Đại số chương 3 lớp 8
TRANSCRIPT
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I. M U V PHNG TRNH
VN I. Chng minh mt s ! nghi"m #$% mt &h'(ng t)*nh
Phng php: Dng mnh sau:x0 l nghim ca phng trnh A x B x( ) ( )= A x B x0 0( ) ( )=
x0 khng l nghim ca phng trnh A x B x( ) ( )= A x B x0 0( ) ( )
+!i ,. Xt xem x0 c l nghim ca phng trnh hay khng?
a) x x3(2 ) 1 4 2 + = ; x0 2= b) x x5 2 3 1 = + ; x03
2=
c) x x3 5 5 1 = ; x0 2= d) x x2( 4) 3+ = ; x0 2=
e)x x
7 3 5 =
; x0 4
= ) x x2( 1) 3 8
+ = ; x0 2
=g) x x5 ( 1) 7 = ; x0 1= h) x x3 2 2 1 = + ; x0 3=
+!i -. Xt xem x0 c l nghim ca phng trnh hay khng?
a) x x x2 3 7 1 2 + = + ; x0 2= b) x x2 3 10 0 = ; x0 2=
c) x x x2 3 4 2( 1) + = ; x0 2= d) x x x( 1)( 2)( 5) 0+ = ; x0 1=
e) x x22 3 1 0+ + = ; x0 1= ) x x x24 3 2 1 = ; x0 5=
+!i . !m gi" tr# k$a% ch% phng trnh c nghim x0 &'c ch( ra
a) x k x2 1+ = ; x0 2= b) x x k x(2 1)(9 2 ) 5( 2) 40+ + + = ; x0 2=
c) x x x k2(2 1) 18 3( 2)(2 )+ + = + + ; x0 1= d) k x x x 5( 3 )( 1) 4(1 2 ) 80+ + + = ; x0 2=
VN II. / nghi"m #$% mt &h'(ng t)*nhPhng php: Dng mnh sau:
Phng trnh A x B x( ) ( )= v nghim A x B x x( ) ( ), Phng trnh A x B x( ) ( )= c v s nghim A x B x x( ) ( ),=
+!i ,. *h+ng t, c"c phng trnh $a- . nghima) x x x2 5 4( 1) 2( 3)+ = b) x x2 3 2( 3) =
c) x 2 1 = d) x x2 4 6 0 + =+!i -. *h+ng t, r/ng c"c phng trnh $a- c . $0 nghim
a) x x x4( 2) 3 8 = b) x x4( 3) 16 4(1 4 ) + = +c) x x2( 1) 2 2 = d) x x=
e) x x x2 2( 2) 4 4+ = + + ) x x x2 2(3 ) 6 9 = =+!i . *h+ng t, r/ng c"c phng trnh $a- c nhi1- hn m2t nghim
a) x2 4 0 = b) x x( 1)( 2) 0 =
c) x x x( 1)(2 )( 3) 0 + = d) x x2
3 0 =e) x 1 3 = ) x2 1 1 =
VN III. Chng minh h%i &h'(ng t)*nh t'(ng 0'(ng
CHNG III1 PHNG TRNH +2C NHT M3T 4N
!rang 34
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i s 8 Trn S Tng
"# ch$ng minh hai phng trnh tng ng% ta c th# s& '(ng m)t tr*ng cc cch sau:+h$ng minh hai phng trnh c cng t,p nghim-.& '(ng cc ph/p 0i1n 2i tng ng # 0i1n 2i phng trnh n3 thnh phng trnh kia-4ai 5ui t6c 0i1n 2i phng trnh:
7 Qui tc chuyn v: !r*ng m)t phng trnh% ta c th# chuyn mt hng t t v ny sangv iav !"i #$uh8ng t& -
7 Qui tc nh%n: !r*ng m)t phng trnh% ta c th# nh%n c& hai v v'i cng mt s h(c )-
+!i ,. Xt xem c"c phng trnh $a- c tng &ng hay khng?a) x3 3= . x 1 0 = b) x 3 0+ = . x3 9 0+ =c) x 2 0 = . x x( 2)( 3) 0 + = d) x2 6 0 = . x x( 3) 0 =
+!i -. Xt xem c"c phng trnh $a- c tng &ng hay khng?
