Vềph ơngtrìnhvàbấtph ơngtrình

A.Mởđầu:
1.Nhữngđiềuphảinóingay:
!"#$"#%&'%(%)&*+,% /&01$"23&*-(4&"5$-(6"78+9-$"23&*-(4&"5+$-(6:-(2;<
+9-=>+,% /&&,.?,+@&$"#%-%A$<B&0,*%#%CD8A-&)&-(#%CD7E<FD"G%=%&"
H%I&J7DK
5LM6 Có &"N&* đốit ợngđặcbiệtnàoxuấthiện -(.&*+,% /&<O7<"P&*
-7Q
5LR6!/<HS%-2T&*H'<'liênquangìđặcbiệt 0;%&"7DQ
5LE6!U& sửdụngcôngcụ *4HI="7%-"/<CD7&"VHW="/?$"/-()&Q
2.Bảnchấtbàitoánptr,bptr X"%CD7&-(Y<<DZ<JS&*0;%&"%1DHS%-2T&*
<.&&*2[%&#8J%&"</<J\J.J/&"*%/-(]<'R^S%(_-(.&*-2`D8<.&&*2[%="%J.
J/&"
M6X"a&*H]&"J\<")&"^V<"*%/-(]*%N7</<HS%-2T&*HWCD7&-(Y<="a&*H]&"&,8
<'-b&"HP&*J7%?%&"+@<"-@.&)&</<Ha&*-"c<0,+9-Ha&*-"c<
R6d"%A-^B$</<H%1D=%V&J.J/&"0,=%A?-4?*%/-(]<O7HS%-2T&*-"78He%^,?-"G7
?W&J\J.J/&"HW-"%A-^B$H' f\J.J/&"H2T<-"%A-^B$(7=.=g?-"h.="a&*H]&"
?,^,HI-@.(7</<?i<-%)D5H=6-4?=%A?-@.&)&</<$"23&*-(4&"+9-$"23&*-(4&"
.Kháiniệmptr1ẩnsố
j,?Z-H=H2T<-"%A-^B$`@&*K
k5l6m*5l6
n;%k5l60,*5l6^,R+%ID-"c<5",?JS6+%A&l:-(.&*$-(&'^,HS%-2T&*<"27+%A-
-(2;<*o%^,p&JS?,-7<U&-4?=%A?5*%#%-"./-6="G%&"N&*(,&*+DZ<HI-"G7?W&
J\J.J/&"
.Chúý
qrAD-"78`9Dsms+u%</<`9DJJ="/<-7<'</<+$-(
qrAD<'&"%1DH=JJH2T<-"%A-^B$5<v&*0;%<'-"IlD9-"%V&&"%1DHS%-2T&*-4?
=%A?6Hw&*-"[%-7<'</<"V5$-(0,+$-(6
r"N&*^x^_$"b7-()&<O7-y%"80o&**%P$</<+@&="y&*+7.*%[",?"w?,
$"/-+%ID(z&*$-("78+$-(&,8^,HP&*"78J7%<#?,<"{<'</<Ha&*-"c<"78+9-

