Shestakova 11 Algebra d

.. , .. , ..

: / .. , .. , .. .; . .. 2- ., . : -, 2004

1. . 1. . 1.1.01.50 1 1 13 = 1,98 :1,1 + (0,592) = 37 2 4 37 198 10 592 50 18 16 = = = 1; 100 11 1000 37 10 20 100 3 1 21 ) 1 + 0,91 :1, 4 + 1 1,911 1 = 2, 66 :1, 4 + (0,711) = 79 4 5 79 266 10 711 100 19 9 = = 1. = 100 14 1000 79 10 10

) 3 1,52 :1,1 + 1 1,842 1

1.1.02. (1) (1) 1 + 2 + 3 + ... + 11 (1 2 + 3 4 + ... + 9 10 + 11) = = 10 10 2 (2 + 4 + 6 + 8 + 10) 60 = = 6; = 10 10 (1) (1) 3 + 5 + 7 + 9 + ... + 27 (3 5 + 7 9 + ... + 23 25 + 27) ) = = 12 12 2 (5 + 9 + 13 + 17 + 21 + 25) 180 = = = 15. 12 12

)

1.1.03. ) 1 + 1, 44 1,75 :1, 2 + (9,1 8,317) =1,2+0,09=1,29; ) 1 + 1, 21 1, 25 :1,1 + (9, 7 9, 416) = 1, 21:1,1 + 0, 284 = 71 71 4 = 1,1 + 0, 04 = 1,14 . 1.1.04. ) +3 + Q 3 P3 Q3 ( P + Q )( P 2 PQ + Q 2 ) + 2 = + 2 2 P PQ + Q P + PQ + Q ( P 2 PQ + Q 2 )2

3 4

10 10 = 1, 44 :1, 2 + 0, 783 = 87 87 10 10

1

( P Q)( P 2 + PQ + Q 2 ) = ( P + Q) + ( P Q) = 2P = 2 (16 x 2 24 x + 9) = ( P 2 + PQ + Q 2 ) 9 3 3 = 2 16 24 + 9 = 2 (9 18 + 9) = 0, x = 0,75 = ; 4 4 16

)

P3 + Q3 P3 Q3 + = ( P + Q) + ( P Q) = 2 P = P 2 PQ + Q 2 P 2 + PQ + Q 2 25 5 = 2 (16 x 2 + 40 x + 25) = 2 16 + 40 + 25 = 16 4 = 1, 25 = 5 . 4

= 2 (25 50 + 25) = 0, 2

1.1.05. )3 5 x1 3 5 x2 3 5 x1 + 3 5 x2 6 5( x2 + x1 ) 6 5 (2) + = = = = x1 + x2 x2 + x1 x2 + x1 x2 + x1 2 5 + 2 1 5 + 2 x2 10 + 2( x1 + x2 ) 10 + 2 20 50 + = = = = 2,5 , x1 + x2 x2 + x1 x2 + x1 20 20

=8, 1+2=2 ; )

1+2=20 . 1.1.06. )5 2u 5 + 4v 5v 2uv + 5u + 4uv 5(u + v) + 2 uv + = = = u v uv uv 2 u+v 1 5 + 2 = 5 + 2 = 5 + 2 = 4,5 , =5 uv 2 4 5 2 4 u+v= , uv= ; 5 5

5 3 + 5u 3 + 4v 3v + 5uv + 3u + 4uv 3(u + v) + = = + 9 = 3 3 + 9 = ) u v uv uv 4 3 15 21 5 4 = + 9 = = 5, 25 , u+v= , uv= . 4 4 3 3

. 1.1.01. )vu 3 uv3 uv(u 2 v 2 ) uv(u v)(u + v) = = = uv(u + v) = vu vu vu

=(3) 6=18, u+v=6, uv=3 ; )vu 3 uv3 = uv(u + v) = (5) 2 = 10 . v u

1.1.02.u v u 2 + v2 u 2 + v 2 + 2uv (u + v)2 + +4 = +4= +2= +2= v u uv uv uv 25 25 3 = + 2 = + 2 = , u+v=5 uv=11; 11 11 11 u v u 2 + v2 u 2 + v 2 + 2uv (u + v) 2 ) + + 12 = + 2 + 10 = + 10 = + 10 = v u uv uv uv 100 100 50 1 = + 10 = + 10 = =3 , 15 15 15 3

)

u+v=10 uv=15.

3

1.1.03.3 2 2 3 2 2

4 3 48 5 u v u v ( uv ) ( u v ) ( uv ) 4 ) = = = = 25 = , 2 2 12 12 ( )( + ) + 5 u v u v u v u v 5 5 12 4 3 u+v= , uv= ; 5 5

2

10 3 u 3v 2 u 2v3 (uv) 2 = ) = = 2 2 4 u+v u v 3

2

10 9 = 10 = 5 , u+v= 4 uv= 10 . 4 12 6 3 3 3

1.1.04. )Q( x) ( x 2 3) 2 ( x 2 + 3) 2 P ( x) = (x2 3)2 = (x2 + 3)2 (x2 3)2 = P( x) ( x 2 3)2 Q( x) ( x 4 4) 2 ( x 2 2) 2 ( x 2 + 2) 2 P( x) = 4 ( x 4 4 x 2 + 4) = 2 P ( x) x 4x + 4 ( x 2 2) 2

= 2 6x2 = 12x2 = 1,08, =0,3 )

( x 2 2) 2 = ( x 2 + 2) 2 ( x 2 2)2 = 8 x 2 = 8 (0,7) 2 = 3,92 , =0,7. 1.1.05. ) P2(Q(x))Q2(P(x))=(P(Q(x))Q(P(x))(P(Q(x))+Q(P(x)))= = 5Q( x) 1 = x +1 1 P( x) + 1 ( x) + 1 5Q( x) 1 + = 5 5

5 x 5x x + 1 1 + = 0 2 x = 0 , =117,399; 5 5

) P6(Q(x))Q6(P(x))=(5Q(x)1)6

P( x) + 1 6 6 = x x =0, =117,277. 5

6

1.1.06. ) (1+3x+2x2)+(1+4x+2x2)+(1+5x+2x2)++(1+17+22)=152x2+15 20 (3 + 4 + 5 + ... + 17) 20 = 2 = = 5; +(3+4+5++17)x+15, 1+2= 2 15 2 15 4

) (2+3+2)+(2+5+2)+(2+7+2)++(2+27+2)= =132+(3+5+7++27)+132,

13 30 (3 + 5 + 7 + ... + 27) 30 = 2 = = 15. 1+2= 13 13 2

1.1.07.

4t 2 (t 2 + 1) 2 (t 4 + 2t 2 1) 2 (t 2 1) 4 = =1; ) p=(7x23y2)2= = 2 2 (1 t 2 ) 4 (1 t 2 ) 4 (1 t 2 ) 2 (1 t )4

2

4t 2 (t 2 + 1) 2 (t 4 + 2t 2 1) 2 (t 2 1) 4 ) p=(5x26y2)2= = =1. = 2 2 (1 t 2 ) 2 (1 t 2 ) 4 (1 t 2 ) 4 (1 t ) 1.1.08. ) =441222+94=(2232)2= = 2t t 2 + 1 + 1 t 1 t (t + 1) 2 1 t 2 2 2

2

( 2 +

32

)(2

2 3 =2

)

2

2t t 2 + 1 t 2 + 2t + 1 = 1 t 1 t 1 t 2

t 2 2t + 1 = t 1

=

(t 1) 2 (1 + t )4 (t 1)4 = = (t + 1)4 ; t 1 (t 1) 4

) =2546022+364=(5262)2= = 2t t 2 + 1 + 1 t 1 t 2 2

(

5 6 2

)(2

5 + 6 2

)

2

=

2t t 2 + 1 (t 1)2 = 1 t 1 t t 1

(t + 1) 2 4 = (t + 1) . 1 t

1.1.09. ) =49242+92+42181=(73)2+6(73)1= (1)2+6(1)1=6, 73=1; ) =81236+42+92+5=(92)2+(92)+5=32+3+5=17, 92=3. 1.1.10.

) 5uv+2(u2+v2)=2(u2+v2+2uv)+uv=2(u+v)2+uv=2 + = 5 5 =2 23 1 = = 0,92; 25 25 32

1

2

5

) 2uv+3(u2+v2)=3(u2+v2+2uv)4uv=3(u+v)24uv== 1,88. =3 4 = + = 5 25 5 25 5 1.1.11.u v (u v ) 2 25 5 4 = 2 4 = = 6, 25; 4 2 2 4 4 2 2 2 2 ) u v 5 = (u v )(u + v ) 5 = u 2 + v 2 5(u + v)2 2uv 5 = u 2 v2 (u 2 v 2 ) 2 49 5 9 7 5 = 2 5 = = . 16 2 16 4 44 4 2 2 2 2 ) u v 4 = (u v )(u + v ) 4 = u 2 + v 2 4 = (u + v)2 2uv = 2 2 2 2

1

27

4

47

1.1.12.

) =

u v u 2 + v2 (u + v) 2 2uv (u + v)2 + + 12 = + 12 = + 12 = + 10 = v u uv uv uv

(7) 2 5 17

+ 10 =

49 17 49 17 + 850 + 10 = ; 85 85

5

) =

u v u 2 + v2 (u + v)2 2uv (u + v) 2 + +4 = +4= +4= +2= v u uv uv uv 2 6 +2 = 36 6 +2 =3 6+2 . 12

(6)2

. 1.1.01. ) ()=3+62+12+19=(3+62+12+8)+11=(+2)3+11=

= 3 11 +11=11+11=0, =2 3 11 ; ) ()=3+92+27+29=(3+92+27+27)+2=(+3)3+2= = 3 2

( (

)

3

)

3

+2=2+2=0, =3 3 2 .

u+v u+v 1 1 = = = = u 3 + v3 (u + v)(u 2 uv + v 2 ) u 2 uv + v 2 (u + v) 2 3uv 1 1 28 28 = = = = ; 2 25 9 + 175 36 211 5 3 + 3 4 7 2 7 u+v u+v 1 1 = = = ) 3 3 = u +v (u + v)(u 2 uv + v 2 ) u 2 uv + v 2 (u + v) 2 3uv 1 1 20 20 = = = . = 2 81 12 405 + 48 453 9 4 + 3 4 5 2 5

1.1.02. ) 12+7z=2(2x5y+z)(3x+2y5z)=243=83=5, 2+5+z=4 3x+2y 5z=3; ) 6x+5y+11z=2(4x+2y+3z)(2xy5z)=231=5, 2xy5z=1 4x+2y+3z=3. 1.1.03.

)

1.1.04.u 3 v3 (u 3 v3 ) 1 1 = 3 3 3 3 = 3 3 = = 6 6 2 u v (u v )(u + v ) u + v (u + v)(u uv + v 2 ) 1 1 1 1 = = = ; = 40 (u + v)((u + v) 2 3uv) (4) ((4)2 3 2) (4) 10

)

u 3 v3 (u 3 v3 ) 1 1 = = 3 3 = = 6 6 3 3 3 3 2 u v (u v )(u + v ) u + v (u + v)(u uv + v 2 ) 1 1 1 1 = = = = . 44 (u + v)((u + v) 2 3uv) (2) ((2) 2 3 (6)) (2) 22

)

1.1.05.

) u3+v3=(u+v)(u2uv+v2)=(u+v)((u+v)23uv)= 3 = 2 4 2 = 13,75; = = = 2 4 4 2 4 4 5 25 3 5 22 55

2 5 5

1

6

) u3+v3=(u+v)(u2uv+v2)=(u+v)((u+v)23uv)= 3 = 2 4 2 = 11, 25. = + 3 = + = = 2 4 4 2 2 4 4 2 42 3 3

2 3 3

7

7

3 9

21

3 30

45

1.1.C06.

) |uv|= (u v)2 = u 2 + v 2 2uv = (u + v)2 4uv = = 4 = 2 22

5

2

1

25 17 17 2 = = ; 4 4 2

) |uv|= (u v)2 = u 2 + v 2 2uv = (u + v)2 4uv = = 4 = +2 = = . 16 16 4 4 41.1.07. ) u4+v4=(u2+v2)22u2v2=((u+v)22uv)22(uv)2= 3 3 1 49 31 4 = 1 2 2 + 2 2 = 2= =3 ; = 3 9 9 9 3 3 3 ) u4+v4=(u2+v2)22u2v2=((u+v)22uv)22(uv)2=2 2 2 2

3

2

9

41

41

5 5 1 121 71 = 1 2 2 = = 2,84 . + 2 2 = 2 5 = 5 25 25 5 5 1.1.C08.2 2

2

2

11 2 6 u v ( u v )( u + uv + v ) ( u + v ) uv 6 6 ) = = = = 2 2 (u v)(u + v) u+v 11 (u v ) 6 121 2 109 6 = 6 = ; 66 11 6 2 15 5 11 11 u 3 v3 (u v)(u 2 + uv + v 2 ) (u + v) 2 uv 11 = = = ) 2 2 = 15 u+v (u v)(u + v) u v 11 225 +5 280 11 56 11 = 11 = = . 15 165 33 113 3 2 2 2

2

1.1.09. ) (23)+(23)=232+232=3(2+2+2)+ +10=3(+)2+10=3121105=413;

7

) (5+2)+(5+2)=5+22+5+22=2(2+22)+14= =2()2+14=281+14(12)=6. 1.1.10. ) (3+2)2+(3+2)2=(9+12+42)+(9+12+42)=9(+)+24+ +4(+)=9(5)+245+45(5)=25; ) (43)2+(43)2=(1624+92)+(1624+92)=16(+)48+ +9(+)=167489+997=247 1.1.11. ) (532)2+(532)2=(25302+94)+(25302+94)=25(+) 30(+)+9(3+3)=25(+)30(+)+9(+)((+)23)= =25330(2)3+9(2)3(9+6)=555; ) (322)2+(322)2=(9122+44)+(9122+44)=9(+) 12(+)+4(3+3)=9(+)12(+)+4(+)((+)23)= =941224+424(166)=260. 1.1.12. ) ()=52()+4()q(x)q2 (x)=(5p(x)q(x))(p(x)+q(x))=2 = +

5 6

5 6

=(236)(+7)=(6)(+6)(+7), 1+2+3=6+(6)+(7)=7; ) ()=82()+7()q(x)q2 (x)=(8p(x)q(x))(p(x)+q(x))=2 = +

145 2 5 71 2 29 2 5 71 + + + + = 6 6 6 6 6 6 6 6 6 6

8

=(216)(+3)=(4)(+4)(+3), 1+2+3=4+(4)+(3)=3. D. 1.1.D01. ) ()=42 ()+3()q(x)q2(x)=(4p(x)q(x))(p(x)+q(x))=2 = +

9

8 9

104 2 8 40 2 13 2 8 40 + + + 9 + 9 = 9 9 9 9 9 9 9 9

4 5

4 5

108 2 4 17 2 27 2 4 17 + + + + = 5 5 5 5 5 5 5 5 5 5

2 2 2 + 2 + 3 =25+25+4=54; =(225)(2)=(+5)(5)(2), 1 ) ()=22()()q(x)q2 (x)=(2p(x)+q(x))(p(x)q(x))= 2 2 2 16 2 2 13 2 8 2 2 13 = + + + = + +

32

3

3

3

3

3 3

3

3

3

3

+ + =( 1)(7)=(1)(+1)(7), 1.1.D02. ) ()=82 ()7()q(x)q2(x)=(8p(x)+q(x))(p(x)q(x))= 8 2 8 136 2 8 8 2 17 2 8 8 = + + + + + = 9 9 9 9 9 9 9 9 9 9 9 9 =(216)(1)=(4)(+4)(1), 123=4(4)1=16; ) ()=32()2()q(x)q2(x)=(3p(x)+q(x))(p(x)q(x))=3 2 3 39 3 25 13 3 25 = + + + + + = 2 2 2

2 1

2 2

2 3 =1+1+49=51.