a) x2 2 0+ = . x x2( 2) 0+ = b) x x1+ = . x2 1 0+ =
c) x 2 0+ = .x
x0
2=
+d) x x
x x
2 1 1+ = + . x x2 0+ =
e) x 1 2 = . x x( 1)( 3) 0+ = ) x 5 0+ = . x x2( 5)( 1) 0+ + =
II. PHNG TRNH +2C NHT M3T 4N
!rang 35
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Trn S Tng i s 8
VN I. Ph'(ng t)*nh 0'% 0'5# 67 89ng &h'(ng t)*nh :;# nh. 6i7i c"c phng trnh $a-
a)x x x2 1 2 7
5 3 15
+ = b)
x x x3 1 51
2 3 6
+ + = +
c)x x x x2( 5) 12 5( 2)
113 2 6 3
+ + + = + d)
x x x xx
4 3 2 2 5 7 2
5 10 3 6
++ =
e)x x x2( 3) 5 13 4
7 3 21
++ = )
x xx
3 1 1 4 9
2 4 8
=
".: a9 < tu= > 09 < tu= > c9 < tu= > '9 v nghim 9 v nghim ;9 v nghim
+!i ?. 6i7i c"c phng trnh $a-
!rang 33
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a)x x x x x x( 2)( 10) ( 4)( 10) ( 2)( 4)
3 12 4
+ + + + = b) x xx
2 2( 2) ( 2)
2(2 1) 258 8
+ + = +
c) x x x x2 2
(2 3)(2 3) ( 4) ( 2)
8 6 3
+ = + d)
x x x x2 2 27 14 5 (2 1) ( 1)
15 5 3
+ =
e)
x x x x x2(7 1)( 2) 2 ( 2) ( 1)( 3)
10 5 5 2
+ + = +
".: a9 x 8= 09 x 9= c9 x 123
64= '9 x
1
12= 9 x
19
15=
+!i @. 6i7i c"c phng trnh $a- 8?i1n 2i @c 0it)
a)x x x x1 3 5 7
35 33 31 29
+ + + ++ = + 84D: +)ng thAm B v* cc h8ng t&)
b)x x x x x10 8 6 4 2
1994 1996 1998 2000 2002
+ + + + = 84D: !rC i B v* cc h8ng t&)
x x x x x2002 2000 1998 1996 1994
2 4 6 8 10
= + + + +
c)x x x x x1991 1993 1995 1997 1999
9 7 5 3 1
+ + + + =
x x x x x9 7 5 3 1
1991 1993 1995 1997 1999
= + + + + 84D: !rC i B v* cc h8ng t&)
d)x x x x85 74 67 64
1015 13 11 9
+ + + = 8+h >: 10 1 2 3 4= + + + )
e)x x x x1 2 13 3 15 4 27
13 15 27 29
= 84D: !hAm h*@c 0Et B v* cc h8ng t&)
".: a9 x 36= 09 x 2004= c9 x 2000= '9 x 100= 9 x 14= -+!i A. 6i7i c"c phng trnh $a- 8?i1n 2i @c 0it)
a)x x x x1 3 5 7
65 63 61 59
+ + + ++ = + b)
x x x x29 27 17 15
31 33 43 45
+ + + + =
c)x x x x6 8 10 12
1999 1997 1995 1993
+ + + ++ = + d)
x x x x1909 1907 1905 19034 0
91 93 95 91
+ + + + =
e)x x x x x x29 27 25 23 21 19
1970 1972 1974 1976 1978 1980
+ + + + + =
x x x x x x1970 1972 1974 1976 1978 1980
29 27 25 23 21 19
= + + + + +
".: a9 x 66= 09 x 60= c9 x 2005= '9 x 2000= 9 x 1999= -
VN II. Ph'(ng t)*nh tB#h
!rang 39
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Trn S Tng i s 8
"# giFi phng trnh tGch% ta p '(ng cng th$c:
A x B x A x( ). ( ) ( ) 0 = h*@c B x( ) 0= A xB x