Ha&*-"c<J7%?,-"y%
q!/<-4&""DS&*l#8(70;%M$-(5+$-(6K
PDF created with pdfFactory Pro trial version www.pdffactory.com
R
-"MKnyr.5= 4?H2T<*%/-(]-"G7H=-"%A-^B$J.J/&"6
-"RK!'`D8&"9-M*-(]-4?H2T<-"G7?W&
-"Eq Kd4?H2T<&"%1D -"B?<"b0yJS
qn1&*"%V?<O7$-(K
H'^,*%/-(]-4?-"98<O7p&JS-"G7H#?J\J.J/&"-(.&*$-(+@&"W8&";-(2;<="%
JJHS%-2T&*JJ<U&$"#%sJS&*H2T<sHW n,H'^,^x`.<U&l/<^B$J\-w&-@%<O7
</<+%ID-"c<+]+AH%?,JJ
n;%"V$-(&"%1Dp&JSr.<O7&'$"#%^,M+Z*%/-(]Hw&*-"[%^,?-"G7?W&J\
JJ-"%A-^B$:-y%$"#%&'%0B804Hy%="%-y%-"98(z&*<'+@&H%=^M"V+B<&"9-Rp&
<'Rr.|
3.BµnvÒsùsos¸nh
$"}$J.J/&"lDy% $"}$J.J/&"&*2T<
m 6=
>
< ¸
$"}$J.J/&"&*2T< $"}$J.J/&"lDy%
q!/<<9D-(P<H2T<-"%A-^B$&)&HIJ.J/&"0;%&"7D<"b&"^,</<+%ID-"c<5",?
JS6?,+%A&JS<"b&"^,*%/-(]<"27+%A (2;<5<U&-4?=%A?6*o%^,p&JS !/<+%ID
-"c<98<"a&*CD7^,</<<y&*-"c<^%)&=A-^@%-~</<$"}$ /&<3+#& n4-"A01
<3+#&<y&*0%V<*%#%$"23&*-(4&"^,*%#%-"./-p&JS="G%&"N&*(,&*+DZ<(Y<(S%
HWH2T<-"%A-^B$:J\*%#%-"./-98Y-"a&$"#%H%=g?0%V<+%A&He%s-/<HZ&*s0,.
HS%-2T&*H2701"4&"-"c<`•="7%-"./-"3&5<3+#&6
qX"y&*$"#%^P<&,.<ò&*<'-"I5"78^,<ò&*&)&6-4?=%A?5*%#%-"./-6-(\<
-%A$p&JS:<'-"I0;%M$-(5+$-(6CD/<w&*=1&"-7&)&-4?=%A?MHS%-2T&*-(D&*
*%7&<"c7p&JS-(\<-%A$Kmp&JS$"i:"V-&"2&*2[%-7-4?0,&*5p&JS-(\<-%A$6:
-"\<(7^,-4?sCDÆ&*<"c70,&*s5p&JS$"i6(w%-~CDÆ&*?,HW%(70,&*M0 !ò&*

$"#%&'%&*78^,<"a&*$"#%<cH/CDÆ&*&,.<ò&*HW%(70,&*:04-"A&ADHÆ-p&JS
$"i-"4$"#%HÆ-H%1D=%V&<".p&$"i
q!ò&*<U&&";(z&*`.<'</<J\J.J/&"*%/-(]KmJ\H/&"*%/ d"A&)&="y&*
H2T<CD)&(z&*Hy%="%0%V<-4?=%A?5*%#%-"./-6$"#%<U&-;%</<=ü&¨&*H/&"*%/
HI</+%V-"'7H%1D=%V&J.J/&"
Tãml¹i:
!'E*%#%$"/$<3+#&-(2;<M$-(5+$-(6
PDF created with pdfFactory Pro trial version www.pdffactory.com
E
5*$M6BK Biếnđổi </<HS%-2T&*<'?ặ "h.&"N&*?i<Hb<"J7DK
5q6Đ2701</<"4&"-"c<<ă&+#&
5q6XD9-"%V&HS%-2T&*-"I"%V&<"D&*5p&JS$"i6
5q6BY-&"F&-ử
5*$R6aKd4?(7HS%-2T&*-"I"%V&<"D&*HI đặtẩnphụ 0;%?i<Hb<"^,?H3&
*%#&=A-<9D5Hặ<+%V-<"PxH%1D&,80;%+,% /&<'-"7?JS6
5*$E6dKĐánhgiá -23&*-"b<"</<HS%-2T&*<'?ặ-HI</+%V-"'7J\J.J/&"
!U&$"#%&Y?H2T<H%1D&,8&AD?DS& xácđịnhmụcđíchbàitoán <ũ&*
&"2"4&"-",&"</< ph ơngángiải
4.Sựgiảithoátkhỏicácphéptoán
D.</<<9D-(P<H2T<-"%A-^B$&)&HIJ.J/&"^,</<+%ID-"c<5",?JS6?,</<
",?JSJ3<9$<O7<"P&*-7^@%^,</<CD8-Y<`2;%"4&"-"c<</<<y&*-"c<<" 23&*
c&**%/-(]H2T<"4&"-",&"-()&</<$"}$ /&<3+#&5CD8-Y<H]&"-(]<3+#&6 d7
HW+%A-</<$"}$ /&<3+#&&"2J7DK
B#&*RK
$"}$ /&lDy% $"}$ /&&*2T<
q -
Ê Ơ
(::)
n
n