3

=(216)(+3)=(4)(+4)(+3), 123=4(4)(3)=48. 8

4

4

4

4

4

4 4

4

4

4

4

4

1.1.D03. ) ()=122()11()q(x)q2 (x)=(12p(x)+q(x))(p(x)q(x))=

=

12 13

2 +

=(29)(+6)=(3)(+3)(+6),

12 36 2 12 81 2 3 2 12 81 + + + + = 13 13 13 13 13 13 13 13 13 13 13

2 2 2 1 2 3 =32(3)2(6)2=542=2916; 2 ) ()=10 ()+9()q(x)q2(x)=(10p(x)q(x))(p(x)+q(x))=

=

10

11

2 +

10 410 2 10 14 2 41 2 10 14 + + + + = 11 11 11 11 11 11 11 11 11 11 11

2 2 2 2 3 =62(6)252=(180)2=32400. =(236)(5), 1 1.1.D04. ) 2()+(7)=+4, 2(7)+(7(7))=7+4, 2(7)+()=11, 3()=2(+4)(11)=33 ()=1; ) 3()+(8)=+5, 3(8)+(8(8))=(8)+5,

3(8)+()=13, 8()=3(+5)(13)=4+2, ()= +1.1.D05. ) ()=2 ()9()q(x)10q2(x)=(p(x)+q(x))(p(x)10q(x))=

2

1 . 4

=

46 11

2

39 26 22 6 15 462 39 26 202 60 150 + + + = 11 11 11 11 11 11 11 11 11 11 11 2 2

2 2 2 2 =(4231)(62916), 1 + 2 + 3 + 4 =

=(1+2)2212+(3+4) 2234= 2 + 2 = 4 4 6 6 =9 1 9 16 53 16 415 + + + = + = ; 16 2 4 3 16 3 48

3

1 9

16

) ()=2()+5()q(x)6q2(x)=(p(x)+6q(x))(p(x)q(x))= = 23 2 12 34 122 30 78 + + 7 7 7 7 7 7 23 2 12 34 2 2 5 13 = 7 7 + 7 + 7 + 7 7 2 2

2 2 2 2 =(526+16)(32+3), 1 + 2 + 3 + 4 =

=(1+2)2212+(3+4)2234= 2 + 5 5 3+ = =9 . 2 = + + + 2 = 25 9 225 225 3 25 5 9 1.1.D06. ) ()=2()3()q(x)4q2 (x)=(p(x)4q(x))(p(x)+q(x))= 3 36 32 1 196 19 2239 214

6

16 1

=

(72+2+12)(2+3+1), 1234=

11 2 14 16 242 4 44 112 14 16 62 11 + + + + + + + = 5 5 5 5 5 5 5 5 5 5 5 5

9

=(12)(34)=

5 12 1 12 =1 ; = 7 1 7 7 2

) ()=2()5()q(x)6q2(x)=(p(x)+q(x))(p(x)6q(x))=13 2 13 33 6 2 13 13 33 6 36 12 = + + + + + = 7 7 7 7 7 2 2

=(22+5)(27+3), 1234= =(12)(34)= = = 7,5. 2 1 2 1.1.D07. ) ()=2 ()7()q(x)8q2(x)=(p(x)+q(x))(p(x)8q(x))= = 31 5 3 15

7

7

7

7

7

7

7

4 26 5 2 5 1 31 2 4 26 40 2 40 8 + + = 2 + 9 9 9 9 9 9 9 9 9 9 9 9

2 2 2 =(423)(2+42), 12 2 3 4 =(12)2 (34)2=

= (2) 2 = 4 = = 2, 25; 16 4 4 ) ()=2()+7()q(x)8q2(x)=(p(x)+8q(x))(p(x)q(x))= = 14 9 2 + 31 34 32 2 32 16 142 31 34 42 4 2 + + = + + 9 9 9 9 9 9 9 9 9 9 9

3

2

9

9

2 2 2 2 2 3 4 =(22+72)(22+34), 1 =(12)2(34)2=12(2)2=4. 1.1.D08. ) 9212+4212+84=(32)24(32)4=((32)2 4(32)+4)8=(322)288, (322)20 ; ) 42+12+9212183=(2+3)26(2+3)3=((2+3)26(2+3)+ +9)12=(2+33)21212, (2+33)20 . 1.1.D09. ) 22+92+10+2=()2+82+10+2=()2+10()+82+

+112=(+5)2+82+1127=(+5)2+8 +11 2 + 0 (+5) 0 ; 16 2

11 25 25 30 30 , 16 32 32

2

) 24+6212+23=(2)2+22 12x+23=(2)212(2)+11 1 1 +22223=(26)2+222239=(26)2+2 99 99 , 11 0 (26)0 . 2 2 2 2

2

2

2

2

1.1.D10. ) 2+2=22+2+2=()2+2=1+2=1+2(+1)=22+2+1= 2 2 2 2

1 1 1 1 , =1 =2 ; + 0 + +

10

) 2+2=(+)22=42=42(2)=224+4=2(1)2+22, +=2 (1)20 . 1.1.D11. ) f(x)=40, 32+16b+8c+4d+2k+m=40, m=0 ( ) 16+8b+4c+2d+k=20, k=0 ( ). 8a+4b+2c+d=10, d=0 ( ). 4a+2b+c=5 ( c=1 ). 4a+2b=5c=4, 2a+b=2, b=0 a=1. , =1, b=0, c=1, d=0, k=0, m=0; ) f(2)=42, 32+16b+8c+4d+2k+m=42, 2(16a+8b+4c+2d+k)+m=221, m=0. 16a+8b+4c+2d+k=21, 2(8a+4b+2c+d)+k=210+1, k=1. 8a+4b+2c+d=10, d=0. 4+2b+c=5, 4a+2b+c=22+1, c=1. 2a+b=2, b=0 =1. , =1, b=0, c=1, d=0, k=1, m=0 1.1.D12. ) f(3)=325, 243+81b+27c+9d+3k+m=325, 3(81a+27b+9c+ +3d+k)+m=3(108)+1, m=1. 81a+27b+9c+3d+k=108, 3(27a+9b+3c+d)+k=336, k=0. 27a+9b+3c+d=36, 3(9a+3b+c)+d=36=312, d=0. 9a+3b+c=12, 3(3a + b) + c =34, =0. 3a+b=4, b=1 =1. , =1, b=1, c=0, d=0, k=0, m=1. ) f(3)=257, 243a+81b+27c+9d+3k+m=257, 3(81a+27b+9c+ +3d+k)+m=385+2, m=2. 81a+27b+9c+3d+k=85, 3(27a+9b+3c+d)+k=328+1, k=1. 27a+9b+3c+d=28, 3(9a+3b+c)+d=39+1, d=1. 9a+3b+c=9, b=c=0, a=1. a=1, b=0, c=0, d=1, k=1, m=2. 2. . 1.2.01.2 2 3 2 3 2 2 1 1 10 = = = = = 6; = ; ) 1 2 3 10 1 2 1 3 1 1 1 3 1 1 10 2 2 2 6 6 2 2 2 2 7 ) = = = = = = 3, = . 1 2 12 7 4 2 2 4 4 2 2 2 2 2 2 2 7 2

1.2.02.

) )

a 2 9b 2 c 2 16d 2 (a 3b)(a + 3b)(c 4d )(c + 4d ) (a + 3b)(c + 4d ) = = .; 2 3b a (c 4d )2 (3b a ) c 8cd + 16d c 4d2

2 25b 2 c 2 4d 2 (a 5b)(a + 5b) (c 2d )(c + 2d ) (a 5b)(c + 2d ) = = c 2d (c 2d ) 2 (5b + a ) c 2 4cd + 4d 2 5b + a

11

1.2.03.

) f(4)=(24)1+341= +1

1 2

3 1 1 1 1 = ; f(6)=(26)1+361= + = ; 4 4 4 2 4

f(f(4))=f(f(6))=f = 2 + 3 = + 12 = 12 ; 4 7 7 4 4 ) f(8)=(48)1+81= + = ; f(4)=(4+4) 1+(4) 1= f(f(8))=f(f(4))=f = 4 + = 8 = 7 . 8 33 33 8 8 1.2.04. ) (14)f(f())=(14)f()(12f())1=(1 4 ) (1 4 ) (1 2 ) 1 (1 4 ) (1 4 ) = 1 2 = = = = 0,03; = 2 1 2 2 1 4 1 2 (1 2 ) 1 1 1 2 1 1 4 1 8 1 8 1 8 1 1 = ; 4 8

1

1

1

1

4

4

1

1

1

1

8

25

) (110)f(f(x))=(110x)f(x)(15f(x))1=

(1 10 ) (1 10 ) (1 5 )1 (1 10 ) (1 10 ) = 1 5 = = = = 0,09. = 5 1 5 5 1 10 1 5 (1 5 ) 1 1 1 5

1.2.05.2 2 2 2 2 2 2 2 2 2 = = 2 2 = ) 2 2 1 1 3 3+ 3 1 3 +1 3 2 3+ 2

=

62 + 2 62 + 2 4 4 4 4 = = = = ; (3 2 1)(3 2 + 1) 9 4 1 9 (0,5) 4 1 9 16 1 143 2 2 2 2 2 2 2 2 2 2 + = + = 2 + 2 = ) 2 2 1 1 +1 1 1+ 1 1 2 1+ 2

=

22 + 2 + 22 2 4 2 4 (0, 2) 2 4 25 100 25 = 4 = = = = . 2 2 ( 1)( + 1) 1 (0, 2) 4 1 54 1 624 156

1.2.06. 1 2 1 = 51; ) 1 + 2 1 1 2 1 = ; 1 2 5 + 1

2 1 = ; + 2 5

5 10 = + 2 ;

=3; 12

1 1

=

1 1 = = = ; 1 3 3

1 1 ) 3 = 41; 1 1

1 3 1 = ; 1 1 4 1 1 1

3 1 = ; 4

4 12 = ;

=

11 ; 3

=

1 3 . = = = 1 11 11 3

. 1.2.01.

) =

2 2 a 2 xy b 2 xy 25c 2 x3 2c 2 x xy (a b)(a + b) 25c 2 x3 = = bx ay + by 10c 4 x 4 (a b) x 10 c 4 x 4 (a + b) y

50c 4 x5 y (a b)(a + b) = 5; 10c 4 x5 y (a b)(a + b) 3c 2 x a 2 xy b 2 xy 4cx 4 3c 2 x xy (a b)(a + b) 4cx 4 = = 3 5 ax bx ay + by 6c x x ( a b ) 6c 3 x 5 y ( a + b )

) =

12c3 x6 y (a b)(a + b) = 2. 6c3 x 6 y (a b)(a + b) 2 x 2 b 2 x3 a 2 x + x 2 a 2 2 = bx + ax ab x 1 x 2 + bx2

1.2.02.

) = ) =

x( x 1) ( x b)( x + b)( x 2 a 2 )( x + 1) = ( x a); ( x b)( x + a)( x 1)( x + 1) x( x + b) 3x 2 6 x x 2 b 2 x 3 a 2 x + 2 x 2 2a 2 2 = x + bx ax ab x 4 x 2 bx2

3x( x 2)( x b)( x + b)( x 2 a 2 )( x + 2) = 3( x + a). ( x + b)( x a)( x 2)( x + 2) x( x b)4ab 3a b 4ab 3a(4a + b) 4a + b + + 4 + = = 2 2 a (4a + b)2 (4a + b)2 a 16a + 8ab + b 4a + b 2 2

1.2.03.

) =

4ab + 12a 2 + 3ab (4a + b) 2 a(12a + 7b) 12a + 7b ; = = a (4a + b)2 a2 a2ab a 2b ab a(5a + 2b) 5 + = 2 2 a (5a + 2b)2 (5a + 2b)2 25a + 20ab + 4b 5a + 2b 22

)

ab 5a 2 2ab (5a + 2b)2 a(b 5a) b + 5a 5a + 2b = = . = a (5a + 2b)2 a2 a2 a

1.2.04.

)

1 1 2 1 4a + 1 1 + + + + = 2a(1 4a) 2a 8a 2 16a 2 4a 1 16a 2 1 + 4a 4a 1

2

13

+

4a + 1 8a + 16a 2 4a 4a + 1 2 1 (4a 1) 2 + = 2a(1 4a) 4a(4a 1)(4a + 1) 4a(4a 1)(4a + 1) 4a 1 2

2 + 1 + 4a 1 4a + 1 1 4a 1 4a + 1 + = = = ; = 2a(1 4a ) 4a (4a 1) 4a (4a 1) 4a(1 4a) 4a 4a 1

)

1 1 2 1 1 4a + 5 + + = 2a(4a 5) 8a 2 10a 16a 2 20a 25 16a 2 25 + 20a 4a 5

2

20a + 25 40a + 16a 2 20a 4a + 5 2 1 (4a 5)2 = 4a 5 (4a 5)(4a + 5) 2a(4a 5) 20a(4a 5)(4a + 5) 4a 5 1 4a + 5 10 4a 5 5 4a 1 4a + 51 = = = . = 2a(4a 5) 20a(4a 5) 20a(4a 5) 20a(4a 5) 20a 4a 5 2

1.2.05.1 ) 3ab

ba 1 3a ba 1 ba 1 + 0,51 = : 3ab 1 + : 1 3 3a + b 3 3

=

b 3a 9a 2 b 2 3a b 3a b : + + 2 : 1 : = b 3a b 3a 3a 3a + b 3ab

9a2 + b2 + 6ab 3a b 3a (3a b)(3a + b) 3ab (3a + b) : = 1; : = 2 ab a a b ab 3 3 3 3 (3a b) + a b (3 )

) = =

5ab 1 9

9ba 1 5a 9ba 1 5ab 1 9ba 1 1 : ( 0,5) : + + = 1 + 5 5a 9b 5 5 9

5a 9b 5a 9b 9b 5a : + 2 : 1 + = 9b 5a 9b 5a 5a 5a 9b

25a 2 81b 2 25a 2 + 81b 2 90ab 5a + 9b 5a : : = 45 ab 45 ab 5 a 5 a 9b (5a 9b)(5a + 9b) 45ab (5a 9b) = 1. = 45ab (5a 9b)2 (5a + 9b)

1.2.06.1 ) =5 ; 1

1

1

1 = 5; 1

= 5;

= 5 ,

1 2 2 2 2 = 3 3 2 2 2 2

2 2 22 25 2 2 2 23 2 = 2 = = ; 2 2 3 2 75 2 2 2 73 2 1 = 2; 1 = 2; = 2 ,

) 14

1 1

1

= 21;

1 + 2 + 3 2 2 = 2 2 2 + 3 2 + 2

3 2 + 3 2 4 2 + 3 2 7 2 = 2 = 2 = . 2 3 2 + 3 8 + 3 2 11 2 2 1

1.2.07.4 133 161 + 5 1 7 ) 3 + 9 0,51

3 2 1 4 3

1 = + 3

133 5 7 2 1 + 16 16 3 3 92 4 2

112 1 1 16 3 1 7 4 1 4 = + = + = = 1; 3 92 4 3 7 3 3 3

3 160 91 + 4 1 7 ) 4 + 4 + 0,1251

2

2 3 1 9 2

1

1 = + 4

160 4 7 2 1 + 9 9 2 2 = 4 + 8 9 3

36 1 1 9 4 1 9 = + = + = 2. 4 4 9 4 4

1.2.08.

) 2 2 2 + 2 = 1 2 2 1 2 + 1 122

2

2

2

2

1 2 = 1 2 2

1 2 1 2+ 2

2

2

=

2

2

= (2 2 1) 2 (2 2 + 1) 2 = 8 2 =2 2

=8(0,5)4=816=128;2 ) 2 2 5 2 5+ 2 2 2

2 2 = 5 1 2

2 2 5+ 1 2

=

=

2 2 2 2 5 1 5 + 1

2

2

=

(5 2 1)2 (5 2 + 1) 2 20 2 = = 5 2 = 4 4 4

=5(0,5)4=516=80. 1.2.09.3 + 4 3 4 128 64 27 + 64 (27 64) 9 2 + 12 + 16 9 2 12 + 16 5 + 4 5 4 + 3 3 3 3 2 2 + + 20 + 16 = 125 + 64 + 125 64 = 250 = 125 . 25 20 16 25 ) 3 3 5 + 4 5 4 128 64 125 + 64 (125 64) 25 2 + 20 + 16 25 2 20 + 16 + 2 2 54 3 27 3 27 3 + 64 + 27 3 64 ) 9 + 12 + 16 9 12 + 16 = = ; = 3 3 3 + 4 3 4

15

1.2.10. 2 2 + 7 x 7 y ( )( + ) ( ) 9( p + q) : = = 2 q 2 + q + p 9q + 9 p ( p q)( p + q) + ( p + q) 7( ) ( )( + 1) 9( p + q) 9( + 1) = ; = ( p + q)( p q + 1) 7( ) 7( p q + 1)

)

)

2 2 + + 9 x + 9 y ( + )( + 1) 4(q p) 4( 1) : = = . 2 q 2 q + p 4q 4 p ( p q)( p + q + 1) 9( + ) 9( p + q + 1)12 9 6b a 36b 2 + 12ab + a 2 36b + + 2 : +2+ = 2 : 6b ab(a + b) a + ab a + b b + ab a

1.2.11.

) :

36b 2 + 12ab + a 2 6 6 = = 2; = 6 3 + ab a b b 16 64 a 8a b 2 + 16ab + 64a 2 b ) 2 + 2 : +2+ = : b ab(a b) a ab a b b ab 8a

b2 + 16ab + 64a 2 8 8 2 : = = 2 . =

1.2.12.

8ab

a b

3

3

) 6 m 5n + = =

120mn 6m 5n 60mn + : = 6m 5n 6m 5n 5n + 6m 36m 2 25n 2

36m 2 60mn + 25n 2 + 120mn 6m(6m + 5n) 5n(6m 5n) + 60mn = : 6 m 5n 36m 2 25n 2 (6m + 5n)2 (6m 5n)(6m + 5n) (6m + 5n) 2 (6m 5n)(6m + 5n) = = 6m+5n=4; 2 2 (6m 5n) (36m + 60mn + 25n ) (6m 5n)(6m + 5n)2 160mn 5m 8n 80mn = 25m 2 64n 2

) 5m + 8n : 5m + 8n 5 m + 8n 8n 5m

= =

25m 2 + 64n 2 + 80mn 160mn 5m(5m 8n) + 8n(5m + 8n) 80mn = : 5m + 8n 25m 2 64n 2 (5m 8n) 2 (5m 8n)(5m + 8n) (5m 8n)2 (5m 8n)(5m + 8n) = = 5m8n=3. (5m + 8n)(25m 2 80mn + 64n 2 ) (5m + 8n)(5m 8n) 2

. 1.2.01.3y 1 3y 3y = = 1 xyz + x 3z 3z yz + 1 xyz + x 3z x z+ x 1 y yz + 1 y+ z yz + 1 3y 3y 3y 3y = = = 0; xyz + x 3z yz + 1 xyz + x 3z xyz + x 3z xyz + x 3z

)

1

3

3

16

6y 2 3y 6y = = 2 xyz 2 x + z z yz xyz 2 2x + z x+ z x+ 2 y yz 2 y z 2 ( yz 2) 3y 6y 6y 6y = = 0. = xyz 2 x + z yz 2 xyz 2 x + z xyz 2 x + z xyz 2 x + z

)

1

1

3

1.2.02.