( ) 0
( ) 0
= =
!a giFi hai phng trnh A x( ) 0= v B x( ) 0= % rHi lI3 tIt cF cc nghim ca chng-
+!i ,. 6i7i c"c phng trnh $a-a) x x(5 4)(4 6) 0 + = b) x x(3,5 7)(2,1 6,3) 0 =c) x x(4 10)(24 5 ) 0 + = d) x x( 3)(2 1) 0 + =e) x x(5 10)(8 2 ) 0 = ) x x(9 3 )(15 3 ) 0 + =
".: a9x x4 3
;5 2
= = 09 x x2; 3= = c9 x x5 5
;2 24
= = '9 x x 1
3;2
= =
9 x x2; 4= = ;9 x x3; 5= = +!i -. 6i7i c"c phng trnh $a-
a) x x2(2 1)( 2) 0+ + = b) x x2( 4)(7 3) 0+ =
c) x x x2
( 1)(6 2 ) 0+ + = d) x x x2
(8 4)( 2 2) 0 + + =".: a9x
1
2= 09 x
3
7= c9 x 3= '9 x
1
2=
+!i . 6i7i c"c phng trnh $a-a) x x x( 5)(3 2 )(3 4) 0 + = b) x x x(2 1)(3 2)(5 ) 0 + =c) x x x(2 1)( 3)( 7) 0 + = d) x x x(3 2 )(6 4)(5 8 ) 0 + =e) x x x x( 1)( 3)( 5)( 6) 0+ + + = ) x x x x(2 1)(3 2)(5 8)(2 1) 0+ =
".: a9 S 3 45; ;
2 3
=
09 S
1 2; ; 5
2 3
= =
c9 S
1;3; 7
2
=
'9 S
3 2 5; ;
2 3 8
=
9 { }S 1; 3; 5;6= ;9 S 1 2 8 1; ; ;2 3 5 2 = +!i =. 6i7i c"c phng trnh $a-
a) x x x x( 2)(3 5) (2 4)( 1) + = + b) x x x x(2 5)( 4) ( 5)(4 )+ =
c) x x x29 1 (3 1)(2 3) = + d) x x x x22(9 6 1) (3 1)( 2)+ + = +
e) x x x x2 227 ( 3) 12( 3 ) 0+ + = ) x x x x216 8 1 4( 3)(4 1) + = +
".: a9 x x2; 3= = 09 x x0; 4= = c9x x1
; 23
= = '9x x1 4
;3 5
= =
9 x x x 4
0; 3;
9
= = = ;9 x 1
4
=
+!i >. 6i7i c"c phng trnh $a-
a) x 2(2 1) 49 = b) x x2 2(5 3) (4 7) 0 =
c) x x2 2(2 7) 9( 2)+ = + d) x x x2 2( 2) 9( 4 4)+ = +
e) x x2 24(2 7) 9( 3) 0+ + = ) x x x x2 2 2 2(5 2 10) (3 10 8) + = +
".: a9 x x4; 3= = 09 x x 10
4;9
= = c9 x x 13
1;5
= = '9 x x1; 4= =
9 x x 23
5;7
= = ;9 x x 1
3;2
= =
+!i ?. 6i7i c"c phng trnh $a-a) x x x x2 2(9 4)( 1) (3 2)( 1) + = + b) x x x x2 2( 1) 1 (1 )( 3) + = +
c) x x x x x x2 2( 1)( 2)( 3) ( 1)( 4)( 5) + = + d) x x x4 3 1 0+ + + =
!rang 3:
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i s 8 Trn S Tng
e) x x3 7 6 0 + = ) x x x4 34 12 9 0 + =g) x x x5 35 4 0 + = h) x x x x4 3 24 3 4 4 0 + + =
".: a9x x x2 1
; 1;3 2
= = = 09 x x1; 1= = c9 x x x 7
1; 2;5
= = =
'9 x 1= 9 x x x1; 2; 3= = = ;9 x x1; 3= =
g9 x x x x x0; 1; 1; 2; 2= = = = = h9 x x x1; 1; 2= = =+!i @. 6i7i c"c phng trnh $a- 8"@t Jn ph()
a) x x x x2 2 2( ) 4( ) 12 0+ + + = b) x x x x2 2 2( 2 3) 9( 2 3) 18 0+ + + + + =
c) x x x2( 2)( 2)( 10) 72 + = d) x x x x2( 1)( 1) 42+ + + =
