6 ly.
Vay phuong trinh da cho Ichong c6 nghiem nguyen.
V i du 9. Giai phuong trinh
= ~x^ - 4x + 3 .
L&igidi. Taco
x' =-x'
-4x + 3 ^x'
'x'+l
=
+2x^ +l^x'
x-2
4-l = - ( x - 2 )
-4x + 4 c^[x^ +\f
- x + 3 = 0 (vonghiem)
={x-2)
-1±V5
<=>x =
x'+x-l =0
Vay nghiem ciia phuomg trinh la x =
2
-1±V5
25