p
:::
cos(:::) arccos(:::)
sin(::) arcsin(:::)
a
(:::)
log
a
(:::)
$"}$ /&&*2T< $"}$ /&lDy%
!/<$"}$ /&-/<HZ&*-(\<-%A$0,.+%A&JSH'<Z&*0;%</<$"}$"T$",?JS
*%P$"4&"-",&"&)&</<^;$",?JSJ3<9$J7DK
M:L,?H7-"c<
R:L,?$"F&-"c<
E:L,?0y-ỷ
4:L,?^2T&**%/<
5:L,?JS?ũ0,^.*7(%-
6:!/<",?-e&*"T$-~<# 6 $"}$-()&
d(.&*0%V<*%#%$-(5+$-(6+@&<U&&Y?<"Y<H%1D&,8HI<'H2T<"V-"S&*01</<
"4&"-"c<$-(5+$-(60,x-2u&*5*%#%$"/$6<3+#&01=ỹ&ă&*lử^x
nb`iMK
G%#%$"23&*-(4&"K
2:
p
3x Ă2 Ă 3:
3
p
x Ă 1=1:
PDF created with pdfFactory Pro trial version www.pdffactory.com
4

d(2;<="%+Y-HUD^[%*%#%<U&<'&"N&*&"B&l}-J7DK
!'K
<ă&+B<R5^%)&CD7&KHA&$"}$^ũ8-"~7+B<R60,+B<E5&*"ĩ HA&$"}$^ũ8-"~7
+B<E6<O7&"N&*^2T&*+B<&"9-5(9-`"'7*%#%&"[$"}$&"F&<"%7<"."z&*JS
0,$"}$<Z&*-(~6
d@.CD7&"VK
(
p
3x Ă 2)
2
mEl - R
(
3
p
x Ă 1)
3
ml-M
E5l - M6 - 5El - R6m-M$(
p
3xĂ2)
2
- E(
3
p
x Ă 1)
3
mM5*6
r"20B8&AD(P-+;-MHS%-2T&*<ă&0`K
p
3x Ă 2 m

3
3
p
x Ă 1+1
2
-7-4?(7HS%
-2T&*-"I"%V&<"D&*5p&JS$"i6 (
3
p
x Ă 1) m-Hh?-"780,.5*6HI<'
$-( $ Ă4t
3
+3t
2
+2tĂ1=0HF8^,M$-(+B<E&"2&*<' n
o
Hặ<+%V mM
-
P"#%&'%^Dy&&"2T<H%I?<O7<"23&*-(4&"$"e-"y&*"%V&-@%="%0Y&*?ặ-"7%
2viênthuốcgiải <".R$"}$ /&^2T&**%/<H'^,$"}$^98 arccos 0, arcsin
<"#="/<&,.Hẻ<.&*/%(75&"N&*Hc7<.&*/%J%)D0%V-6(w%HI&'A<"w&*| P"/
?9-J\",%"ò7<O7<9D-(P<
4.Vềbàitoáncóthamsố
X%&CD78^@%J\0b0.&0%V<*%#%$-(&"2^,H%-4?0,&*:J\=%A?-4?="y&*<"{
$"i-"DZ<</<"-"c<:<y&*&*"V-4?=%A??,<ò&$"i-"DZ<&"%1D8AD-S G%#%
$"23&*-(4&"<ũ&*-"A:-(.&*&"N&*H%1D=%V&-"\<-A="/<&"7D?Z-HS%-2T&*J_
<'&"N&**%#%$"/$=%A?-4?0,*%#%-"./-="/<&"7D n%V<H'="y&*<"{$"i-"DZ<
HS%-2T&*-(\<-%A$5p&JS6-7H%-4??,<ò&$"i-"DZ<0,.&"%1D8AD-S:&"N&*8AD
-S$"i*%75*%S&*&"2<"9-lP<-/<-(.&*$"#&c&*H%1D<"A<"9-"'7"o<6<'-"I
<"27-2[&*?%&"*%/-(]5<ũ&*&"2?Z-+%A&JS-(.&*+%ID-"c<",?H2T<-"%A-^B$