e) x x x x( 1)( 3)( 5)( 7) 297 0 + + = ) x x x4 22 144 1295 0 = ".: a9x x1; 2= = 09 x x x x0; 1; 2; 3= = = = c9 x x4; 4= = '9 x x2; 3= =
9 x x4; 8= = ;9 x x5; 7= =
VN III. Ph'(ng t)*nh #h% n D mEF+c 0Ec giFi phng trnh ch$a Jn K mLu:
*+'c ,-!m iu kin
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Trn S Tng i s 8
a)x
x xx x26 1 5 3
2 57 10
++ =
+b)
x x
x x x xx22 1 4
0( 2) ( 2)4
+ =
+
c)x x
x x x x x
2
2
1 1 ( 1)
3 1 3 2 3
=
+ d)
x x x x21 6 5
2 3 6 =
+
e)x
x x x x
2
3 2
2 2 16 5
2 8 2 4
+ =+ + + )
x x x
x x x x x
2
2 2 6
1 1 2( 2)
1 1 1
+ + =+ + +
".: a9x 9
4= 09 v nghim c9 x
3
5= '9 x 4=
9 v nghim ;9 x 5
4=
+!i =. 6i7i c"c phng trnh $a-
a)x x x x
8 11 9 10
8 11 9 10+ = +
b)
x x x x
x x x x3 5 4 6 =
c) x x x x2 2
4 31 0
3 2 2 6 1 + = + + d) x x x x1 2 3 6
1 2 3 6+ + =
".: a9 x x 19
0;2
= = 09 x x 9
0;2
= = c9 x x0; 3= = '9 x x6 12
;5 5
= =
+!i >. 6i7i c"c phng trnh $a-a) b)
".: a9
III. GII TN +JNG CCH K2P PHNG TRNH
!rang 3
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i s 8 Trn S Tng
+c 0Ec giFi t*n 0Vng cch l,p phng trnh:*+'c ,-9:; ;h+i&i ;h+- tPng $0 b# chia l=n 54 . gi7m $0 chia &i m2t nHa th hi- ca
hai $0 mAi l 94E !m hai $0 &E".: 3:; CE
+!i >. F2t &2i cng nhGn $Ha m2t &%Qn &Rng tr%ng 9 ngyE Jgy th+ nhMt &2i $Ha &'c1
3&%Qn
&Rng ngy th+ hai &2i $Ha &'c m2t &%Qn &Rng b/ng4
3&%Qn &'c lm &'c tr%ng ngy
th+ nhMt ngy th+ ba &2i $Ha C4m cSn lQiE !Tnh chi1- di &%Qn &Rng m &2i ph7i $HaE".: 9t r/ng n>- x%" ch` $0 3 & th $0 My gi7m &i 344E
!rang 3D
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Trn S Tng i s 8
".: 333E+!i ,O.6ia &nh % c : ngRi b0 m b Fai . %E !-Li tr-ng bnh ca c7 nh l 39E J>-
.i>t th=m ch` $0 4 .% b=n ph7i t-Li b Fai th &'c t-Li ca b0 t-Li ca m b/ng9
10t-Li
b0 . gMp 9 l@n t-Li ca %E !m t-Li ca mYi ngRi tr%ng gia &nh %E".: !u2i ca 0% m^% 0/ _ai v "* l[n lQt l: :4 9 &>n bQn c-0i c]ng th+ n nhn n.i=n k% . &'c lMy th=m1
11$0 k% cSn lQiE
U,i phGn &2i & c ba% nhi=- &2i .i=n . mYi &2i .i=n nhn ba% nhi=- .i=n k%E".: 54 )i viAn m`i )i vin nh,n 54viAn k^*-
+!i ,-. F2t ngRi b"n $0 $@- ri=ng th- h%Qch &'c nh $a-
B @n th+ nhMt b"n \ tr"i .1
6$0 $@- ri=ng cSn lQiE
B @n th+ hai b"n 5C tr"i .1
6$0 $@- ri=ng cSn lQi mAiE
B @n th+ ba b" 3D tr"i .1
6$0 $@- ri=ng cSn lQi mAi .E.EEE