) + + + = z y x z y x z y x z y x = = = =( y 2 z 2 ) 2 ( z 2 x 2 ) 2 ( x 2 + y 2 ) 2 ( y 2 z 2 )( z 2 x 2 )( x 2 y 2 ) + + = xy xz yz z2 y2 x2 z 2 y2 x2 y 4 x2 2 x2 y 2 z 2 + z 4 x2 + z 4 y 2 2 x2 y 2 z 2 + x4 y 2 + x4 z 2 + 2 x2 y 2 z 2 + y 4 z 2 ( xyz ) 2 ( y 2 z 2 )( z 2 x 2 )( x 2 y 2 ) = ( xyz )2 y4x2 + y4z2 + z4 x2 + z4 y2 + x4 y2 + x4z2 2x2 y2z2 ( y4 x2 y4z2 + z4 y2 z4x2 + x4z2 x4 y2 ) = (xyz)2 y 2 z 2 x2 2( y 4 z 2 + z 4 x 2 + x 4 y 2 ) 2 x 2 y 2 z 2 = 2 2 x2 + y2 + z 2 2; ( xyz ) y z2

y

z

2

z

x

2

x

y

2

y

z z

x x

y

) + + + = z y x z y x z y x z y x = = =( y 2 z 2 ) 2 ( z 2 x 2 ) 2 ( x 2 y 2 ) 2 ( y 2 z 2 )( z 2 x 2 )( x 2 + y 2 ) + + = z2 y2 x2 z 2 y2 x2 ( xyz )2x2 y 4 + x2 z 4 2 x2 y 2 z 2 + y 2 z 4 + y 2 x4 2z 2 x2 y 2 + z 2 x4 + z 2 y 4 2 x2 y 2 z 2 ( xyz )2

z

x

2

x

y

2

y

z z

x x

y

y 4 z 2 y 4 x2 z 4 x2 z 4 y 2 + x4 z 2 x4 y 2 + 2 x2 y 2 z 2 ) = ( xyz )2 y2 z 2 z 2 x2 2 y 4 x2 + 2 z 4 x2 + 2 z 4 y 2 + 2 x4 y 2 8x2 y 2 z 2 = 2 8. 2 2 2 z 2 + y 2 + x2 + z 2 x y z ( x + 2a)( x + 2b) ( x + 2b)( x 2c) ( x 2c)( x + 2a) + + = (c + a)(c + b) (a b)(a + c) (b + c)(b a) ( x + 2a)( x + 2b)(a b) + ( x + 2b)( x 2c)(c + b) ( x 2c)( x + 2a)(a + c) = (a + c)(a b)(b + c) (a b)x2 + 2(a2 b2 )x + 4ab(a b) + (c + b)x2 + 2(b2 c2 )x 4bc(c + b) (a + c)x2 (a + c)(a b)(b + c)

1.2.03.

) = =

17

2(a 2 c 2 ) x 4ac(a + c) 4(a 2b ab 2 bc 2 b 2c + a 2c + ac 2 ) = 2 = 4; (a + c)(a b)(b + c) a b ab 2 + a 2c abc + abc cb 2 + c 2 a c 2b ( x 5a)( x + 5b) ( x + 5b)( x 5c) ( x 5c)( x 5a) + + = (c a)(c + b) (a + b)(a c) (b + c)(b + a)(a + b)(c a )(c + b)

)

= ( x 5a)( x + 5b)(a + b) ( x + 5b)( x 5c)(b + c) + ( x 5c)( x 5a)(c a) =2 2 2 2 2 2 2 = x (a + b) + 5x(b a ) 25ab(a + b) x (b + c) 5x(b c ) + 25bc(b + c) + x (c a) (a + b)(c a)(c + b)

=

5 x(c 2 a 2 ) 25ac(c a) ac bc ab = 25 + = (a + b)(c a)(c + b) + + + + ( a b )( c b ) ( a b )( c a ) ( c a )( c b ) 25(ac 2 a 2c + bc 2 + b 2c a 2b ab 2 ) = 25 . (ac 2 a 2c + abc a 2b + bc 2 abc + b 2c ab 2 )

1.2.04.x 2 + y (3x + 11y ) = 5 , 2+3+112=5+102, 22+2=0, xy + 2 y 2 x3 2 xy 2 3 x 2 y + 7 y 3 x3 2 x3 3 x3 + 7 x3 ()2=0, =, = = 3; x3 2 y 3 x3 2 x3 x 2 + y (7 x + 10 y ) = 3 , 2+7+102=3+62, 2+4+42=0, ) xy + 2 y 2

)

(+2)2=0, =2, x3 + 3xy 2 + 3x 2 y 3 y 3 ( + )3 4 3 3 4 3 = = = 1. 3 3 3 3 x + 13 y x + 13 8 3 + 13 3

1 6 1 36 1 1 12 + 36 2 + 2 12 6 2 2 = 36 2 + 2 12(6 ) + = = = 2 3 (6 )2 (6 )3 2 2 (6 ) 2 (6 )3 1 1 6 6 4 2 2 4 2 2 2 = 36 + 1 12(6 1) = (36 + 1)(6 1) 12 (6 1) = (6 2 1) 2 (6 2 1)3 (6 2 1)3 6 4 2 2 3 216 108 + 18 1 (6 1) = = =1; (6 2 1)3 (6 2 1)3 1 ) ()7=1, =1, = . 1 16 1 4 2 + 2 8 + = (4) ( +16 )=8(4) ( +4 )= 2 3 1 1 4 4 2 2 2 3 1 1

1.2.05. ) ()5=1, =1, = 1 , : (6)2(2+362)+12(6)3(161)

18

4 2 4 2 2 2 = 16 + 1 8 (4 + 1) = (16 + 1)(4 1) 8 (4 + 1) = 2 2 2 3 2 36 4 2 2 3 = 64 48 + 12 1 = (4 1) = 1. 2 3 2 3

(4 1)

(4 1)

(4 1)

(4 1)

(4 1)

1.2.06.

)

4 2 + 4 2 = 0,8 ; 42+42=3,222,41,62; 4 2 + 3 + 2 22

7,22+6,4+0,62=0; 362+32+32=0; 36 + 32 + 3 = 0 ;

= 16 148 = 16 2 37 ; 1,2y

x x y , , x > 0 ,

y 1,2

< 0 , ,

. )3 2 3 4 2 = 0, 6 ; 3(22+5+42)=5(32342); 2 2 + 5 + 4 2 2

21282=0; 21 = 8 ;

8 x ; x y , , > 0 , = y 21 1,2x 2 . =2 y 21

, 1.2.07.2

) 2+ 3+

9 3 9 3 =16, + = 2 + 2 + 6 = 22 , + = 22 ; 2

27 3 9 = + 2 3 + 2 = 22(16 3) = 13 22 ; 3 16 4 16 4 = 9 ; + = 2 + 2 + 8 = 9 + 8 = 17 , + = 17 ; 2 2

) 2+ 3+

64 4 16 = + 2 + 2 4 = 17(9 4) = 5 17 . 3

1 7 1 + = 2 + 7 + 6 2 6 2 + 37 + 6 2 2 + 6 2 + 7 6 + + + 6 7 = = ( + 6 )( + )( + 6 ) 6 2 + 37 + 6 2

1.2.08.

)

( + )(6 2 + 37 + 6 2 ( 2 + 7 + 6 2 )(6 + ) = (6 2 + 37 + 6 2 )( 2 + 7 + 6 2 )(6 + ) 63 + 372 + 62 + 62 + 372 + 62 63 422 362 2 72 63 ; = 7 = 7 0 = 0 (62 + 37 + 62 )( + )(6 + )( + 6)

= 7

19

1 5 1 2 + 2 = 2 2 + 5 + 4 + 4 2 + 5 4 + 17 + 4 4 + + + 4 5 5( + ) = = ( + )(4 + )( + 4 ) 4 2 + 17 + 4 2 ( + )(4 2 + 17 + 4 2 )

)

2

5 =0. 4 2 + 17 + 4 2 18 3 + 3 2 27 13

1.2.09.

) = = =

3 2 + 3 + 1 3 2 + 13 1 + = 9 2 + 3 + 1 3 2 +

18 3 + 3 2 (3 2 + )(3 1) 3 2 + + (3 + 1) 2 3 2 13 = 27 3 1 3 2 +

18 3 + 3 2 9 3 + 3 2 + + 9 2 + 6 + 1 3 2 13 = 27 3 1 3 2 + (9 2 + 3 + 1) (9 2 6 + 1) 3 1 = ; 2 (3 + 1) 3 + 1 (3 1)(9 + 3 + 1) 14 3 + 7 2 13 3

) = = =

7 2 + 7 + 1 7 2 + 11 2 1 + = 7 2 + + 1 7 + 7

14 + 7 2 7 ( + 1)( 1) 7 2 + 7 + ( + 1)( + 1) 7 2 11 = 7 ( + 1) ( 1)( 2 + + 1) (14 3 + 7 2 7 3 + 7 ) (7 2 + 7 + 2 + 2 + 1 7 2 11) = 7 ( + 1) ( + 1)( 2 + + 1) 7 ( 2 + + 1) ( 1) 2 1 = . ( 1)( 2 + + 1) 7 ( + 1) + 1

1.2.10. ) 16x2+9x2+3=(4x3x1)2+24+3=62+27=63.; ) 252+29=(5+1)2+1=25+1=26. 1.2.11.3 6 2 40 ( 2 6 40) ( + 4)( 10) = = = (| + 4 | +10) + 40 (| + 4 | +10) + 40 | + 4 | +10 + 40 x, x < 4 = x( x 10) ; x + 10 , x 4 24 10 (16)(30) 240 480 40 40 = = 20 = ; d(20)d(20)= 24 + 10 + 2 16 + 10 2 36 24 6 3

) d(x)=

) d(x)=

3 + 2 56 ( 7)( + 8) = (| + 8 | +7) + 56 | + 8 | +7( + 8)

20

d(14)d(14)=1.2.12.

14 (7) (22) (14)(21)(6) 14 28 = 14 = . 14 22 + 7 22 (14) 6 + 7 (6) 3 31

) ()2=

3 +

= 3 ; ()2=3 1

+ = 3 ; +=9, 31

(1+1)(33)1(33)= + =

1 1 1 ( 3 3 ) = 3 3

( + ) 3 3 3 3 ( 3 3 ) = ()2(+)=39=27; 7 1

) ()3=

= 1 ; ()3=1;

= 1 ;=7, 71

1 1 1 1 4 4 (11)(44)1(44)= 4 4 ( ) =

=

( ) 4 ( 4 4 ) = ()()3=7. 4

4 4

D. 1.2.D01.

) f(x)=

3 2 8 9 3 8 2 9 2 + + ; f(x)= = +2+4++3=2+3+7, 2 3 2 3 2 3

(; 2]. f(x) = x2 + 3x + 7 x (; 2], , min f(x) = f(2) = 5, , [5; +) 2.3 2 27 1 + . 3 1 3 1 3 27 2 1 + = 2+3+9++1=2+4+10, x (; 3]. f(x)= 3 1

) f(x)=

f(x) = x2 + 4x + 10 x (; 3], , min f(x) = f(3)=7, , [7; ) 5. 1.2.D02. ) (2+2)1= = + 2 2 1

=

2 2 ( )2 = = 3 3 + ( + )( 2 + 2 )

( ) 2 1 1 = = ; ( + )(( + )2 3 ) 4(16 + 3) 76

) (22)2=

2 2

2

3 3 = 2 2

2

=

( )4 = ( 3 )23

21

=

( ) 4 ( ) 4 1 1 . = = = 2 2 2 2 2 2 196 (( )( + + )) (( )(( ) + 3 )) (2 (4 + 3)) 7 + 7 , 234()3=7+7; 27=7+7; 7=7; ( ) 3 2 + 2 + 2 2 2 + 2 2 + 2 2 5 1 = = =2 ; 2 2 3 + 4 2 2 3 2 + 4 2 2

1.2.D03.

) 234=

=, ) 24=

5 + 5 ; 24()1=5+5; 25=5+5; 5=5; =, ( ) 1

2 + 4 + 2 2 2 + 4 2 + 2 2 7 3 = = =1 . 4 2 2 + 3 2 2 2 2 + 3 2 4

1.2.D04.

) 1+1=

26 26 26 ; ; +1 = 0 ; + = 5 5 5

2

1 13 144 , =5 = , .. =5 =5. = 5 25 5 1,2

:

3 2 2 4 2 75 2 10 2 4 2 61 = = 94 4 2 2 100 2 5 2 2

3 2 2 4 2 3 2 10 2 100 2 107 22 11 = = =3 =3 ; 26 26 13 4 2 2 4 2 5 2 25 2

) 1+1=

5 5 ; + = ; 2 5 +2=0; 2 2

2

1 53 , =2 = , =2 =2; = 4 2 1,2

:

5 2 + 4 3 2 20 2 + 8 2 3 2 25 12 = = =1 13 13 2 2 + + 3 2 8 2 + 2 2 + 32

5 2 + 4 3 2 5 2 + 8 2 12 2 1 = 2 = . 2 2 2 2 16 2 + + 3 2 + 2 + 12

1.2.D05.

) 151=4 ; 1 51 =4; 5 =4; =1; =, 3

2

2

2

3 2 + 4 + 2 2 3 2 + 4 2 + 2 2 9 3 1 = 2 = = =1 ; 6 2 2 2 + + 4 2 + 2 + 42

22

) 1+41=5 , 1

2

2 2 +41 2 =5; +4 =5, 2

3

4 2 2 42 2 2 2 1 = 2 = = . =1; =, 2 2 2 2 6 3 3 + + 2 3 + + 2

1.2.D06.

) f(x)=

2 + 10 + 61 ( + 5)2 + 36 36 = = ( + 5) + . ( + 5) +5 +5

f(x)=, (+5)+

36 = , ( + 5)

(+5)2(+5)+36=0. , 0, 24360, 2144, ||12. |f(x)| 12; .. f(x) (; 12] [12; + ), , 5. ) f(x)= 2 4 + 29 ( 2)2 + 25 25 = = ( 2) + . 2 2 2 25 f(x)=, (2)+ =, (2)2(2)+25=0. 2

, 0, 24250, 2100, ||10. |f(x)| 10, .. f(x) (; 10] [10; +), , 7. 1.2.D07. ) 1+1=2, + = 2 ; 2 + 2 1 = = ; + 2 +1=0, =1, =. 4 3 4 + 3 7 2

) 1+1=2; + = 2 ; 2 +1=0; =1, =, 1.2.D08.5 + 3 5 + 3 = = 8 . 3 4 3 4 2

2

) 1211=4; 21 = 4 ; + 4 21=0; =7 ( (;) ). =7 + 2 7 + 2 5 = = ; 2 + 3 14 + 3 11

) 1401=3; 40 = 3 ; 3 40=0; 23

2

=5 ( (;) ). =5 1.2.D09.3 15 16 . = = 4 3 20 3 23 2

) 1241=2; 24 = 2 ; 2 24=0; =6 ( (;) ). =6 + 6 + 7 1 = = = ; 3 4 18 4 14 2

) 1401=3; 40 = 3 ; 3 40=0; =8 ( (;) ). 2 8 2 6 =8 = = . 2 3 16 3 13

2

1.2.D10.

) 1+121=7; + 7 +12=0; =3 =4. =3 =4 3 + 2 9 + 2 7 3 = = = 1 3 4 4

2

3 + 2 12 + 2 10 = = = 2; 4 5

) 1+61=5; + 5 +6=0; =2 =3. + 3 2 + 3 1 + 3 3 + 3 = = = = 0. 2 5 6 5 2 5 4 5 9 2 + + 5 2 1.2.D11. ) = . 2++52=24+42; ( 2 )2

2

=2 =3

2(1)(4+)+4252=0. , 0: =(4+)24(1)(4252)=1622+82+21622+202+162 202=2 (4419)0 19 . 44

2 + + 5 2 19 ; , 44 ( 2 )2

4. 24

)

2 + + 4 2 = , 2++42=()2; ( )2

2(1)(2+)+242=0. , 0. =2(2+1)242(4)(1)=2(42 + 1 + 4 42+16 +4a16) = = y2(24a 15) 0 a , , 1.1.2.D12.( + 2)3 ( 1) 2 8 1 ( + 2)3 8 ( 1) 2 1 + = + = 2 2 2 3 + 6 2 + 12 2 2 + = 2+6+12+=2+7+12=(+3)(+4). = 2 5 8

) f(x)=

f(x) [3;+). f(x)f(3)=42, , 22.( + 3)3 ( + 1)2 27 1 ( + 3)3 27 ( + 1) 2 1 + = = +2 +2 +2 3 2 2 + 9 + 27 + 2 = = 2+9+27=2+8+27. f(x)f(5)=92 ( +2

) f(x)=

f(x) [5;+ )). , 48. 3. . 1.3.01. 1 3 ) 1 9 9 1 9 4 1 9 2 9 1 11 4 2 4 1 2 3 9 9 9 4 2 = = = = = = 1

1 1 = = 4 = 2; 0, 25

1 a4 ) 1 a16

16

6 1 1 1 a 6 = =5. = 4 = a ,2 0 a 1 1 1 + + 2 10 30 19 30

1

1.3.02.

)

=3

5

=

30 30 = 5 19 = 51 = 0, 2; 14 14 1 = 5 9 = 51 = = 0, 2. 5 9

19

)

=7

4

1 1 1 + + 2 8 56

=

9 14

25

1.3.03.

) = =

9 3

27 ( 3 )( + 3 ) = 9 3 = +3 = + 3 + 9 +3 =1

( 3 )( + 3 + 9 ) ( 3 )( + 3 ) +3 15 1 + 3 250 10 =

= =

+ 6 + 9 ( + 3 + 9 )

3 +3

3 25 10 + 3 250

15 10 15 10 = ; 1 + 150 151

)

4 +2

+ 8 ( 2 )( + 2 ) = 4 +2 = 2

( + 2 )( 2 + 4 ) ( + 2 )( 2 )

2 + 4 2

=

( ==

2

)

2

( 2 + 4 )

2 = 10 2 10 2 = . 1 20 19

=

2 2

=

2 25 21 2 50

=

10 2 1 2 100

1.3.04.