J.J/&"6&"2&*-7="y&*<.%&'^,p&JS504<'$"#%?i<Hb<"^,H%-4?&'HFD|6?,
*o%^,-"7?JS0,c&*0;%?ỗ%-4&""DS&*Hặ<+%V-l#8HA&0;%-"7?JS-4&""DS&*
0,H%1D=%V&*%#%-"./-^@%="/<H%+DZ<-7$"#%^DB&l} 2[&*?%&"5CD/-(4&"+%V&
^DB&6 n4^x` ()&-y%&*"ĩ 01+#&<"9-
thamsố ^,K
sốthamgia 0,.J\-"%A-^B$J.J/&"-(.&*$"23&*-(4&"<'-"I*F8HA&&"N&*
*%#%$"/$="/<&"7D<".0%V<-4?=%A?p&JS
X%&Hặ-?4&"0,.lP<<#?<O7?Z-&*2[%<"{"D8?Z (B&H/&"="%*ặ$+,% /&
+%V&^DB&&'%(%)&*"78<'-"7?JS&'%<"D&* LW8Hặ-(7&"N&*<"Nsnếus<'^x
5?Z-</<"-\&"%)&6<".&"N&*-4&""DS&*l#8HA&<" (B&H/&"s*%#%-"./-<O7
PDF created with pdfFactory Pro trial version www.pdffactory.com
5
+@&s; H'-"\<<"9-^,?Z-CD/-(4&"`%•&`]<"</<x-2u&**%#%-"./- dy%&"B&-"98
<'&"N&*<FD"G%-"2[&*`%•&(7-\&"%)&0;%?4&"&"2J7DK
5BM6 BF8*%[J_-9&<y&*-"A&,8 &"2&*^ì -"4J7.Q0B8&)&l}-
5BR6rAD5H%1D=%V&6-"DB&^T%<".$"}$s-9&<y&*=%7s="y&*l#8(7-"4J7.Q 5<"o&
</"-9&<y&*="/<-(/&"H%<@?+É8HW&"B&(7"78H%1D=%V&="y&*<".$"}$0~7
&"B&(76
5BE6 QD78^@%5BM6
!c-"ACD/-(4&"^DB&l}-0}-"A-H2T<</<-4&""Dw&*l#8HA&HI<DS%<v&*<".
-7?Z-=A-^DB&-e&*"T$
r'%<"D&*-y%"3%="'"%ID^,MJS"J(9-&*/&0%V<*%#%0,+%V&^DB&$-(:+$-(
<'-"7?JS?.&*+@&"W8+4&"-"#&"3&-"4-S- d"\<(7^.@%+,%-B$&,8<U&u+@&
$"p?<"9-<O7M+/<&y&*`F& r'=.="'&"2&*(Y<(S%04&"N&*JD8^DB&0i&0Æ-
=%ID0}-<@&:&*,8<ß&"J-y%-"2[&*HS%`%V&0;%&'&"2-"I<U&H%CD7?Z-<.&
H2[&*?27^U80B8 !ß&+@&"W8lD9-$"/-0%V<s*%#%0,+%V&^DB&s-~-"DB-*%#%0,
="%&,.</%-"7?JSH/&**"}-=%7&*¨&<#&-b&"<'^x<O7*%#%-"DB "4-7J_0B-&'
(7?,?elÎ0,lh?l}- 0B8-"y%|
nDMK
G%#%+%V&^DB&0;%?^,-"7?JS$-(K

m + x =1+m
2
x
—————
$-( $ (m ¡1)=(m
2
¡1)x
rAD m
2
¡ 1=0$(m¡1)(m +1)=0$m=1".Æ< m = ¡1
m =1$-( $ 0x =0&"B& RK-B$&*"%V?
m = ¡1 $-( $ 0x = ¡2K0y&*"%V?
rAD m
2
¡ 1 6=0$m6=M0, -M
$-( $ x =
m ¡1
m
2
¡1
$ x =
1
m +1
nB8K
qn;% m =1-B$&*"%V?$-(^, R
qn;% m = ¡1 $-(0y&*"%V?
qn;% m 6= M0, -M$-(<'&*"%V?`D8&"9- x =
1
m +1
PDF created with pdfFactory Pro trial version www.pdffactory.com

6
!ũ&*$"#%&'%-")?^,="%`v&*=ỹ&ă&*Hặ-p&JS$"i*%#%$"23&*-(4&"5+$-(6
<'-"7?JS+@&<U&Hặ-H%1D=%V&<".p&JS$"i=ẻ.="y&*</%H%1D=%V&s-2u&*+us
?,+@&^DB&(7<".&*"%V?<O7$"23&*-(4&"="y&*l/<-"\<
nDEK
d4?-"7?JS m HI$"23&*-(4&"K
cos
2
x Ă 2m:cosx +1m0<'R&*%V?$"F&+%V- x 2 50;
ẳ
2
6

d7%&@&l#8HA&^,Hặ-cosx = t (w%-"4$-(<'&*%V? $ $-( t
2
Ă2m:t +1=0<'
R&*%V?$"F&+%V-5-"B?<"bR&*"%V?$"F&+%V- 2 (0;
ẳ
2
)6|
lờigiảiđúngphảilà:
Hặ- cosx = t 2 (0;1) ="% x 2 50;
ẳ
2
6?,0;%?ỗ%t2(0;1) -"4 cosx = t J_J%&"(7
`D8&"9-HP&*Mn
0
x2(0;
ẳ
2