Ai c"ch & th b"n l@n $a- c]ng l .Na h>t . $0 $@- ri=ng b"n mYi l@n &1- b/ng nha-EU,i ngRi & &K b"n ba% nhi=- l@n . $0 $@- ri=ng th- h%Qch &'c l ba% nhi=- tr"i?".: 33 tri%0n l[n-
+!i ,.fa lAp j f * gp $"ch tng c"c bQn hc $inh .]ng kh khPn tMt c7 &'c 9C c-0nE !( $0
$0 c-0n $"ch ca lAp j $% .Ai lAp f l6
11E !( $0 $0 c-0n $"ch ca lAp j $% .Ai lAp * l
7
10E
U,i mYi lAp gp &'c ba% nhi=- c-0n $"ch?".: YEp : C: cunb lEp ?: 5: cunb lEp +: 534 cun-
+!i ,=._Gn $0 t(nh j hin nay l
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i s 8 Trn S Tng
. c hai chZ s c '8ng: xy x y10= + - "iu kin: x y N x y, ,0 9,0 9 < -
. c 0a chZ s c '8ng: xyz x y z100 10= + + - "iu kin: x y z N x y z, , ,0 9,0 , 9 < -
+!i ,. !m m2t $0 t nhi=n c hai ch` $0 bi>t r/ngB !Lng hai ch` $0 l 53
B J>- &Li chY hai ch` $0 th &'c m2t $0 mAi lAn hn $0 & l 9- th=m ch` $0 5 .% b=n ph7i hay b=n tr"i $0 & ta &'c m2t
$0 c < ch` $0E fi>t r/ng n>- .i>t th=m .% b=n ph7i $0 & th &'c m2t $0 lAn gMp ba l@n $0nhn &'c khi ta .i>t th=m .% b=n tr"i $0 &E !m $0 &E
".: :3CDE+!i =. F2t $0 c hai ch` $0 tr%ng & ch` $0 hng chc gMp 9 l@n ch` $0 hng &n .#E J>- &Li chY
hai ch` $0 ta &'c m2t $0 c hai ch` $0 nh, hn $0 ban &@- 5C &n .#E !m $0 &E".: 95E
+!i >. F2t $0 t nhi=n c hai ch` $0 c tLng c"c ch` $0 b/ng DE J>- th=m ch` $0 4 .% gi`a haich` $0 ta &'c m2t $0 c 9 ch` $0 lAn hn $0 &K ch% l 5C4E !m $0 &E
".: 3E+!i ?.
".:
VN III. KL9i !m #hFng S !m )ing mt 6i"#Shi cng vic khng Qc * 0Vng s lQng c( th#% ta c*i t*n 0) cng vic l m)t n vMcng vic% 0i#u thM 0Ki s B-dng suIt lm vic l ph[n vic lm Qc tr*ng m)t n vM th\i gian-OWi l khi lQng cng vic% n l ndng suIt% t l th\i gian lm vic- !a c: A nt= -!2ng ndng suIt riAng 0Vng ndng suIt chung khi cng lm-
+!i ,. Uai ngRi c]ng lm m2t cng .ic tr%ng 3: giR th x%ngE JPng $-Mt ca ngRi th+ nhMt b/ng3
2
nPng $-Mt ca ngSi th+ haiE U,i n>- mYi ngRi lm m2t mnh c7 cng .ic th ph7i mMt
thRi gian ba% lG-?".: ef gi\b f gi\-
+!i -. F2t bon ch+a c &t hai .Si nAc ch7y .% . m2t .Si th"% nAc raEB fon tr0ng khng n>- mV ri=ng .Si th+ nhMt th $a- : giR bon &@y nAcEB fon tr0ng khng n>- mV ri=ng .Si th+ hai th $a- < giR bon &@y nAcEB fon tr0ng khng n>- &ong thRi mV c7 ba .Si th $a- D giR 53 phZt bon &@y nAcEU,i n>- bon ch+a &@y nAc mV ri=ng .Si th"% nAc th $a- ba% lG- $ th"% h>t nAc ra?