) =

19 9 70 19 9 70 14 + 5= + 2 2 14 5 5 14

((

5 + 14 14 5 5 14 14 5 5 14

)(

)=)

19 9 70 70 14 70 + 5 70 + 70 19 19 + = +0 = ; 2 2 2 14 5 5 14

)

17 5 66 17 5 66 6 + 11 11 6 6 11 11 + 6= + = 2 2 11 6 6 11 11 6 6 11 17 5 66 66 + 6 66 11 66 + 66 17 17 = + = +0 = . 2 2 2 11 6 6 11

)(

1.3.05.

)

1 1 3 5 3+ 5

(

3+ 5 3+ 5 5 + 45 = 2 5 1+ 9 = 32 5

)

( )

(

)

=

2 5 5 4 = 10; 4 1 1 ) 2 3 2+ 3

(

2+ 3 2+ 3 12 75 = 3 43

)

(

4 25 =

)

= 2 3 3 (2 5) = 18. 26

1.3.06. 1 ) 1 4 +9 4 81 2 4

4 9 16 2 81 2 1 4 4 +9 4 4 9 = = 1 1 9 1 = = =2 ; 4 4 4 9

4

=

16 2 81 2 16 2 81 2

= 1 =

1 ) 1 8+9

8

81 2 8

8 9 1 64 2 81 2 8 8 +9 8 8 9 = = 1

8

=

64 2 81 2 64 2 81 2

8 = 1 = 9

=

9 1 =1 . 8 8

. 1.3.01.

)

+6 +5 +1

+ 6 1 + 4 1 +1

=

((

+5

)(

+1

+1 1 +1

))=

(

1 + 6 1 + 5 1 +1

)

2

= +5

)(

1 + 5

1 + 12

= + 5 1 5 = 1; ) +6 +8 +4 2 +6 2 +6 2 +4 =

( ) = )( +4

+6 +8 +4

(

)

2

+6 2 +8

(

+2

+4

2 +4

)((+

2+2

)(

2+4

2 +4

)=

= + 4 2 4 = 2. 1.3.02. ) 18 4 14 + 18 + 4 14 = )21 4 17 + 21 + 4 17 =

(

14 2

)

2

+2

14 + 2

)

2

= 14 2 + 14 + 2 = 2 14;2

(

17 4

)

(

17 + 4

)

=

= 17 4 + 17 + 4 = 2 17. 1.3.03. ) 13 + 4 3 + 13 4 3 =

(2

3 +1 +

)

2

(2

3 1

)

2

= 2 3 + 1 + 2 3 1 = 4 3;

27

)

21 + 4 5 + 21 4 5 = 6 14 2 7 + 142

(2

5 +1 + 2 25 2

)

2

(2

5 1

)

2

= 2 5 + 1 + 2 5 1 = 4 5. 6 2 2+ 2 22 2

1.3.04.

) 6 + 2 12,5 + =

= 6+=

+

6 14 7 2+ 2

( 6 + 5 2 )( 2 + 2 ) + 62+ 2

(

)

= 6+5 2 +22 1 + 2

=

12 + 16 2 + 10 + 6 2 2 1+ 2

(

)

=

( )= 2 (1 + 2 )

= 11 2;

) 5 + 8 4,5 =

5 10 2 5 10

= 5+2 =

8 9 2

5 10 5 2 2

(5 + 12 2 )( 2 2 ) 52( 2 1)

(

)

= 5 + 12 2 = 14

5 2 2 2

== 7 2.

10 + 19 2 24 5 2 2

1.3.05.

(

2 1

)

( 2(

2 1

)= 2 1)

14 2

) 2 =

3

1 2

1 2 1 : 2 1 = : ( 3 ) 2 =

((

))

( 2 1)

( 2 1)

=

1 = 1 = (491 )1 = 49;

) 2 +

5

3 2

3 3 5 4 + 1 : ( + ) 2 = 3 : 4 + 1 2 = 2

((

))

3

=

( 4 + 1) 23 2

=

( 4 + 1)

1 = 1 = (641 )1 = 64.

1.3.06. 1 6 1 2

) 1

+

1 3 2

1 6

2 2 1 3 + 1 3 1 3 = = 1 1 1

64 4

6

2

=1 1 + 3 1 3 = 1 1 + 3 = 3 ;4 10 3 2 5 1 + 5 = 1

) 1

1 10

3 3 1 1 10 5 + 2 = 1 1 10

2

1 =1

2 5

1 + 3 5

2 5

3 5

4 4 = 1 1 5 = 5 .

28

2 1 5 ) 16(2,72,3)(3,3+2,9+2,5)=16 2,7 3 2 4 2 6 3 5 1 5 5 1 + + 6 = 1 = 5; 3,3 6 1 + 0,4 6 3,5 3,1 2,5 ) 1 ( + )( 2,1+1,7)=16 3,5

1.3.07.

(

)

(1

0,4

+

0,8

2,5

) = 1 (1+ ) (1 6 0,4

0,4

+ 0,4

( ))2

6

= 1 1 + 0,4

(

( ) ) = 3

1,2

.

1.3.08.

)

15 +1 4

3 2 + +1

=

( 15)(2 + + 1) ( 3)( + 1 4) ( + 1 4)(2 + + 1)

=

= 2 + + 1 30 15 + 1 + 1 + 4 + 3 + 1 12 = 6( 2 + 1 7) = 6; )2 +1 8 + +1 4 +1 2 + 1 7) 12 ( 4)(3 + 3) ( 12)( 3 + 1) = = 3 +1 3 + 3 ( 3 + 1)(3 + 3) = 3 12 + 3 4 3 3 + 12 3 + 12 = 2( + 4 3) = 2. 3 3 + 3+ 3+ 3 + 4 3)

4

1.3.09.1 1 1 1

) f(3+x)f(3x)= (3 + ) 6 (3 ) 6 (3 ) 6 (3 + ) 6 = = (3 + ) 6 (6 (3 + )) 6 = f2(3+x); (f(3+x)f(3x))3= (3 + ) 6 (3 ) 6 = (3 + )(3 ) = 9 2 = =9 7 1 7 2 = 9 7 2 = 9 7 1 = 8 6 ;7 1

1

1

2

2

2 3

2

1

2

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

1 ) 4 (4 (2 + 1 ) 4 (2 + 1 ) 4

1 )) 4

1

1

(2 ) 4 (4 (2 )) 4 =2

(2

1 1 = (2 + ) 4 (4 (2 + )) 4 = f2(2+x); 2

(f(2+x)f(2x))2= 1 = 4 2 1 7 2 2

1 1 (2 + ) 2 (2 ) 2 = (2 + )(2 ) = 4 2 = 7 9 1 2 = 4 2 7 = 4 = = 2 . 4 4 4

(

)

29

1.3.10.

) f(6+x)f(6x)= 5 (6 + )3 (12 (6 + ))3 5 (6 )3 (12 (6 ))3 = = 5 (6 + )3 (6 )3 5 (6 + )3 (6 )3 = (f(6+x)f(6x))5=

(

5

(6 + )3 (12 (6 + ))3

) )

2

= f2(6+x);

( (

5

(6 + )6 (6 )6

)

5

36 = (6+x)6(6x)6=(36x2)6=

(

2 = 1; 35

)

6

) f(4+x)f(4x)= 3 (4 + ) 2 (8 (4 + ))2 3 (4 )2 (8 (4 )) 2 = = 3 (4 + ) 2 (4 )2 3 (4 + ) 2 (4 ) 2 = (f(4+x)f(4x))3=1.3.11.3

(

3

(4 + ) 2 (8 (4 + ))24 4

2

= f2(4+x);2 4

(4 + )4 (4 ) 4

) = (4+x) (4x) =(16x ) = 16 ( 15 ) 3

2 4

= 1.

) 11 4 7 11 + 4 7 = ( 7 2) 2 ( 7 + 2) 2 = = 7 2 ( 7 + 2) = 4; (4)216=1616=0, , x2 16 = 0; ) 17 12 2 17 + 12 2 = (3 2 2) 2 (3 + 2 2)2 = = 3 2 2 3 2 2 = 4 2; (4 2 )232=3232=0, , x2 32 = 0.1.3.12.0,5

)

3

3 2 +3 +2 1 1 4 + 0,5 3 = 0,5 0,5 3 + 2 3 2 3 +2 3 2

(

)(

)

6 9 4 9 4 = 2 = 2 16 = 8; = 4 3 9 3

)

0,5

2

2 3 2 3 1 1 9 0,5 2 = 0,5 0,5 2 +3 2 3 2 + 3 2 3

(

)(

)

4 9 6 4 9 = 3 = 3 81 = 27. = 2 2 4 9

. 1.3.01.82 7 161 72 5 7 1 94 5 8+ 2 7 161 + 72 5 = ( 7 1)2 (9 4 5)2

) =

=

( 7 + 1)2 (9 + 4 5) 2

=

7 +1 9+4 5

( 7 1)(9 + 4 5) ( 7 + 1)(9 4 5) = 81 16 5

30

= 9 7 9 + 4 35 4 5 9 7 9 + 4 35 + 4 5 = 8 35 18 ; ) =12 2 11 17 12 2 12 + 2 11 17 + 12 2 = ( 11 1)2 (3 2 2) 2 ( 11 + 1) 2 (3 + 2 2) 2 =

11 1 3 2 2

11 + 1 3+ 2 2

=

( 11 1)(3 + 2 2) ( 11 + 1)(3 2 2) = 9 42

= 3 11 3 + 2 22 2 2 3 11 3 + 2 22 + 2 2 = 4 22 6.1.3.02. b b 2ba a 2ab b b 2a b = ( a b) ( a b) = ) 2b 1 1 2b a 2a b a ab b + ab 2ab 2ab a b a b 2ab( a b ) = ab ; = 2 ab ( b a ) + b + b 10b + b 10a b ) 1 1 a + ab b + ab + 10ab 10ab 1 1 1 1

10ab = a+ 10b a+

a 10ab + b a+ a 10a + b a+

b 10ab( a + b ) b = = ab . 10 ab ( b + a ) b b

1.3.03.

) (3 )1 3 33 9 + 27 = (3 )1 2 ( 3) 9( 3) = = (3 ) 1 ( 3)( 2 9) = (3 ) 1 ( 3)2 ( + 3) = =3 + 3 = + 3 , >3; 3

) (4 ) 1 3 9 3 + 24 16 = (4 ) 1 ( 1)( 4) 2 = = (4 ) 1 ( 4) 1 = 1 , >4.1.3.04.

) 16 2 8 + 1 2 4 + 4 = (4 1)2 ( 2)2 = |41||2|= =14(2)=13, x0.1

, ()=1 ( + 2) 4 = 2, 1.3.D09. ) ( + 52 + 20)

= 2 + 24 = 14.

3 3 3 + + ... + = 17 + 14 + 49 + + 52 20 + 17 3( 20 17) 3( 17 14) 3( + 49 + 52) = ( + 52 + 20) + + ... + = 20 +17 + + 49 52 17 14 = ( + 52 + 20) ( 17 20 + 14 17 + ... +

35

+ + 52 + 49) = ( + 52 + 20)( + 52 20) =

= ( + 52 + 20) = 72; ) ( + 51 + 23) 2 23 + 21 + 2 17 + 14 + ... + = + 49 + + 51 2

2( 21 23 2( 19 21 2( + 51 + 49 = ( + 51 + 23) = 21 + 23 + 19 + 21 + ... + + 51 49 =

( + 51 + 23) ( 21 23 + 19 21 + ... + + + 51 + 49) = ( + 51 + 23)( + 51 23) =

= ( + 51 + 23) = 74.1.3.D10. ) 2a 4b a a: +4 b = 2 8ab + 16b 2 + 2 a b 2a ab ab 1 8 8a b 1

= (a 4b) 2 + : =| a 4b | + = 2 a b 8 a (2 a b ) =|3,7818,48|+ =14,7+ =14,825; ) a 5b a a: +5 b = 9 2 6ab + b 2 + + + a b a ab ab 5a b 1 1 8

: = (3a b)2 + =| 3a b | = a+ b 5 a + ab

=|3,34,62|

1 =1,320,2=1,12. 5

1.3.D11. ) + 6 9 6 9 = ( 9 + 3)2 ( 9 3)2 =

= 9 + 3 | 9 3 |= 9 + 3 (3 9) = 2 9 , 9 0 2 x + 5x 6 = y 2.3.D07. ) ; 2 ; 2 2 2 ; 2 x + x = y 5 6 y + 5 y 6 + (5 x 6) = y y + 5y 6 = x y 2 + 5 y 6 = x2 x2 + 5x 6 = y 2 x > 0 294 6 y > 0 49 x = 5 x = 5 ; ; ; 12 49 y = 294 y = 6 x + y = 5 5 5 37 x 12 y = 30

x 0 x 0 y 0 y 0 ; x2 y2 + 4x 7 = 0 ; 2 2 x 4 x 7 y + = ( x 2 + 4 x 7) + 4 y 7 = x 2 x + y = 14 4 14 30 14 (xy)(x+y)+4x7= (xy)+4x7= x y7=0; 4 4 4 308 7 14 44 x = x= x + y = 4 ; 4 . ; 4 308 7 y= 30 x 14 y = 28 44 y = 4 4 x2 + 4 x 7 = y ) ; 2 y + 4y 7 = x

2.3.D08. )

| x 3 |= 3 y + 2 | y + 2 |= 3 x 3

( y + 2) = 3 3 y + 2 y = 7 y + 2 = 0 y + 2 = 9 y = 2 ; ; x 3 = 0 x 3 = 9 x = 3 x = 12 | x 1 |= 5 y 2 | y 2 |= 5 x 1

;

( x 3) = 3 y + 2

;

( x 3) = 3 y + 2

4 ( y + 2) = 729( y + 2)

;

)

( y 2) = 5 5 y 2 y 2 = 0 y 2 = 25 y = 2 ; x 1 = 0 ( x 1) = 25 x = 3

;

( x 1) = 5 y 2

;

( x 1) = 5 y 2

4 6 ( y 2) = 5 ( y 2)

;

y = 27 . x = 26

2.3.D09. )

3 x 2 + 35 x 11 4 x + 1 5x + 1 x 1

= 0;

3x 2 + 35 x 11 = 4 x 1 ; 5x + 1 x + 1

3x 2 + 35 x 11 = 16 x 2 8 x + 1 4 x 1 0, 5 x + 1 0, x + 1 0 ; 2 5 x + 1 ( x + 1)

90

13x 2 43x + 12 = 0 1 ; x 4 5 x + 1 ( x + 1) 2

4 x = 3 x = 13 1 4 ; x= ; x 4 13 5 x + 1 ( x + 1)2

)

4 x 2 + 40 x 11 = 3x + 2 =0; ; 11x + 9 x 3 11x + 9 x + 3 2 3 3x + 2 0, x + 3 0, 11x + 9 0 x x = 5 x = 5 3 3 2 2 ; ; ; x= . 2 4 x + 40 x 11 = 9 x + 12 x + 4 2 5 x 28 x + 15 = 0 5 x 2 2 11x + 9 ( x + 3) 3 11x + 9 ( x + 3) x + ( x + 3)2 11 9 x + 4 y = 28 2.3.D10. ) ; xy+4( y + x )=0; ( x + y )( x y + 4) = 0 y 4 x = 28 4 x 2 + 40 x 11 3x 2 x = y x = y 4 ; y = 28 + 4 x y = 28 + 4 x 16 x = y = 0 x = y 4 ; 2 0 = 28 + 0 ( y ) 4 y 12 = 0 y = 6 x = 4 ; ; x = 2 y = 36

)

x + 3 y = 37 y 3 x = 37

; (xy)+3( x + y )=0; ( x + y )( x y + 3) = 0

x = y 3 x = y = 0 x = y 3 x = y ; ; y = 37 + 3 x y = 37 + 3 x 0 = 37 + 0 y = 37 + 3 y 9 x = y 3 2 ( y ) 3 y 28 = 0 y = 7 y = 49 ; . x = 4 x = 16

2.3.D11. )

x + 5 y + x 2 25 y 2 36 = 6

x 2 25 y 2 36 = 0 x + 5y = 6 ( x 2 36) x 2 25 y 2 36 = 0

;

x 2 36 = 0 6( x 5 y ) 36 = 0 x = 6 x = 6 ; ; ; y = 0 y = 0 y = 0 x + 5y = 6 2 2 x + 5 y + x 25 y 36 = 6

)

x 2 y + x 2 4 y 2 49 = 7 ( x 2 49) x 2 4 y 2 49 = 0

;

2 2 x 4 y 49 = 0 x 2 y = 7

91

x 2 49 = 0 7( x + 2 y ) 36 = 0 x = 7 x = 7 ; ; . y = 0 2 = 7 x y y = 0 y = 0 x 2 = 7

2.3.D12. )

3 x + 4 y + 4 3 x + 4 y = 5 x+5 + y+3 = 4

;

( 3 x + 4 y ) 2 + 4( 3x + 4 y ) 5 = 0 x+5 + y+3 = 4

;

3x + 4 y = 1 ; x+5 + y +3 = 4

3 x + 4 y = 1 3( x + 5) + 4( y + 3) = 28 ; ; x+5 + y+3 = 4 x+5 + y+3 = 4

x+5 = 4 y+3 ; 2 7 y + 3 24 y + 3 + 20 = 0

(

)

x+5 = 2 + = 3 2 y

x+5 = 4 y+3 x+5 = 4 y +3 ; 10 y+3 = y+3 = 2 7 18 79 x+5 = x= 7 x = 1 49 ; ; 10 y = 1 y = 47 y +3 = 7 49

)

3 x + 2 y + 7 3 x + 2 y = 8 x +1 + y + 2 = 2

;

( 3 x + 2 y ) 2 + 7( 3x + 2 y ) 8 = 0 x +1 + y + 2 = 2

;

3x + 2 y = 1 ; x +1 + y + 2 = 2

3 x + 2 y = 1 3( x + 1) + 2( y + 2) = 8 ; ; x +1 + y + 2 = 2 x +1 + y + 2 = 2 x +1 = 2 y + 2 x +1 = 2 y + 2 2 y+2 = y+2 = 2 5 8 39 x +1 = x= 5 x = 1 25 ; . 2 y = 2 y = 46 y+2 = 5 25

x +1 = 2 y + 2 ; 5 y + 2 12 y + 2 + 4 = 0 x +1 = 0 + = y 2 2

4. . 2.4.01. ) (2cos x+1)(3sin x4)=01 cos x = 2 x = + 2k , k Z ; .. x = + 2k , k Z ; , 3 3 sin x = 4 , .. | sin x | 1 3

) (2sin x+1)(4cos x+5)=01 sin x = 2 x = 3 + 2k , k Z 3 , ; .. x = + 2k , k Z 2 3 5 2 3 cos x = , .. | cos x | 1 4

92

2.4.02. ) 4sin2x12sin x+5=0

t=sin x, 4t212t+5=0, t1 = , t2 =x=

5 2

1 1 ; .. |t|1, sin x = , 2 2

+ 2k , k Z ; 2 3 1 2 3 2

) 4cos2x+4cos x3=0; t=cos x, 4t2+4t3=0, t1 = , t2 = ; .. |t|1, cos x = 1 , x = + 2k , k Z . 2 3 3 , 2 x = + 2k + , x = + k + , k Z ; 2 6 12 2 2 ( cos 2 x sin 2 x ) = 1 ,

2.4.03. ) 2sin2x=2cos2x+ 3 , 2(cos2xsin2x)= 3 ,cos 2 x =

)

2 cos 2 x = 2 sin 2 x + 1 , 1 2 , 2x =

+ 2k , x = + k , k Z . 4 8 1 1 1 2.4.04. ) cos x sin( x) = , sin 2 x = 2 2 2 2 2 1 3 3 sin 2 x = , 2x = + 2k , k Z , x = + k , k Z ; 4 8 2 4 2 cos 2 x =

) sin x cos( x) =

3 1 3 , sin 2 x = , 4 2 4 3 3 3 sin 2 x = , 2x = + 2k , k Z . + 2k , k Z , x = 4 6 2 2 3

2.4.05.