)
nB8&)&HI$-(H1(7<'&*%V?&"28)D<UD$ $-( t
2
Ă 2m:t +1=0<'R&*%V?
$"F&+%V- t 2 (0;1) HA&HF8+@&"W8lh?^@%nDR
d~nDE<'-"I(P-&*78(7?Z- bàihọclà:
d(.&*0%V<lử^x0%V<<'&*"%V?5"78^DB&l}-JS</<&*"%V?6?Z-$"23&*-(4&"
<'-"7?JS?,-7Hặ-p&JS$"iK
5q6Hặ-H%1D=%V&<".p&JS$"i-"h.H%1D=%V&<O79&JS*S<:<y&*0%V<&,8<"b&"^,
-4??%1&*%/-(]<O7",?-m$5l6Kp&$"i-()&?%1&l-"G7H%1D=%V&p&JS
5q6 MDS&l}-JS&*"%V?<U&l}-<#-b&"<#?J%&"&*"%V?l-~&*"%V?-<O7
$"23&*-(4&"p&$"i5$"}$*%#%&*2T<6
B.Mộtsốgiảiphápchitiết:
LW8lh?^@%+#&*5R6</<$"}$ /&<3+#&HI"%IDx-2u&*"4&"-",&"</<=ỹ
&ă&*<3+#&?,-y%+,8(7J7DHF8
1.Ph ơngtrình(bptr)bậcnhất:
dy%lh?-(o&*^.@%&,8="%`@8<"."/J?9-<ă&+#&5$"U&&"%1D-(.&*&"N&*
Hc7-(ẻ -~&*&*w%-(2;<-y%6="y&*$"#%04&'`?,04&'+/.HZ&*H2T<<".<"P&*
H%1D<"P&*HW?9-
M M:X"/%&%V?K
PDF created with pdfFactory Pro trial version www.pdffactory.com
7
j,+,% /&^%)&CD7&-;%H%1D=%V&J.J/&"0;%</<+%ID-"c<"4&"-",&"+u%E$"}$
/&5q6:5-60,5 60;%"z&*JS5</<",?+B<&"9-6
M R:Đ%1D<3+#&<U&+%A-K
5-6J\*%#%-"./-K
5lh?+#&*R6
5-6^DB-^VK
r";^,5q60,5-6-"40y-2&"2&*5K6-"4<U&JS<"%75?ẫDJS$"F&-"c<6 6= 0
2.Ph ơngtrìnhđathứcbậccaohơn1:

ĐF8^,&1&-#&*<".</<-"I^.@%$-("78+$-(<'-"I*%#%H2T<J7D&,8
M M:X"/%&%V?K
j,+,% /&^%)&CD7&-;%H%1D=%V&J.J/&"0;%</<+%ID-"c<"4&"-",&"+u%E$"}$
/&5q6:5-6:5 60;%"z&*JS0,</<",?^ũ8-"~7+B<-\&"%)&5</<",?H7-"c<6
R R:Đ%1D<3+#&<U&+%A-K
5-6f\*%#%-"./-K
d4?</<"$"F&-b<"-(%V-HIH7-"c<01-b<"</<&"F&-ử+B<&"9-0,+B<R0y&*"%V?
5-6jDB-^VK
5q6!U&&Y?0N&*</<=%A&-"c<01-7?-"c<+B<R
5q6Xỹ &ă&*$"F&-b<"&"F&-ử*w?
5=&*06!/<"&"p?&*"%V??y-H7-"c<+B<^;&"3&R5?ò (7*%/-(]^,?H7-"c<
-(%V %)D6
5=&*M6Đ]&"^x B3zDK
rADH7-"c< P5l6<'&*"%V?^,<5-c<^, P5<6m06-"4 P5l6<'&"F&-ử5l - <6:
5&*"ĩ7^, P5l6m5l - <6 Q5l60;% Q5l6KH7-"c<6
5=&*R6KRD@&*$"F&-b<"<3+#&<O7M-7?-"c<+B<R*w?$"F&-b<"<"b&"-Y<
0,$"F&-b<"&"F&-ử
5q6P"23&*$"/$l}-`9D+z&*+#&*0,-(i<H7&`9D
PDF created with pdfFactory Pro trial version www.pdffactory.com