".: T gi\ T pht-+!i . F2t cng nhGn ph7i lm m2t $0 $7n phqm tr%ng 5C ngyE _% &K .'t m+c mYi ngy $7n
phqm n=n $a- 5< ngy anh &K lm x%ng . lm th=m 34 $7n phqm n`a ng%i k> h%QchE !Tnh
xem mYi ngy anh &K lm &'c ba% nhi=- $7n phqmE".: DsFn phJm-
VN IV. KL9i #hFn 0ng 07F
!rang 3\
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Trn S Tng i s 8
OWi ' l 5uUng \ng )ng t& i% v l v,n tc% t l th\i gian i% ta c: d vt= -,n tc n a &im f .Ai .n t0c 4 km^h roi tN f O-ay ngay .1j .Ai .n t0c :4 km^hE *7 &i . .1 mMt m2t thRi gian l giR 3: phZtE !m chi1- di O-Kng&Rng tN j &>n fE
".: km120 -+!i -. F2t xe &Qp khVi hnh tN &im j chQy .Ai .n t0c 34 km^hE Wa- & 9 giR m2t xe hi &-Li
the% .Ai .n t0c 4 km^hE U,i xe hi chQy tr%ng ba% lG- th &-Li k#p xe &Qp?".: 2gi\-
+!i . F2t ngRi &i xe gn m"y &i tN a &im j &>n a &im f tr=n m2t O-Kng &Rng di km35 EZc trV .1 ngRi & &i the% c%n &Rng kh"c di km42 .Ai .n t0c km hn .n t0c l't &i l
< km^hE !hRi gian l't .1 b/ng3
2thRi gian l't &iE !m .n t0c l't &i . l't .1E
".: ,n tc lQt i l Tf kmhb v,n tc lQt v l e kmh-
+!i =. F2t xe t7i &i tN j &>n f .Ai .n t0c 4 km^hE i &'c 3: phZt th gp &Rng xM- n=n .nt0c tr=n O-Kng &Rng cSn lQi gi7m cSn :4 km^hE .y &K &>n ni chm mMt 5C phZtE !mchi1- di O-Kng &Rng tN j &>n fE
".: km80 -+!i >. Zc < giR 5 phZt m2t t &i tN j & &=n f .Ai .n t0c D4 km^hE hi &>n f t ngh( 5 giR
rsi roi O-ay .1 j .Ai .n t0c n j lZc 55 giR c]ng ngyE !Tnh O-Kng &RngjfE
".: 54km-+!i ?. Ung ngy !-Mn &i xe &Qp &>n trRng .Ai .n t0c 53 km^hE W"ng nay d% dy m-2n !-Mn
x-Mt ph"t chm 3 phZtE !-Mn nhqm tTnh & &>n trRng &Zng giR nh hm trAc th !-Mn ph7i&i .Ai .n t0c 5 km^hE !Tnh O-Kng &Rng tN nh !-Mn &>n trRngE
".: 3 km-+!i @. F2t ngRi &i xe m"y tN thnh ph0 !hanh U%" . thnh ph0 inhE J>- chQy .Ai .n t0c 3
km^h th $ m-2n $% .Ai d nh l 3 giRE J>- chQy .Ai .n t0c 94 km^h . gi`a &Rng ngh( 5giR th cng m-2n mMt 3 giRE U,i & &>n ni &Zng giR m dc &Rng khng ngh( th xe ph7ichQy mYi giR ba% nhi=- kilmet?