) tg 2 x = 3tg( x) , tg x tg x + 3 = 0 x = k , k Z tg x = 0 , ; tg x = 3 x = 3 + m, m Z

(

)

)

2.4.A06. ) tg2x+3tg x+2=0, tg x=t, t2+3t+2=0, t1=1, t2=2, .. tg x = 2 tg x = 1 x = + k , k Z ; 4 x m m Z arctg( ) , = + 2

tg x = 0 x = k , k Z 3tg 2 ( x) = tg x , tg x( 3tg x 1) = 0 . . , tg x = 1 x = + m, m Z 3 6

) tg2x3tg x4=0, tg x=t, t23t4=0, t1=1, t2=4 tg x = 1 x = + k , k Z . 4 tg x = 4 arctg , = + 4 x m m Z

93

94

. 2.4.B01. ) 3sin2x5sin xcos x+2cos2x=0, 3sin2x3sin xcos x2sin xcos x+2cos2x=0, 3sin x(sin xcos x)2cos x(sin xcos x)=0, (3sin x2cos x)(sin xcos x)=0,2 2 x = arctg + k , k Z 3 sin x = 2 cos x tg x = 3 ; , 3, sin x = cos x tg x = 1 x = + m, m Z 4

) 2sin2x5sin xcos x7cos2x=0, .. cos x=0, : 2tg2x5tg x7=0, tg x=t, 2t25t7=0 2t2+2t7t7=0, 2t(t+1)7(t+1)=0, (2t7)(t+1)=07 7 x = arctg + k , k Z 2t 7 = 0 tg x = 2 . 2 , t + 1 = 0 , tg x = 1 x = + m, m Z 4 cos 4 x = 1 cos 4 x + 1 , =0, 2.4.B02. ) sin x + 4 0 sin x + 4 k 4 x = + 2k , k Z x = + , k Z 4 2 , x + l , l Z x + l , l Z 4 4 x = + k , k Z 4 sin 4 x 1 = 0 sin 4 x 1 =0, ) , cos x 8 0 cos x 8 k 4 x = + 2k , k Z x = + , k Z 8 2 2 , ; x = + k , k Z . 8 x + l , l Z x + + l , l Z 8 2 8 2

2.4.B03. ) 3tg2x5tg x+2=0, 3tg2x3tg x2tg x+2=0, 3tg x(tg x1)2(tg x1)=0 (3tg x2)(tg x1)=02 2 x = arctg + k , k Z 3tg x 2 = 0 tg x = 3 3, tg x 1 = 0 , tg x = 1 x = + l , l Z 4 3 .. ; 4

95

) 2tg2x3tg x+1=0, 2tg2x2tg xtg x+1=0 2tg x(tg x1)(tg x1)=0, (2tg x1)(tg x1)=01 1 x = arctg + k , k Z 2 tg x 1 = 0 tg x = 2 2, tg x 1 = 0 , tg x = 1 x = + l , l Z 4

2.4.B04. ) 2(cos3xsin3x)=1,5(cos xsin x) 2(cos xsin x)(cos2x+cos xsin x+sin2x)=1,5(cos xsin x)

3 . 4

cos x sin x = 0 tg x = 1 x = 4 + k , k Z . , , 1 1 sin 2 x + = 0 sin 2 x = 3 + 2m, m Z 2 2 2 x = 2 3 x = 4 + k , k Z .. ; 2 x = 3 + m, m Z 4 6

(cos xsin x)(2+sin 2x1,5)=0, (cos xsin x) sin 2 x + =0 2

1

) 2(cos3x+sin3x)=2,5(cos x+sin x) 2(cos x+sin x)(cos2xsin xcos x+sin2x)=2,5(cos x+sin x) (cos x+sin x)(2sin 2x2,5)=0 cos x + sin x = 0 tg x = 1 , x = 3 + k , k Z sin 2 x = 1 4 6 2 x = 4 + m, m Z . x = 3 + k , k Z 4 6 2.4.B05. ) tg x = tg ( x + ) 1 < x < 5 ; 2 3 x 2 x = 2 + 3 + k x = 3 + 2k 1 1 2 2 1 ; x + k , k , l ; 1 < + 2k < 5 ; 1 < 2k < 4 . x + k 3 3 3 2 2 x 1 1 + l x + l 3 2 3 2 2 2 2 2 , k = 0; 1; 2; x = ; x = + 2 = 2 ; x = + 4 = 4 . 3 3 3 3 3 x ) tg x = tg ( ) ; 3 < x < 6 ; 6 3

96

2 6k x = 5 + 5 2 6k 2 6k 2 1 0; tx 3 2 x 3 2

1 t

t2-2t-48=0; t1=8, t2=-; 8x-1 8; 8x-1=81; x-1=1; x=2. t1=3, t2=-6; 3x+5 6; 3x+5=31; x+5=1; x=-4. 2.5.C05. ) 9x-24 3 32x33x24 3 3 8 3x 32

48 -2=0; t

18 +3=0; t2+3t-18=0; t

=33-x; 32x-24 3 2

-33-x=0;

-33-x=0; 3x 0 ;8 3

31,5x-3=0; 31,5x=t; t2-

t-3=0; D=

10 2 64 100 =( ); + 12(3 = 3 3 3

8 10 + 3 =3 3 , t = 4 5 =- 1 0 ; t2-

5x

5x

2.5.C06. )

645 3 x = 3 168 + x ; 215-9x= 2 3

4

(8 + x )

; 15-9x=

4 4 8+ x 3 3

31 13 13 x= ; x= . 3 3 31

)

279 5 x = 3 97 + x ; 3

27 15 x 2

=3

14 + 2 x 3

; 81-45x=28+4x; 49x=53; x=

53 . 49

2.5.C07. )1 4x 4 x + 24 4 x (4 x + 3) = x ; =0; (4 x + 3)(4 x + 24) 4 + 3 4 + 24x

42x+24x-24=0; 4x=t>0; t2+2t-24=0; t1=4, t2=-6; 4x 6; 4x=4, x=1. )1 3x = ; 3x+18=32x+43x; 32x+33x-18=0; 3x=t>0; 3x + 4 3x + 18

t1=3, t2=-6; 3x 6; 3x=3, x=1. 2.5.C08. a)1 1 2 1 2 2 x ; 5 + 5 x = 0 ; +5 = 5 5 25 5 25x 2 x x

255-x+5 5 t1= , t2=)1 5

-2=0; 5x

x 2

= t > 0 ; 25t2+5t-2=0; t2+x

t 2 =0; 5 25

2 2 x ; 5 2 - ; 5 2 = 51 ; = 1 ; x=2. 5 5 2

4 1 1 -x 4 2 1 1 2 4 x 2 + 6 = ; 6 + 6 - =0; 6 = t > 0 ; t + t = 0 ; 3 6 4 3 4 3 4 4 5 + 1 16 25 3 3 ; t1= 3 3 = ; t2=- ; 6-x - ; D= + 1 = 2 6 9 9 2 2

x

x

x

6

x 2

= 61 ;

x = 1 ; x=2. 2x y 4 5 = 20 ; 2 x y 1 4 5 = 16

x y 4 5 = 20 2.5.C09. x y 1 ; 16 5 = 16

x 20 4 = y 400 y 5 ; 5 = 16 ; 2 5 20 5 y 1 = 16 52 y

5-y=

16 1 -y -1 20 = ; 5 =5 ; y=1; 4x= = 4 ; x=1. : x=1, y=1. 80 5 5

115

5x 2 y = 10 ) x y 1 ; 25 2 = 25

y 10 2 = x 10 5 ; 5x=5; x=1; 2y= ; y=1. : x=1, y=1. 2x 5 5 10 = 25 x 5 2

2.5.C10. a)

64 2 x 2 y = 12 y 2 y = 12 ; 2 ; x + y = 6 x = 6 y

22y+122y-64=0; 2y=t; t2+12t-64=0; t1=4, t2=-16; 2y 16; 2y=4; y=2; x=6-2=4. : x=4,y=2. ) 35 3x 3 y = 78 y 3 y = 78 2y ; 3 ; 3 -783y-243=0; 3y=t>0; t2-78t-243=0; t1=-3, x y + = 5 x = 5 y x +1 y+2 x +1 x 2 3 = 2 -x x 2 3 = 2 ; ; 2 =3 ; x=0; y=-2. y = x 2 x y = 2

t2=81; 3y=81; y=4; x=5-4=1. : x=1, y=4. 2.5.C11. a)

: x = 0, y = 2. ) 3 x + 3 2 y 3 = 3 3 x + 3 2 x + 2 = 3 ; ; y = x + 5 x y = 5

3x+2=2-x-2; x+2=0; x=-2; y=3. : x = 2, y = 3. 2.5.C12. ) 32x+1=27+533x+32x; 32x(3-1)-533x-27=0; 232x-533x-27=0; 3x=t>0; 2t2-53t-27=0; D=2809+827=(55)2; t= ) 52x+1=25+745x+252x; 352x-745x-25=0; 5x=t>0; D=5476+300=5776=(76)2; t1= D. 2.5.D01 a) 121 13x 121 13 11 13xx 92 2

53 55 53 + 55 1 ; t1= =27, t2=- 3x; 3x=27;x=3. 4 4 2

74 76 1 = 5x; t2=25; 5x=25; x=2. 6 38

9

-13 11x

2

=169 11xx2 92

2

92

-11 13x ; );x2 9

2

8

;

-143 112

x 9

2

=169 112

x 9

2

-143 13102

x 9

11(11 132

x2 9

-13 1192

x2 9

)=13(13 11102

-11 135

9

-13 11x2

=0; 13x5

= 11x

; x2-10=0; x= 10 . ; );2

) 169 8 x 13(13 8 13 88 x 8 x2 2

6

-8 13x2

=64 13xx2 72

62

-13 8 x

x 6

-8 132

x 6

)=8(8 13

x 6

-13 8

x2 6

x2 6

-8 13

x2 6

=0; 8

13

x2 7

= 0 ; x -7=0; x= 7 .2

2

2.5.D02. a) 81+ x 8 81- x = 56 ; 64 81+ x -8 81 x =56;+1

=t >0;

2 64 3t = 56 ; t2+7t-8=0; t1=1, t2=-8 8 x +1 ; t

2

+1

= 80 ; x2-1=0; x= 1 .

116

) 51+ x -5 51 x =20; 25 51+ x -5 51 x =20; 51 x =t>0.2 2 25 5t = 20 ; t2+4t-5=0; t1=1, t2=-5 51 x ; 51 x =50; x2=1; x= 1. t

2

2

2

2

2

2.5.D03. a) 25252 x +3

x +3

6 52x +3

x +3

+ 5 = 0 ; 542 x +3

x +3

6 52

x +3

+5 = 0 ; 3 4

=t>0; t -6t+5=0; t1=5, t2=1; 50

2

=5 ; 2 x + 3 =1;

1

1 3 x+3= ; x1=-2 ; 52 4 4

=5 ; x+3=0; x2=-3. : x1 = 2 , x2 = 3.x2

) 9

x2

-4 3x2

x2

+3=0; 3

=t>0; t2-4t+3=0; t1=3, t2=1; 3

x2

=31;

x 2 =1;

x1=3; 3

=3 ; x-2=0;x2=2. : x1 = 3, x2 = 2.

0

4 y 1 4 y 1 y +3 4 y 1 = 8 x 3 = 2 4 y 1 y + 3 x 2.4.D04. a) y + 3 ; y +3 ; x 3 =x 4 ; ; = 3 4 x 16 = 4 =2 x

16y-4=3y+9; 13y=13; y=1; x4-1=8; x=2. : x = 2, y = 1. x3 y 11 = 16 ) y 3 ; =4 x

3y-2y=-6+11; y=5; x2=4; x=2. : x = 2, y = 5. x+2 x + 2 0,5 y 32 x 4 26 3x + 2 4,5 0,5 x = 3 x + 2 x2 =3 2.5.D05. a) 9 26 3 ; ; y = 9+ x y x = 9 |3-2+x; 32x-4-2630,5x-2,5-3-x+2=0 32x-426 3 3 3 3 26 28 + 2 676 + 4 27 784 28 3 3 3 3 = 54 ; 0>t 31,5x-3; D= = = 2 ; t1= 2 27 27 3 3 23 3 27 1,5x-3 1,5x-3 1,5

3 y 11 3 11 y 3 11 x2 2 = 4 ; x 2 2 = x y 3 ; y = y 3 ; 2 2 y 3 =4 x

31,5x-3-1=0; 31,5x-3=t>0; t2-

26

t-1=0;

3

=

3 3

;3

=3 ; 1,5x-3=1,5; x=3; y=12. : x = 3, y = 12.

)

43x-3+6341,5x-3-1=0; 43x+6341,5x-64=0; 41,5x=t>0; t2+63t-64=0; t1=1, t2=-64 41,5x; 41,5x=40; x=0; y=6. : x = 0, y = 6. 2.5.D06. a) 25 25 49 3 x

16 x 1 + 63 4 x +1 0,5 y = 4 x +1 42 x 2 + 63 40,5 x 2 = 4 x +1 ; ; y = 6 + x y x = 6

3 x

+ 35

3 x

= 49

3 x

. 49

3 x

;

5 + 7

3 x

5 1 = 0 ; 7 3 x

3 x

= t > 0 ; t2+t-1=0; D=1+4=5; 3 x

1 + 5 5 , 0>t2 t1= 2 7

5 ; 7

=

5 1 3 5 1 ; = log 5 ; x= 2 x 2 7

3 log 57

5 1 2

.

117

36

33

33

3

) 49 x 42 x = 3 36 x . 36 x ; 7 x 7 x 7 x 2 3 = 0 ; = t > 0 ; t -t-3=0; D=1+12; 6 6 6

2(x+3)2=0; x=3. ) 34( x + 2) =32( x + 2)22

1 + 13 1 + 13 1 + 13 6 7 x 7 x 1 + 13 3 t1= , 0>t2 ; = ; = log 7 ; x = 3log . 6 6 2 2 2 x 2 6 2 3 2.5.D07. a) 24( x + 3) = 2 4( x + 4)( x + 2) ; 2 2 2 3 1 3 4( x + 3)2 2( x + 4)( x + 2) 2 + 22 = ; 24( x + 3) + 22( x + 3) = 0 ; 2 2 2 2 3 3 2 1 2( x + 3)2 2( x + 3)2 2 = t > 0 ; t + t - =0; t1=1, t=- 2 ; 22( x + 3) = 20 ; 2 2 22 2 4 1 4 3 9( x + 3)( x +1) ; 34( x + 2) + 32( x + 2) = 0 ; 3 3 3 4 4 2 1 2( x + 2)2 = t > 0 ; t + t = 0 ; t1=1, t2= 3 ; 3 3 3

3

3

7

2(x+2)2=0; x+2=0; x=-2. 2.5.D08. a) 8 93 2x2

27 4

x

x +1

=

lg 27 3 3 ; lg 9 2 2

3x

2 x 2

= log9 27 = 1,5 ;

=x

3 ; x-2=1; x=3. 2x +1

) 4 27

9 8

=

lg 64 3 ; lg16 2 x1 2

2 x 3x 3

=

3 ; -x-3=1; x=-4. 2 7 2x 3x 2 = 3 (3x+13 ); 2 31

2.5.D09. a) 4x-13 3

=3

x+

1 2

-7 22 x 1 ; 22x+1

x2 x2 4x 3x 9 2x 16 1 1 2 = 3 3x; = ; 4 2 = 3 2 ; x-2 =0; x=2 . 2 16 9 3 2 3 2 2

) 9x-2 71 2x 3 = 39

x

1 2

=7 1

x+

1 2

4 32 x 1 ; 32x+x3 2

4 2x 2 3 = 7 (7x+ 7 x ); 3 7 3 3 =0; x= . 2 2

7 7

7x ; 9

=7

x

3 2

; x-

2.5.D10. a)

22 x 3 y = 17 y 2 x 3 2 y

= 1

;

22 x 22 x 2 2 x 1 = 17 x 2 3 = 2 + 1 y

;

22x=16; x=3; 3 2 = 8 + 1 ; 118

y = 2 ; y=4. : x = 3, y = 4. 2

) y 52

32 x 5 y = 16 y 3x 5 2

= 2

;

32 x 32 x 4 3x 4 = 16 x 2 5 = 2 + 3 y

; 3x=3; x=1;

y =1; y=2. : x = 1, y = 2. 2 2 2 2 1 2 2.5.D11. a) 9sin x + 72 = 3( )cos x 3 ; 32 sin x 34 cos x + 72 = 0 ; 3

= 2+3;

32 sin x 9 32 sin x + 72 = 0 ; 8 32 sin x = 72 ; 32 sin x = 32 ; sin2x=1; sin x = 1 sin x = 1 ; x= + k , k Z. 2 2 2 1 2 sin 2 x + 12 = 4( )cos x 2 ; 22 sin x 24 cos x + 12 = 0 ; ) 4 2 22 sin 22

2

2

2

2

2

x

8 2sin

2

x

+ 12 = 0 ; 2sin

2

x

= t > 0 ; t2-8t+12=0; t1=6, t2=2;

= 6 ; sin x=log26 , .. log26>1; 2sin x = 2 ; sin2x=1; x= + k , k . 2 7 x 2 7 x 5 = ; 9(7x-27-x)=5(7x+27-x); 2.5.D12. a) x 7 + 2 7 x 9 1 47x=(10+18) 7-x; 7x=7-x+1; 72x-1=1; 2x-1=0; x= . 2 1 5 x 2 5 x 3 x -x x -x x ) x = ; 7(5 -25 )=3(5 +25 ); 45 =205-x; 52x-1=1; x= . 2 5 + 2 5 x 7

sin 2 x

2

6. . 2.6.A01 a) log5(x-3)=2; (x-3)=52; x-3=25; x=28. : x=28. ) log3(x+1)=4; x+1=34; x+1=81; x=80. : x=80. 2.6.A02 a) log4(3x-4)=log4(x+1); 3x-4=x+1; 2x=5; x=2,5. : x=2,5. ) log2(5x+4)=log2(x+5); 5x+4=x+5; 4x=1; x= 2.6.A03 a) log2(x2-2x+8)=4; x2-2x+8=16; x2-2x-8=0; x1=-2, x2=4. : x1=-2, x2=4. ) log4(x2+2x+49)=3; x2+2x+49=43; x2+2x-15=0; x1=3, x2=-5. : x1=3, x2=-5. 2.6.A04. a) log3 ( x + y ) = 4 x + y = 81 ; . x y = 85 x y = 85 x + y = 26 x + y = 64 ; . x y = 60 x y = 601 1 . : x= . 4 4

(2) (1): 2y=-4, y=-2; x=81-y x=83. : x=83, y=-2. ) log 2 ( x + y ) = 6 ; x y = 60

2y=4 y=2; x=64-y x = 64 2 = 62 . : x=62,y=2. 119

2.6.A05. a)

log 6 (3x y ) = 2 ; log18 (6 x + y ) = 1

3x y = 36 . 6 x + y = 18

. : 9x=54; x=6; y=18-6x; y=-18. : x=6,y=-18. ) log 7 (2 x y ) = 2 2 x y = 49 ; ; log14 (7 x + y ) = 1 7 x + y = 14

: 9x=63; x=7; y=14-7x; y=14-77; y=-35. : x=7,y=-35. 2.6.A06. a) log2(5x-73)-2=log23; log2(5x-73)-log24=log23; log2 5 x 73 =log23;5 x 73 = 3 ; 5x-73=12; x=17. : x=17. 4 5 4

) log5(9x-124)-1=log54; log5(9x-124)-log55=log54; log5 9 x 124 =log54; 9x-124=20; 9x=144; x=16. : x=16. . 2.6.B01. a) log7x2+log7x4+log7x5=log7x(x+33); log7x40-log7x(x+33)=0; log7x 40 =0;x + 33

40 = 1 x+33=40; x=7. : x=7. x + 33

) log4x2+log4x4+log4x6=log4x(x+44); log4x48=log4x(x+44); x+44=48; x=4. : x=4. 2.6.B02.log 2 x 3 y = 13 log 2 x = 13 + 3 y ; a) 3log 2 x + y = 1 3(13 + 3 y ) + y = 1

;

39+9y+y=-1; 10y=-40; y=-4; log2x=13-12; log2x=1; x=2. : x=2,y=-4. ) log 6 x 2 y = 3 ; 2log 6 x + y = 1 log 6 x = 3 + 2 y ; 2(3 + 2 y ) + y = 1

6+4y+y=1; 5y=-5; y=-1; log6x=3-2; log6x=1; x=6. : x=6,y=-1.log 2 x + log 2 y = 5 log 2 x y = 5 x y = 32 ; ; ; x 3 y = 20 x 3 y = 20 x = 20 + 3 y 20 28 ; y1=8, (3y-20)y=32; 3y2-20y-32=0; D=400+4332=282; y= 6 4 y2=- - ; xy=32 x=4. 3

2.5.B03. a)

: x=4,y=8.log3 x + log3 y = 3 log3 x y = 3 x y = 27 ; ; ; x y = 6 x y = 6 x y = 6 6 12 ; y1=9, (y-6)y=27 y2-6y-27=0; D=36+427=122; y= 2

)

y2=-3 ; xy=27 x=3. : x=3,y=9. 120

5log 1 x + 3log 2 y = 11 2 ; 2.6.B04. a) 4log 1 x + log 2 y = 13 2 5log 1 x 39 12log 1 x = 11 ; 7 log 12 2 2

5log 1 x + 3(13 4log 1 x) = 11 2 2 ; log 2 y = 13 4log 1 x 2 x = 28 ; log 1 x = 4 ;2

1 x=( )-4= x=16; log2y=-13+44; log2y=3 y=8. : x=16,y=8. 2 3log 1 x log5 y = 13 3log 1 x + 13 = log5 y 2 2 ) ; ; + = x y 2log 3log 5 1 5 2log 1 x + 3(3log 1 x + 13) = 5 2 2 2-4 2 log 1 x + 9 log 1 x + 39 = 5 ; 11log 1 x = 44 ; log 1 x = 4 , x=( 1 ) ;2 22

x=2 =16; log5y=3(-4)+13; log5y=1 y=5. : x=16,y=5.4

2

2

log 1 (8 x 3 y ) = 1 8 x 3 y = 5 5 ; ; 2x + 3y = 4 log 2 (2 x + 3 y ) = 2 9 18 : 10x=9; x= ; 3y=4-2x; 3y=410 10 22 11 11 , y = . : x=0,9, y = . 3y = 10 15 15 log 1 (9 x + 2 y ) = 3 9 x + 2 y = 8 5 ; ; 6x=5; x= ; ) 2 + = x y 3 2 3 6 log3 (3x + 2 y ) = 1

2.6.B05. a)

15 5 1 ; 2y=0,5; y=0,25. : x= ,y= . 6 6 4 2.6.B06. a) log 5 x 3 32 = 5 ; |5x-3|5=25; |5x-3|=2;

2y=3-3x; 2y=3-

5 x = 5 x = 1 5 x 3 = 2 1 1 . : x1= ,x2=1. 5 x 3 = 2 ; 5 5 1 x x = = 5

) log|2x+13|27=3; |2x+13|3=33; |2x+13|=3; 2 x + 13 = 3 2 x = 10 x = 5 2 x + 13 = 3 ; 2 x = 16 x = 8 . : x1 = 5, x2 = 8.

2.6.B07. a) log6(x2-3x+32)=2; x2-3x+32=36; x2-3x-4=0; x1=4, x2=1. : x1=4,x2=1. ) log3(x2+7x+37)=3; x2+7x+37=27; x2+7x+10=0; x1=-5, x2=-2. : x1=-5, x2=-2. 2.6.B08. a) log (3x + 2 x 3) = x ; 3x+2x-3=3x; x=1,5. 1 : x=1,5.3

121

) log 1 (73 x 5 x 7) = 3x ; 73x-5x-7=73x; -5x=7; x= 7 . : x= 7 .7

5

5

2.6.B09. a) log3(x2+5x+5)=log3(x2-x+5); x2+5x+5=x2-x+5; 6x=0; x=0. : x=0. ) log7(x2-3x+3)=log7(x2+x+3); x2-3x+3=x2+x+3; -4x=0; x=0. : x=0. 2.6.B10. a) 2log2 (3 x ) = x + 24 ; 3x2+x-24=0; D=1+4324=172;x=2

1 17 ; x1=-3, x2=2 2 . : x1=-3, x2=2 2 . 3 3 6 2 ) 5log5 (2 x ) = 13x 21 ; 2x2-13x+21=0; D=169-4221=1; x= 13 1 ; x1=3, x2=3,5. : x1=3, x2=3,5. 4 5 x = 25 x = 5 2 5 x = 27 ; . 2.6.B11. a) log 1 | 2 5 x | =-3; |2-5x|=27; = 2 5 x 27 5 x = 29 x = 29 3 5 5

: x1=5, x2= 29 . ) log 1 | 19 5 x |= 2 ; |19-5x|=4; ; 19 5 x = 42

19 5 x = 4

5 x = 15; x = 3 . 5 x = 23; x = 23 5

23 . 5 2.6.B12. a) log 1 ( x 2 6 x + 22) = log 1 (6 x 5) ; x2-6x+22=6x-5;

: x1=3, x2=5

5

x -12x+27=0; D=144-427=36; x=

2

) log ( x 2 9 x + 52) = log (5 x + 4) ; x2-9x+52=5x+4; 1 13 3

12 6 ; x1 = 3, x2=9. : x1=3,x2=9. 2

x -14x+48=0; D=14 -448=4; x= : x1=6,x2=8. 2.6.C1. a) y log 2 ( x + 1) = 64 2

2

2

14 2 ; x1 = 6, x2=8. 2

2-y; 2-2y-162-y+64=0; (2-y-8)2=0; 2-y=8; -y=3; y=-3; log2(x+1)=23; x+1=28; x+1=256; x=255. : x = 255, y = 3. ) y log3 ( x 2) = 25 5

2

y

+ log 2 ( x + 1) = 16

;

6+ y log 2 ( x + 1) = 2 y

2

y y + log 2 ( x + 1) = 16 2 + 64 2 = 16

;

6+ y log2 ( x +1) = 2

; -

5-2y-105-y+25=0; (5-y-5)2=0; -y=1; y=-1; log3(x-2)=5; x-2=35; x=243+2=245. : x = 245, y = 1. 2.6.C02. a) log2,1 16 5 x =log2,1(2x-5);16 5 x =2x-5; 16-5x=4x2-20x+25; 4x2-15x+9=0;

5

y

y y + log3 ( x 2) = 10 5 + 25 5 = 10

;

log3 ( x 2) = 25 5 y

;

122

16 5 x 0 15 9 3 ; x1=3, x2= . ; D=225-169=9 ; x= 8 4 2 x 5 02

1 x3 5 . x 2 1 2

, x2 . : x=3. ) log1,4 18 + 11x =log1,4(2x-9);18 + 11x =2x-9; 11x-18=4x2-36x+81; 4x2-47x+99=0; 47 25 11 18 D=2209-1699=252; x1= = , x2=9; 11x-18 0; x ; 2x-9 0; 8 4 11 1 x 4 . , x1 . 2

: x=9. 2.6.C03. a) 3log82(3x+79)-14log8(3x+79)+16=0; log8(3x+79)=t; 3t2-14t+16=0; D=196-1216=4; t1=8 3 14 2 8 = 2 , t2= ; 3 68

log8(3x+79)=2; 3x+79=64; x=-5; log8(3x+79)= ; 3x+79= 83 = 28, x=256 79 , x = 59. : x = 5, x = 59. 3

) 3log82(5x+89)-16log8(5x+89)+20=0; log8(5x+89)=t; 3t2-16t+20=0; D=256-240=16; t1=16 4 10 = 2 , t2= ; 3 610

log8(5x+89)=2; 5x+89=64; 5x=-25; x=-5; log8(5x+89)=10 210 89 ; 5x= 8 3 -89; x= , x = 187. : x = 5, x = 187. 3 5

2.6.C04. a) lg(x+3)=-lg(2x+5); lg[(x+3)(2x+5)]=0; (x+3)(2x+5)=1; 2x2+11x+14=0; D=121 4214 = 9; x = 11 3 , x1=2, x2= 7 ;2 4 x > 3 x + 3 > 0 ; 5 , x2 . : x = 2. 2 x + 5 > 0 x > 2

) lg(x+8)=-lg(3x+22); (x+8)(3x+22)=1; 3x2+46x+175=0; D=21162100=16; x1,2= 46 4 ; x1 = 7, x2 = 25 ; x+8>0; x>-8; 3x+22>0; x>2.6.C05. a) 2log2x+log8x-log16x=25 1 1 25 ; 2log2x+ log2x- log2x= ; 3 3 4 36 3 22 . , x2 . : x=7. 3

24 + 1 25 log2x= ; log2x=4; x=16. 12 3 17 1 1 17 ) 2log3x+log9x+log27x= ; 2log3x+ log3x+ log3x= ; 2 2 3 2

123

12 + 5 17 log3x= ; log3x=3; x=27. 6 2

2.6.C06. a) log 1 (1 + 3x) = 6 7log7 4 ; log 1 (1 + 3x) = 6 4 ; 1+3x= ; x=6 6

1 6

5 . 18

) log 1 (3 + 2 x) = 8 5log5 4 ; log 1 (3 + 2 x) = 8 4 = 4 ; (3+2x)=22

2

1 11 ; x=- . 4 8

2.6.C07. a) ( x + 3)log x+3 ( x + 2) = 9 ;

x + 3 1 x 2 ; ; x + 3 > 0 x > 3

(x+2)2=9; x+2=3; x=1, x=-5 . : x=1. ) ( x + 2)log x+2 ( x +1) = 16 ; 2

x + 2 1 x 1 ; ; x + 2 > 0 x > 2

(x + 1)2 = 16, x + 1 = 4, x1 = 3, x2 = 5 . : x = 3. 2.6.C08.log5 x + log 2 y 4 = 13 log x + 4log 2 y = 13 a) log x 4 + log y = 1 ; 5 ; 5 1 4log 5 x log 2 y = 1 2

17log5x=17; log5x=1; x=5; 4-log2y=1; log2y=3; y=8. : x = 5, y = 8. ) log 2 x + log 6 y 3 = 7

12-log6y=11; log6y=1; y=6. : x = 16, y = 6. 2.6.C09.

log x + 3log 6 y = 7 ; 2 ; 10log2x=40; log2x=4; x=16; log 2 x3 + log 1 y = 11 3log 2 x log 6 y = 11 6

19 19 x+ > 0 x> 19 5 4 a) ln(x+ )=ln ; ; 4 ; 4 4x 5 x>0 > 0 4x 19 5 2 x+ = ; 4x +19x-5=0; D=361+80=212; 4 4x 19 + 21 1 19 21 1 x1= x2= = 5 . : x = . = 8 4 8 4 14 14 x > 0 x> 14 5 14 3 ; ; ) ln(x- )=ln ; 3 ; x> 3 3x 5 3 x > 0 > 0 3x 14 5 ; 3x2-14x-5=0; D=196+60=162; x = 3 3x 14 + 16 14 16 1 x1= = - . : x=5. = 5 , x2= 6 6 3

2.6.C10. a) log7(x+9)+log7(5x+17)=2; log7(x+9)(5x+17)=log749. 124

x + 9 > 0 2 : ; 17 ; x>-3 ; 5 5 x + 17 > 0 x > 5

x > 9

(x+9)(5x+17)=49; 5x2+62x+104=0; D=3844-2080=1764=422; x1=62 + 42 62 42 = 2 ; x2= = 10, 4 - . 10 10

: x=-2. ) log3(x+4)+log3(5x+8)=2.x + 4 > 0 3 : ; 3 ; x>-1 ; 5 5 x + 8 > 0 x > 1 5 x > 4

log3(x+4)(5x+8)=log39; 5x2+28x+32-9=0; 5x2+28x+23=0; D=784-460=324=182; x1= : x=-1. 2.6.C11. a)

28 18 28 + 18 = 1 . = 4,6 , x2 = 10 10

log 3 3 x + log 3 3 y = 3 x > 0 ; ; x+y=4; x=4-y; log 3 3 (4 y ) y = log 3 3 3 ; y > 0 log 2 ( x + y ) = 2

4y-y2=3; y2-4y+3=0;

y1 = 1 y = 3 2 . x 3 = 1 x2 = 1

: x1=3, y1=1; x2=1, y2=3. ) log 4 2 x + log 4 2 y = 4 x > 0 ; ; x+y=3; y=3-x; log 4 2 x(3 x) = log 4 2 2 ; y > 0 log 3 ( x + y ) = 1 x = 2 x1 = 1 2 . : x1 = 1, y1 = 2; x2 = 2, y2 = 1. y = 2 1 y2 = 117 x> 3 ; 16 x 3

x2-3x+2=0;

2.6.C12. a) log3x+17(3x2+2)=log3x+17110.3x + 17 > 0 : 3x + 17 1 ; 2 3 x + 2 > 0

log3x+17(3x2+2)=log3x+17110; 3x2+2=110; 3x2-108=0; x2-36=0; x= 6; x=-6 . : x=6. ) log3x+8(2x2+3)=log3x+8353 x + 8 > 0 D: 3x + 8 1 ; 2 2 x + 3 > 0 8 x> 3 ; x 7 3