".: 9D km-+!i A. Uai t khVi hnh c]ng m2t lZc & &i tN U-> . JungE n t0c xe th+ nhMt l :4 km^h
.n t0c xe th+ hai l n Jung ngh( nHa giR roi O-ay lQi U-> th gpxe th+ nhMt V c"ch Jung 54 kmE !Tnh O-Kng &Rng U-> v JungE
".: 554 km-
+!i . w-Kng &Rng j_ di \ km gom &%Qn jf l=n d0c &%Qn f* n/m ngang &%Qn *_ x-0ngd0cE F2t ngRi &i b2 tN j &>n _ roi O-ay trV .1 j h>t tMt c7 9 giR :5 phZtE !Tnh O-Kng &Rngf* bi>t .n t0c lZc l=n d0c ca ngRi & l : km^h lZc x-0ng d0c l < km^h . lZc &i tr=n&Rng n/m ngang l km^hE
".: :km-+!i ,O.F2t xe t7i &i tN j &>n f .Ai .n t0c : km^hE Wa- & m2t thRi gian m2t xe c%n cng x-Mt
ph"t tN j .Ai .n t0c - khng c g thay &Li th &-Li k#p xe t7i tQi fE Jhng $a-khi &i &'c nHa O-Kng &Rng jf th xe c%n tPng .n t0c l=n D km^h n=n $a- & 5 giR th&-Li k#p xe t7iE !Tnh O-Kng &Rng jfE
".: :4 km-+!i ,,. F2t &S m"y x-i dSng tN b>n j &>n b>n f mMt : giR . ng'c dSng tN f .1 j mMt giRE
n t0c ca dSng nAc l 3 km^hE !m chi1- di O-Kng &Rng tN j &>n fE".: km80 -+!i ,-.F2t ca n x-i dSng tN j &>n f mMt giR . ng'c dSng tN f &>n j mMt < giRE !Tnh
kh%7ng c"ch jf bi>t .n t0c dSng nAc l 3 km^hE
!rang 94
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i s 8 Trn S Tng
".: 534 km-+!i ,.Uai b>n $ng j . f c"ch nha- :4 kmE *]ng m2t lZc .Ai ca n x-i dSng tN b>n j c m2t
chi>c b tri tN b>n j .Ai .n t0c 9 km^hE Wa- khi &>n f ca n trV .1 b=b$ j ngay . gp bkhi b &K tri &'c C kmE !Tnh .n t0c ca ca nE
".: 3D kmh-+!i ,=.F2t chi>c th-y1n &i tN b>n j &>n b>n f h>t giR tN b>n f &>n b>n j h>t D giRE U,i m2t
&"m b% tri the% dSng $ng tN j &>n f h>t ba% lG-?".: 9gi\-+!i ,>.
".:
VN V. KL9i #W ni 8Fng h*nh hX#4nh chZ nh,t c hai kGch thEc a% 0- Din tGch: S ab= b +hu vi: P a b2( )= +
!am gic vung c hai c8nh gc vung a% 0- Din tGch: S ab1
2=
+!i ,. *h- .i m2t kh- .Rn hnh ch` nht b/ng m60 hi- &2 di ca chi1- di . chi1- r2ng lm20 E !m &2 di c"c cQnh ca hnh ch` nhtE
".: m m5 ;25 -+!i -. F2t thHa &Mt hnh ch` nht c ch- .i l m56 E J>- gi7m chi1- r2ng m2 . tPng chi1- di
m4 th din tTch tPng th=m m28 E !m chi1- r2ng . chi1- di thHa &MtE
".: m m12 ;16 E+!i . F2t kh- .Rn hnh ch` nht c chi1- di b/ng 9 l@n chi1- r2ngE J>- tPng mYi cQnh th=m
m5 th din tTch kh- .Rn tPng th=m m2385 E !Tnh &2 di c"c cQnh ca kh- .RnE".: m m18 ;54 E
+!i =. Ui- $0 &% ch- .i ca hai hnh .-ng l m32 . hi- $0 &% din tTch ca chZng l m2464 E!m $0 &% c"c cQnh ca mYi hnh .-ngE
".: c8nh hnh vung nh l m25 b c8nh hnh vung lEn l m33 -
+!i >. F2t kh- .Rn hnh ch` nht c ch- .i l m450 E J>- gim chi1- di &i1
5chi1- di c .
tPng chi1- r2ng th=m1
4chi1- r2ng c th ch- .i hnh ch` nht khng &LiE !Tnh chi1- di .