2x2+3=35; 2x2-32=0; x2-16=0; x= 4; x=-4 D. : x=4. D 2.6.D01. a) log8log9log7x+6((7x+6)9+x2-x-56)=0; log9log7x+6((7x+6)9+x2-x-56)=1; log7x+6((7x+6)9+x2-x-56)=9; (7x+6)9=(7x+6)9+x2-x-56; 125

x2-x-56=0; D=1+456=225; x=

1 15 ; x1=8, 2

x2=-7 . : x=8. ) log6log7log3x+14((3x+14)7+x27x30=0 log7log3x+14((3x+14)7+x27x30)=1 log3x+14((3x+14)7+x27x30)=7 2 x 7 x 30 = 0 x = 10, x = 3 1 3x + 14 > 0 1 3x + 14 > 0

: 3; 10. 2.6.D02. a) log2003(2x3+x2+4x-34)=log2003(2x3-x+2); 2x3+x2+4x-34=2x3-x+2; x2+5x-36=0; D=25+436=132; x=5 13 8 ; x1= = 4 , 2 2

x2=-9 . : x=4. ) log2002(2x3+x2-x-48)=log2002(2x3+3x-3); 2x3+x2-x-48=2x3+3x-3; x2-4x-45=0; D=16+445=142; x=4 14 ; x1=9, 2

x2=-5 . : x=9. 2.6.D03. a) log ( x + 3)2 ( x3 9 x 2 10 x) = log x + 3 x3 10 x 2 x + 22 ;log ( x + 3)2 ( x3 9 x 2 10 x) = 2 log ( x + 3)2 x3 10 x 2 x + 22 ; 9 13 ; x1=11, x2=-2 2

x3-9x2-10x=x3-10x2-x+22; x2-9x-22=0; D=81+422=132; x= . : x=11. ) log ( x + 6)2 ( x3 + 3x 2 4 x) = log x + 6 x3 + 2 x 2 7 x + 10 ;log ( x + 6)2 ( x3 + 3x 2 4 x) = 2 log ( x + 6)2 x3 + 2 x 2 7 x + 10 ;

x3+3x2-4x=x3+2x2-7x+10; x2+3x-10=0; D=9+410=49; x=3 7 ; x1=-5 . x2=2. 2 y2 =0 lg x3 ; log ( x 2 + y 2 + 23) = 2 6 y2 =1 ; x3 x 2 + y 2 + 23 = 36

: x=2. 2.6.D04. a)

y = x 3+ 2 ; x2+(x-1)2=13; x2+x2-2x+1=13; 2x2-2x-12=0; 2 2 x + ( x 1) + 23 = 36

x2-x-6=0; D=1+46=25; x=

, .. x-3 0, x2 = 2; y=x-1 y=-3. : x=-2,y=-3. 126

1 5 ; x1=3 2

) x 52 2

y +1 =0 lg log ( x + y + 38) = 3 42 2

; x5

y +1 =1 x 2 + y 2 + 38 = 64

;

y = x 6 ; 2 2 x + ( x 6) = 26

x +x -12x+36=26; 2x -12x+10=0; x2-6x+5=0; D=36-45=16; x= x-5 0; x2=1; y=x-6 y=-5. : x=1, y=-5. 2.6.D05. a) log 2 ( y x) = 4 2 3 = 486x y

2

64 ; x1=5 , .. 2 y x = 4 ; x 4+ x 2 3 = 486

;

2x343x=486; 812x3x=486; 6x=6 x=1; y-x=4 y=5. : x=1, y=5. ) log 3 ( y x) = 2 3 4 = 768 x y

;

y x = 3 ; x (3 + x ) = 768 3 4

3x434x=768; 12x=12 x=1; y=3+x y=4. : x=1, y=4. 2.6.D06. a) log3(2x+89)+log3(x+34)=3+log320; log3(2x+89)(x+34)=log327+log320; log3(2x2+68x+89x+3026)=log3540; 2x2+157x+2486=0; D=(157)2-422486=(69)2; x=157 69 ; 4

x1 = 22, x2 = 56,5 . : x=22. ) log5(2x+81)+log5(x+38)=2+log521; log5(2x+81)(x+38)=log525+log521; log5(2x2+76x+81x+3078)=log5525; 2x2+157x+2553=0; D=(157)2242553=(65)2; x=157 65 ; 4

x1=55,5 , x2=23. : x=23. 2.6.D07. a) log3(5x+1)+log5x+13=log32 (5 x + 1) log3 3 17 17 ; log3(5x+1)+ ; = log3 (5 x + 1) 4 4

17 log3 (5 x + 1) + 1 = 0 . 4 17 t+1=0; 4

log3(5x+1)=t, : t2D=(

17 2 289 4, 25 3, 75 ) -4= -4=3,752; t= ; t1=0,25; t2=4; 4 16 24 1 1 ; 5x+1= 3 4 ; 5x= 4 3 -1; x= 3 ; log3(5x+1)=4; 5x+1=81; 5 4

1

log3(5x+1)=

5x=80; x=16; 5x+1>0; 5x>-1; x>- . : x1=

3 1 ,x2=16. 5 log 4 4 10 10 ) log4(3x+1)+log3x+14= ; log4(3x+1)+ ; = log 4 (3x + 1) 3 3 1 54

127

10 log4(3x+1)+1=0. log4(3x+1)=t, 3 10 8 10 2 100 64 8 2 2 10 -4= =( ) ; t= 3 3 ; t1=3, : t - t+1=0; D=( ) -41= 3 3 9 9 3 2 2 1 1 1 t2= = . log4(3x+1)=3; 3x=63; x=21. log4(3x+1)= ; 3x+1= 3 4 ; 3 2 3 3

log42(3x+1)-

3 4 1 4 1 . : x=21, x= . 3 3 x x 2.6.D08. a) log 7 (3 1) + log 7 (3 2) = log 7 (3x + 23) ;

3x= 3 4 -1; x=

3

log 7 (3x 1)(3x 2) = log 7 (3x + 23) ; 32x-23x-3x+2=3x+23;

32x-43x-21=0. 3x=t, : t2-4t-21=0; D=16+421=102; t=4 10 ; t1=7, 2

t2=-3 . : x=log37. ) log 3 (2 x 1) + log 3 (2 x 3) = log 3 (2 x + 69) ;log 3 (2 x 1)(2 x 3) = log 3 (2 x + 69) ; 22x-32x-2x+3=2x+69;

22x-52x-66=0. 2x=t, : t2-5t-66=0; D=25+466=289; t=2 log 5

t2=11, 2x = 11, x = log211. : x=log211. 2.6.D09. a)x

5 17 ; t1=6 , 2

2 = log3 2 log32 x + log x 3 ; 3 log3 x 4 2 1 4 2 ; log x 3 log x 3 = ; log x 3 log x 3 = log3 2 3 5log x 3 5 3 log3 32 5 3

1 2 1 log x 2 3 log x 3 = 0 ; 3logx23+10logx3+3=0; D=100-36=64; 5 3 5 1 log x 3 = 3 ; x = 27 ; 3 x = 3 log x 3 = 3 2 8 4 8 log 2 log5 x log x 5 = 0 ; ) log x 5 + = log5 2 log8 x + log x 5 ; log x 5 + 5 3 9 log5 8 3 9 1 8 1 1 1 8 1 log x 5 + =0 |logx5; log x 2 5 + log x 5 = 0 ; 3 9 3 log x 5 3 9 3 log 5 = 3logx25+8logx5-3=0; D = 64 + 36 = 100; x 3 ; log x 5 = 3 1 x = 125 . x = 1 3 5

2.6.D10. a) log8(x+6)2+log8(x+4)2= 128

2 ; log3 8

log8(x+6)2+log8(x+4)2=2log83; log8(x+6)2+log8(x+4)2=log89; (x+6)2(x+4)2=9; ((x+6)(x+4))2=32; x2+4x+6x+24=-3 x2+10x+24=3; x2+10x+27=0; D=100-427 0 3 15 x2= e , 5 , , x2 = e15 lnx= 15 ln x 4 > 0 ln x > 4 5

. : x=e. ) log 3 (ln x3 5) + log 3 (ln x5 9) = 0 ; log 3 ((3ln x 5)(5ln x 9)) = 0 ,4 4 4

ln x > 3 ln x 5 > 0 , 5 ln x 9 > 0 ln x >

5 9 3 , ln x > ; (3lnx-5)(5lnx-9)=1; 9 5 5

15ln2x-27lnx-25lnx+45=1; 15ln2x-52lnx+44=0; D=522-41544=2704-2640=82; lnx= : x = e2.52 8 22 ; lnx=2 x1=e2; lnx= . 30 15 log x x 1 2 log x y + 2 log y = 2 ; x 5 x y = 4

1 log x y + 2 log y x = 2 2.6.D12. a) 2 ; 5 x y = 4 1 2 log x y + 2 log x x 2 log x y = 0 ; 2 5 x y = 4

1 2 log x y 2 log x y + 2 = 0 ; 2 5 x x = 4

129

2 (log x y 2) = 0 ; 5 x y = 4

log x y 2 = 0 ; 5 x y = 4

2 y = x ; 2 5 x x = 4

-x+5 x =4; x-5 x +4=0. 2

x =t, :

53 t -5t+4=0; D=25-44=9; t= ; t1=4, x=16, y=256; 2

t2=1, x=1, y=1 . : x=16, y=256. ) 5 log x y + log y x = 2 ; x > 0, y > 0, x, y 1 3 x y = 2 1 5 log x y + log y = 2 ; x 3 x = 2 + y

5 2 5 log x y log x y + 1 = 0 2 ; logxy = t; t2 t + 1 = 0; t1 = 0,5, t2 = 2; 2 9 x = 4 + y + 2 y y = x log x y = 0,5 y = x 2 ; ; ; y = x 2 = y x log x y = 2 3 x y = 2 y = x 3 y y 2 = 0 ; 2 y = x x 3 x + 2 = 0 y = x 3 y y = 2 ; 2 y = x 3 x x = 2

y = x y =1 y = x2 ; x = 2 x =1

x = 1 y =1 . : (4, 16). y = 16 x = 4

3. 1. . 3.1.A01. a) (1-4x)22(1-4x); (1 4x)(1 4x 2) 0; (1 4x)(1 4x) 0; (1 4x)(1 + 4x) 0. 1 4 1 4x

: x

) (1-3x)2 5(1-3x); (1 3x)(1 3x 5) 0, (1 3x)(4 3x) 0; (1 3x)(4 + 3x) 0; 4 3 1 3x

1 4

1 . 4

: x

4 3

1 . 3

130

2 2 x 2 x 20 0 x x 20 0 x x 20 0 ; ; ; 2 x < 0 x < 0 x 4 < 4 x 4 x 5 1 9 x2-x-200; x2-x-20=0; D=81l x= ; x1=5, x2=-4; . 2 x < 0

3.1.A02. a)

-42

02

5

x2

: [ 4;0 ) .

)

x + 3x 10 0 5 x 2 x + 3x 10 0 ; ; ; 2 x 0 < x < 0 x < 0 3 7 x2+3x-10=0; D=9+410=49; x= ; x1=-5, x2=2. 25 0 2 x

x + 3x 10 0 ; x +1 < 1 x

: [ 5;0 ) .7 < x < 5 ; 1 x 5

3.1.A03. x 2 + 2 x 35 < 0 2 x + 2 x 35 < 0 ; ; 1 10 x 2 x 5 2 12 2 x +2x-35=0; D=4+435=144; x= ; x1=-7, x2=5. 22 35 2 x x > 0 ; 5 x + 1 1 5 x

7

1 5

5

x

: 7; . 5 4 < x < 6 ; 1 x 2

1

x 2 2 x 24 < 0 2 24 + 2 x x 2 > 0 x 2 x 24 < 0 ; ; ; ) 1 4 x 2 2 x + 1 1 2 x x 2 2 10 2 x -2x-24=0; D=4+424=100; x= ; x1=6, x2=-4. 24

1 2

6

x

: 4; . 2

1

3.1.A04. a) 3(x-5)(5x+4)3. x=17 . 4

x = 4. : x = 4. 3.1.B03. a) (x2+2x-35)(x2-7x-8)0; x1=-7, x2=5; x2-7x-8=0; D=49+48>0; x1=8, x2=-1; (x+7)(x-5)(x-8)(x+1) x ; 2 ; 3.1.B05. a) 2 25 x < 16 25 x < 16 x( x 5) < 0 0 < x < 5 ; 4 4. 2 16 x < 5 < x < 5 25

133

-

4 5

0

4 5

x 5

: 0; . 50 < x < 7 3. 3 x ; 2 9 ; ) 2 16 x < 9 x < 16 3 40

x( x 7) < 0 3 ; 3 0 x > 0 5 x + 1 > 1 4 x 3.1.B07. a) |5x+1|>1-4x; ; ; . 5 x + 1 < 0 1 x < 2 x 1 4 x 5 x < 2

: (; 2) (0; +).

7 x 6 6 x + 7 0 x > 0 x > 0 + > 6 7 7 2 x x ; ; ) |6x+7|>7-2x; 7. 6 x + 7 < 0 x < 7 x < 2 6 6 x 7 > 7 2 x 7 x < 2

: (; 7 ) (0; +).2

134

3.1.B08. a) x(x-1)212(x-1); x(x-1)2-12(x-1)0; (x-1)(x(x-1)-12)0; (x-1)(x2-x-12)0; x2-x-12=0; D=1+412=49; x= x2=-3; (x-1)(x-4)(x+3)0. + -3 1 4 + x

1 7 ; x1=4, 2

: [ 3;1] [ 4; + ) . ) x(x+3) 10(x+3); x(x+3) -10(x+3)0; (x+3)(x(x+3)-10)0; (x+3)(x2+3x-10)0; (x+3)(x+5)(x-2)0.2 2

-

+

-

+

-5

-3

2

x

: [ 5; 3] [ 2; + ) .

3.1.B09. a) (5x2+

1 1 1 1 6 x-2)3(x2+ x+4)3; 5x2+ x-2x2+ x+4; 4x26; x2 ; 3 3 3 3 4 6 6 6 6 x . : ; . 2 2 2 2

) 4 x 2 + x 2 3x 2 + x + 4 ; 6 6

1 1 1 2 2 2 4 x + x 2 3x + x + 4 ; x 6; 6 6 1 6 x 6 . : [ 6; 6] .

3

3

3.1.10. ) x 5 2 4 x 5 + 2 + x 5 + 2 0 ; 4x2 5 + 2 x 5 8 x 5 4 + 5x2 + 2 2 x 5 + 4 0; 20x2 6 x 5 + 5x2 + 4 x 5 0; 25x2 2 x 5 0;x 25 x 2 5 0 . +0

(

)(

) (

)

2

(

)

+

2 5 25

: 0;

2 5 . 25

) x 7 3 7 x 7 + 3 + x 7 + 3 0 ; 7x 7 + 3x 7 21x 7 9 + x 7 + 6 x 7 + 9 0; 56x2 12 x 7 0; 14x2 3x 7 0; x 14 x 3 7 0 .2 2

(

)(

) (

)

2

(

)

135

+ 0

+

3 7 14

:

3 7 0; .3.1.11. 14

)

2 2 x 2 + 2( x 2 6 x + 9) 13x + 20 x + 2( x 3) 13x + 20 ; ; 2 2 2 2 2 x > 5 x ( x + 2) 5 x ( x + 2) 2 x < 0 x 2 + 2 x 2 12 x + 18 + 13x 20 0 2 3 x 2 + x 2 0 ; 3 x + x 2 0 ; ; 2 2 2 x (5( x + 2) 2) < 0 x (5 x + 10 2) < 0 x (5 x + 8) < 0 1 3 1 2 3x2 + x 2 = 0; D = 1 + 4 2 = 9; x = ; x1 = ; x2 = ; 6 3 3

2 3

1 38 5

x

5x + 8 = 0; 5x = 8; x = ; x = 1 .3 52 2

3 5

12

x

: ; 1 . 5

3

)

x + 2( x 1) 2 x + 7 2 2 x > 4 x ( x + 3)

;

x + 2( x 2 x + 1) + 2 x 7 0 2 2 2 4 x ( x + 3) x < 0

;

2 3 x 2 x 5 0 ; 2 x (4 x + 12 1) < 0

3 x 2 2 x 5 0 ; 2 x (4 x + 11) < 0

3x2 2x 5 = 0; D = 4 3 5 4 = 64; x = 1

28 2 ; x1 = 1; x2 = 1 ; 6 3

1

2 3

x

4x + 11 < 0; 4x + 11 = 0; x = +

11 3 x = 2 . 4 4x

2

3 4

: ; 2 . 4

3

3.1.12. ) (5x2 + 0,7x 2,7)7 (x2 + 4x 2,7)7; 5x2 + 0,7x 2,7 x2 + 4x 2,7; 4x2 3,3x 0; x(4x 3,3) 0.+ + 0

33 40

x

. : (;0] 33 ; + 40

) (3x2 + 0,7x 2,8)5 (x2 + 5x 2,8)5; 3x2 + 0,7x 2,8 x2 5x 2,8; 136

2x2 4,3x 0; x(2x 4,3) 0.+ + 0

43 20

x

: 0;

43 . 20

. 3.1.01. ) 5(1 x) 4(1 x)2 < (1 x)3; (1 x)((1 x)2 + 4(1 x) 5) > 0; (1 x)(1 x + 5)(1 x 1) > 0; x(x 1)(6 x) > 0; x(x 1)(x 6) < 0; + + x 0 6 1 x (; 0) (1; 6) + +

) 3(2 x) 2(2 x)2 < (2 x)3; (2 x)((2 x)2 + 2(2 x) 3) > 0; (2 x)(2 x + 3)(2 x 1) > 0; (2 x)(5 x)(1 x) > 0; (x 2)(x 5)(x 1) < 0.1 2 5 x

x (; 1) (2; 5). 3.1.02. ) (x + 2) 11x 2; x 7x + 6 0;2 2 2 4 2

(x2 6)(x2 1) 0; (x 6 )(x + 6 )(x 1)(x + 1) 0.+ + 1 1 +

x [ 6 ; 1] [1; ) (x2 2)2 4x2 11; x4 8x2 + 15 0; (x2 5)(x2 3) 0; ( x 5)( x + 5)( x 3)( x + 3) = 0 .+ + +

6

6

x

6 ].