chi1- r2ng kh- .RnE".: m m100 ;125 E+!i ?. F2t kh- &Mt hnh ch` nht c chi1- di hn chi1- r2ng l 54mE J>- chi1- di tPng th=m
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Trn S Tng i s 8
+!i ,. 6i7i c"c phng trnh $a-
a) x x x x x26 5 3 2 3 (3 2 ) + = b)x x x
x2( 4) 3 2 1
4 10 5
+ = +
c)x x x2 3 5 3(2 1) 7
3 4 2 6
+ = d)
x x xx
6 5 10 3 2 12
2 4 2
+ + + = +
e) x x x x 2( 4)( 4) 2(3 2) ( 4) + = ) x x x x3 3 2( 1) ( 1) 6( 1)+ = + +
".: a9 x 3
2= 09 x 5= c9 x
17
19= '9 x
1
2= 9 x 14= ;9 x
2
3=
+!i -. 6i7i c"c phng trnh $a-
a) x x x x(4 3)(2 1) ( 3)(4 3) = b) x x x225 9 (5 3)(2 1) = + +
c) x x2 2(3 4) 4( 1) 0 + = d) x x x x4 3 22 3 8 4 0+ =
e) x x x2( 2)( 2)( 10) 72 + = ) x x x3 22 7 7 2 0+ + + =
".: a9 S 3
; 24
= 09 S 3 4
;5 3
= c9 S
2
;65
= '9 { }S 1; 2;2=
9 { }S 4;4= ;9 S 1
2; 1;2
=
+!i . 6i7i c"c phng trnh $a-
a)x x x x2 4 6 8
98 96 94 92
+ + + ++ = + b)
x x x x2 2 45 3 8 4 69
13 15 37 9
+ + + ++ = +
".: a9 x 100= 09 x 15= +!i =. 6i7i c"c phng trnh $a-
a)x x x2
2 3 4
2 1 2 1 4 1
=+
b)x x
x xx x2
2 18 2 5
1 32 3
+ =
+ ++
c)x
x x x x
2
3 2
1 2 5 4
1 1 1
+ =
+ +
".: a9 x 9
2= 09 x 1= c9 x 0=
+!i >. !hng ca hai $0 b/ng 9E J>- tPng $0 b# chia 54 &n .# . gi7m $0 chia &i m2t nHa th $0th+ nhMt th- &'c lAn hn $0 th+ hai th- &'c l 94E !m hai $0 ban &@-E
".: 3: vCE+!i ?. *h- .i ca m2t hnh ch` nht b/ng 5:4 m hi- gi`a $0 &% chi1- di . chi1- r2ng l 54 mE
!m $0 &% c"c cQnh ca hnh ch` nhtE".: 94 m v :4 mE
+!i @. !h]ng th+ nhMt &ng :4 lTt d@- th]ng th+ hai &ng C lTt d@-E th]ng th+ hai lMy ra m2tl'ng d@- gMp 9 l@n l'ng d@- lMy ra V th]ng th+ nhMtE Wa- & l'ng d@- cSn lQi tr%ng th]ngth+ nhMt gMp &i l'ng d@- cSn lQi tr%ng th]ng th+ haiE U,i &K lMy ra ba% nhi=- lTt d@-?
".: 3< lGt v DC lGt-+!i A. *h- .i b"nh xe lAn ca m2t &@- m"y xe lHa l < m . ca b"nh xe nh, l 3: mE hi xe
chQy tN ga j &>n ga f th b"nh nh, &K lPn nhi1- hn b"nh lAn l :444 .SngE !Tnh O-Kng&Rng jfE
".: 5
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i s 8 Trn S Tng
&@- .Ai .n t0c hn d nh l 54 kmh. &i nHa O-Kng &Rng cSn lQi .Ai .n t0c thMp hnd nh l < kmhnhng t &K &>n &Zng thRi gian &K nhE !Tnh thRi gian t &K d nh &iO-Kng &Rng tr=nE
".: 3gi\E+!i ,,. F2t xe t &i tN U J2i .1 !hanh U%"E Wa- khi &i &'c :9 km th dNng lQi :4 phZtE .1
&>n !hanh U%" &Zng giR &K nh n ph7i &i .Ai .n t0c b/ng 53 l@n .n t0c trAc &E !Tnh
.n t0c lZc &@- bi>t r/ng O-Kng &Rng U J2i v !hanh U%" di 5