5

3

3+

5

x

x [ 5; 3] [ 3; 5] .

3.1.03. ) |5x + 7|(5x + 4) 0; 7 51

4 54 3

x

x ; + .

7 5

4 5

) |3x + 7|(3x 4) 0; x ; + . +

7 34 3

3.1.04. ) |4x 5| (4x 5)2, |4x 5| (4x 5)2 0; 0 |4x 5| 1; 1 4x 5 1; 1 x 3 3 . : 1; . 2 2 2 . : 3 2 0; 3 .

7 3

x

) |3x 1| (3x 1)2; (3x 1)2 |3x 1| 0, 0 |3x 1| 1; 1 3x 1 1; 0 x

137

t 2 16 t 0 ; 3.1.05. ) 18 3 | t + 2 |> 1 8 3

t 2 + 6t 16 0 ; t > 1 t 3 < 1 2

8 t 2 t > 1 ; t < 3 x

8. t 2 28 t 0 ; ) 9 3 | t + 3 |> 2

t 2 + 3t 28 0 ; t > 1 < 5 t

(t + 7)(t 4) 0 ; t > 1 t < 5

t 5 7 1 4 t 7 t 4 . .

3.1.06. )

9 x 3 3+ x 6 ; 2 x 6 ; | 2 x + 5 | 11 2 x 16 8 3

9 x 3 ; x 3 x 8 x

9

x [9; 8] {3}.

13 x 11 1 + x 12 ) ; x 11 ; x [13; 2] {11}. | 2 x 9 | 13 x 2

3.1.07. ) (x2 8x + 48)2 (x2 8x 50)2 < 0; 98(2x2 16x 2) < 0; x2 8x 1 < 0; D = 64 + 4 = 68;x= 8 2 17 = 4 17 ; 4 17 < x < 4 + 17 . 2

) (x2 6x + 52)2 (x2 6x 50)2 < 0; 102 (2x2 12x + 2) < 0; x2 6x + 1 < 0; D = 36 4 = 32;x= 64 2 = 3 2 2 ; 3 2 2 < x < 3+ 2 2 . 2

3.1.08. ) (x2 2x + 32)4 > (x2 2x 50)4; (x2 2x + 32)2 (x2 2x 50)2 > 0; 82(2x2 4x 18) > 0; x2 2x 9 > 0; x > 1 10 ; x (; 1 10 ) (1 + 10 ; +). x < 1 + 10

) (x2 10x + 30)4 > (x2 10x 56)4; (x2 10x + 30)2 (x2 10x 56)2 > 0; 86(2x2 20x 26) > 0; x2 10x 13 > 0; D = 100 + 52 = 4 38;x= 10 2 38 x > 5 + 38 ; ; x (; 5 38 ) (5 + 38 ; +). 2 x < 5 38

138

3.1.09. ) (0,3x2 + 0,5x 5)2 > (0,3x2 + 0,5x + 5)2; 10(0,6x2 + x) > 0; x2 + ) (0,1x2 + 0,3x 5)2 > (0,1x2 + 0,3x + 5)2; 10(0,2x2 + 0,6x) > 0; x2 + 3x < 0; 3 < x < 0. 3.1.10. ) x 2 + x 7 < x 2 + x + 7 ; 5 5 3 3 9 1 2 3 1 2 3 1 2 3 2 x + x 7 x + x + 7 < 0 ; 14 x + x < 0 ; x + x > 0 ; 5 5 5 5 3 3 3 x > 0 4 ; x ; 1 (0; +) . x < 1 4 5 5 2 2

10 2 x < 0; 1 < x < 0; 6 3

1

3

4

1

3

4

) x 2 + x 4 < x 2 + x + 4 ; x 2 + x 4 x 2 + x + 4 < 0 ; 3 3 3 3 2 2 2 2 x > 0 4 4 1 8 x 2 + x < 0 ; x 2 + x > 0 ; 1 ; x ; 1 (0; +) . 3 3 3 x < 1 3

1

2

4

1

2

4

1

2

2

1

2

2

3.1.11. ) x 2 x + 6x2 10x + 3 < 0; 8 22 2

5 3

19 2 5 16 35 2 10 2 x x 1 < 0 ; x x < 0 ; 3 3 3 3 3

2

2

D 5 7 5 7 5+ 7 =7; x= ; . 0 ; x x+ x x > 0 ; 2 5 2 5 5

(

)

D = 1 8 < 0; x (; +). ) x 2 x + 1 3 18 2 1 13 2 1 18 2 1 13 > x x ; x x+ x x > 0 ; 3 5 3 5 5 3 5 4 4 2 2

139

31 2 2 x 4 + 1 > 0 ; D = 8 < 0; x (; +). 2x 5 3 9

D. 3.1.D01. ) (x2 9x)2 + 4x2 36x 140 < 0; (x2 9x)2 + 4(x2 9x) 140 < 0; (x2 9x + 14)(x2 9x 10) < 0;+ + +

x (1; 2) (7; 10). ) (x2 7x)2 + 18x2 126x + 72 < 0; (x2 7x)2 + 18(x2 7x) + 72 < 0; (x2 7x + 12)(x2 7x + 6) < 0; (x 3)(x 4)(x 6)(x 1) < 0; x (1; 3) (4; 6).+ + + 1 3 4 6 x

1

2

7

10

x

3.1.D02. ) 6(4x + 3)(x2 x + 9) < 9(4x + 3)2 + (x2 x + 9)2; (3(4x + 3) (x2 x + 9))2 > 0; x2 + 13x 0; x 0; x 13. , x x = 0 x = 13; x (; 0) (0; 13) (13; +). ) 6(4x + 1)(x2 + 9x + 3) < 9(4x + 1)2 + (x2 + 9x + 3)2; [3(4x + 1) (x2 + 9x + 3)]2 > 0; x2 + 3x 0; x 0, x 3. , x 0; 3. x (; 0) (0; 3) (3; +). 3.1.D03. ) |x 2|(x2 6x 16) 6x2 24; |x 2|(x2 6x 16) 6(x 2)(x + 2) 0; I. x 2; (x 2)(x2 6x 16 6x 12) 0; (x 2)(x2 12x 28) 0; (x 2)(x 14)(x + 2) 0; x {2}[14; +). +

II. x 2; (x 2)(x2 6x 16 + 6(x + 2) 0; (x 2)(x2 4) 0; (x 2)(x 2)(x + 2) 0; x (; 2]{2}. + x 2 2 : x (; 2] {2} [14; +). ) |x 5|(x2 7x 60) 7x2 175; |x 5|(x2 7x 60) 7(x 5)(x + 5) 0; (x + 5)[|x 5|(x 12) 7(x 5)] 0; I. x 5; (x + 5)(x 5)(x 19) 0; x 19; + x 5 19 II. x 5; (x + 5)[(x 5)((x 12) + 7) 0; (x + 5)(x 5)(x 5) 0; x 5. +x 5 5 : x (; 5] {5} [19; +).

2

14

x

140

3.1.D04. )

x 2 | x 2 25 | 9( x 2 25) ; x( x 6) ( x 6) x 52 2 ( x 25)( x 9) 0

; I. x2 25 0; x 5 ( x 1)( x 6) 0 + + + 3 5 3 II. x2 25 0; 5 x 5;2 2 ( x 25)(9 + x ) 0 ; ( x 6)( x 1) 0

x = 5;x

5

2 x 25 0 x = 5. : x = 5. ( x 6)( x 1) 0

)

x 2 | x 2 + 49 | 16( x 2 49) ; x( x 9) x 9

I. x2 49 0;

( x 7)( x + 7)( x 4)( x + 4) 0 ; x = 7; ( x 9)( x 1) 02 2 ( x 49)( x + 16) 0 ; x = 7. : x = 7. ( x 9)( x 1) 0

II. x2 49 0; 7 x 7;

3.1.D05. ) (3x 8)(x2 4x 2) |3x 8| |x2 4x 2|, (3x 8)(x2 4x 2) 0;8 x x 2 6 3 +

( (

)) ( x ( 2 + 6 )) 0 . +

2 6

8 3

2+

6

x

8 x 2 6; [2 + 6; +] . 3 9 5 41 5 + 41 x 0; 2 2

) (4x 9)(x2 5x 4) |4x 9| |x2 5x 4|; x (4x 9)(x2 5x 4) 0; x 4 + +

5

41

2 5 41 9 5 + 41 x ; ; + . 4 2 2

9 4

5+ 2

41

x

0 (4x 9)(x2 5x 4) 0; x : x 5 41 1 5 + 4 ; 2 ; + . 2 4 2

5 + 41 2

3.1.D06. ) (x2 + 1,5x + 0,7)2 + (x2 + 4,2x + 0,862)2 141

(x2 + 2,5x + 0,76)2 + (x2 + 3,2x + 0,802)2; (2x2 + 4x + 1,46)(x 0,06) (2x2 + 7,4x + 1,664)(x 0,06); (x + 0,06)(3,4x + 0,204) 0; (x + 0,06)(x + 0,06) 0; x = 0,06. ) (x2 + 1,7x + 0,9)2 + (x2 + 3,8x + 0,585)2 (x2 + 2,7x + 0,75)2 + (x2 + 2,8x + 0,735)2; (2x2 + 4,4x + 1,65)(x + 0,15) (2x2 + 6,6x + 1,32)(x + 0,15); (x 0,15)(2,2(x 0,15)) 0; x = 0,15. 3.1.D07. ) f(x) = 14x2 + 13. (x, f(x)), OX x = |f(x)|, OY y= |x|. : x y; |14x2 + 13| |x|; (14x2 + 13)2 x2; (14x2 + 13)2 x2 0; (14x2 + 13 x)(14x2 + 13 + x) 0; (x + 1) x + 1

13 13 (x 1) x + 0; 14 14 +

+ 1 x

13 14

13 14

13 13 x 1; ; 14 14

1 .

) x = |13x2 + 12|; y = |x|; |13x2 + 12| |x|, (13x2 + 12)2 x2; (132 + 12)2 2 0; (132 + 12 )(132 + 12 + ) 0; (13x2 + x 12)(13x2 x 12) 0;12 12 ( x + 1) x ( x 1) x + 0 ; 13 13 + + 1

+ 1 x

12 13

12 13

x 1;

12 12 ; 13 13

1 .

3.1.D08. ) f(x) = x4 8|x|3 + 16x2 < 9; I. x 0; x4 8x3 + 16x2 < 9; x2(x 4)2 9 < 0; (x2 4x 3)(x2 4x + 3) < 0; (x 2 + 7 )(x 2 7 )(x 8)(x 1) < 0; 0 1 + 3 +

2+ 7

x

x [0; 1) (3; 2 + 7 ); II. x 0; x4 + 8x3 + 16x2 9 < 0; x2(x + 4)2 9 < 0; (x2 + 4x 3)(x2 + 4x + 3) < 0; 142

(x + 2 7 )(7 + 2 + 7 )(x + 1)(x + 3) < 0;

+

3

+ 1

0 x

2 7

x (2 7 ; 3) (1; 0]. : x (2 7 ; 3) (1; 1) (3; 2 + 7 ). ) f(x) = x4 14|x|3 + 49x2 > 36; I. x 0; x2(x 7)2 36 > 0; (x2 7x 6)(x2 7x + 6) > 0; (x 1)(x 6) x 0 1

7 + 73 7 73 x > 0; 2 2 + 6 +

7+ 2

73

x

x (1; 6)

7 + 73 ; + ; 2

II. x 0; x2(x + 7)2 36 > 0; (x2 + 7x 6)(x2 + 7x + 6) > 0; 7 73 7 + 73 x+ x+ (x + 6)(x + 1) > 0; 2 2 + 6 + 1 0 x

7 73 2

x ; 7 + 73 7 73 ; + . (6; 1) (1; 6) 2 2

7 73 (6; 1) . 2

: ;

3.1.D09. ) f(x) > 0 x, x = 3; f(|x + 3| 17) > 0; f(3) = 0; |x + 3| 17 = 3; |x + 3| = 20; x + 3 = 20 x + 3 = 20 ; x = 17 x = 23 .

f(|x + 3| 17) > 0 x, x = 17 x = 23, , x (; 23) (23; 17) (17; +). ) f(x) < 0, x, x = 5; f(|x 1| + 18) < 0; f(5) = 0; |x 1| + 18 = 5; |x 1| = 13; . f(|x 1| + 18) > 0 x (; +). 3.1.D10. ) f(x) > 0 x, x = 7; f(7) = 0; (x 6)f(x) 0; x 6 0; x 6. x = 7 . : x (; 6] {7}.143

) f(x) > 0 x, x = 9; (x + 7)f(x) 0; f(9) = 0; x + 7 0; x 7. x = 9 . : x (; 7] {9}. 3.1.D11. ) f(x) ; T = 9; f(x) 18; f(x) = 9x x2; x [0; 9]; 9x x2 18; x2 9x + 18 0; (x 6)(x 3) 0; x [3; 6] [0; 9]. , : x [3 + 9k; 6 + 9k], k Z. ) f(x) ; T = 11; f(x) 18; f(x) = 11x x2; x [0; 11]; 11x x2 18; x2 11x + 18 0; 2 x 9 [0; 11]. , x [2 + 11k; 9 + 11k], k Z. 3.1.D12. ) f(|x 1| 1) < f(|5x + 2|); |x 1| 1 > |5x + 2|; I. x 1 0; x 1; x 2 > |5x + 2|; x 2 > 5x + 2; x < 1, , x 1. II. x 1 0; x 1; 1 x 1 > |5x + 2|; x > |5x + 2|; 1) 5x + 2 0; x ; x > 5x + 2; x < = . , x ; . 5 3 5 6 3

2

2

1

2

1

2 1 1 2 2) 5x + 2 0; x ; x > 5x 2; x > . , x ; . 5 2 2 5 1 1 : x ; . 2 3

) f(|x 4| 4) > f(|3x + 5|). f , f(m) > f(n), m < n |x 4| 4 < |3x + 5|; I. x 4 0; x 4; x 4 4 < |3x + 5|; 1) 3x > 5; x > ; x 8 < 3x + 5; x >

5 13 . , x 4; 3 2 5 2) 3x < 5; x < , I. 3

II. x 4; 4 x 4 < |3x + 5|; 1) 3x 5; x 1 ; x < 3x + 5; x > ; x ; 4 ; 3 4 4 2) 3x 5; x ; x < 3x 5; x < .5 5 5 , x ; : x ; ; . 2 4 2 2. 3.2.01. ) x5(7 3x) 0; x 0; x(7 3x) 0; x(3x 7) 0; x < 0 7 . : x (; 0) ; + . 7 x 3 3 3 ) x (4 5x) 0; x(4 5x) 0; x 0; x(5x 4) 0;5 3 5 2 2 5 5

144

x < 0 4 . : x (; 0) ; + . x 4 5 5

3.2.02. ) f ( x) = ( x 2 4 x + 3)

1 3

f(x) x24x+3>0 .. (x1)(x3)>, x (, 1)(3, +); ) f ( x) = ( x 2 5 x + 6 ) 2 f(x) x25x+60 .. (x2)(x3)0, x (, 2][3, +).6 x 13 3.2.03. ) ; x 3 2 x + 13 0 3 18 61

6 13 ; x 3x 18 + 12 x + 13 0

6 x 13 x > 0 ; 15 x 5

6 0 0 5 x 75 + 15 x + 71 0 x 5 1 x > 0 ; x ; 5 1 x 5 5 .

3 x > 2 5 5 x 13 1 3 3.2.04. ) ; x ; 2 2 ; + . >0; 2x 5 2 5 x < 2 1 2 3x 17 2 ) < 0 ; 3 < x < 5 . x+3 3 2 < x 3 x 18 0 3 < x < 6 3.2.05. ) ; x (0; 1). 1 0 < x < 1 x(1 x) < 0

)

8 < x < 9 x 2 x 72 < 0 ; ; x 1 >0 x(3 x) < 0 x3

8 < x < 9 . : x (8; 0) (3; 9). x > 3 x < 0

3.2.06. ) 4 >

1 4 p 1 1 ; > 0 ; p (; 0) ; + . p p 4

) 2 >

1 1 . p < 0, .. p > 0: > 0 . p p 1 2 1 2

p < 0: 2p < 1; p > , , p ; 0 . 145

. 3.2.01. )4 1 + > 12 , x 1; 1 x (1 x) 2

12(1 x)2 4(1 x) 1 < 0; 12t2 4t 1 = 0; D = 16 + 48 = 64;48 1 1 1 1 1 1 1 1 = ; t2 = ; < 1 x < ; 1 < x < ; < x < 1 . 6 6 24 6 2 6 2 2 2 1 1 : x ; 1 1; 1 . 6 2 3 1 ) + > 18 ; x 3; 18(3 x)2 3(3 x) 1 < 0; 3 x (3 x)2 39 1 1 18t2 3t 1 = 0; D = 9 + 72 = 81; t = ; t1 = ; t2 = ; 2 18 6 3 1 2 2 1 1 2 3 < x < 2 ; 2 < x < 3 . : x 2 ; 3 3; 3 . 6 3 3 6 6 3

t1 =

1 12