Разложение на множители, возведение в степень

1.

(7+9х/4)+(2х/3)=7х+1

(28+9х)/4 + 2х/3 = 7х+1

84+27х+8х=84х+12

84-12=84х-27х-8х

72=49х

Х=72/49

 

2.

(7-6х/2)+10х <   (8х+1/2)-12

(14-6х+20х)/2 < (16х+1-24)/2

14+14х < 16х-23

Х ≥ 37/2

Х € [37/2; + ∞)

 

3.

6х²+2х-11=0

D = 4 +264 = 268; √D = 2 √67

X₁= (-2+2√67)/12 = (-1+√67)/6

X₂= (-2-2√67)/12 = (-1-√67)/6

 

4.

(Х+2/х-1)-(х-2/х+1)=(4х²)/2х-1

4/x = 2x-1

2x²-x-4 = 0

D = 1+32 = 33; √D = √33

X₁ = (1+√33)/4

X₂ = (1-√33)/4

 

5.

х²-6х+8>0

х²-6х+8=0

D = 36-32 = 4; √D = 2

X₁ = (6+2)/2= 4

X₂ = (6-2)/2= 2

(x-4)(x-2)› 0

Х € (-∞; 2) U (4; + ∞)

 

6.

2х²-4х+7<0

2х²-4х+7 = 0

D = 16-56 < 0

X € R

 

7.

√(х+2)=3х-4

X+2 = (3x-4)²

X+2=9x²-36x+16

9x²-37x+14=0

D = 1369-504 = 865; √D =√865

X₁= (37+√865)/18

X₂= (37-√865)/18

8.

√(х²+3х-3)=2х-3

х²+3х-3=(2х-3)²

3x²-15x+12=0

D = 225-144 = 81; √D = 9

X₁=(15+9)/6=4

X₂=(15-9)/6= 1

 

9.

25^0.3*5^1.4*625^0.25= 5^2*0.35^1.45^4*0.25= 5^0.6+1.4+1=5³=125

 

10.

49^(-2/3)*7^1/2*7^(-3/4)= 7^{(-4/3+1/2-3/4)}= 7^(-19/12)

 

11.

216^(-1/3)*1/6^(-2)=5^(-1)*1/25^(-1/2)

6^3*(-1/3)*6^(-1)*(-2)= 5^(-1)*5⁽¹⁾

6=1

 

12.

(4/5)^x =25/6

2(2/5)^x = (5/3)*(2/5)^-1

log (2/5)^2x= log(2/5)^-5/3

2x=-5/3

X = -5/6

 

13.

(1/6)^(4x-7) =6^(x-3)

(1/6)^(4x-7)= (1/6)^(3-x)

4x-7=3-x

5x=10

X=2

 

14.

2^(3x+6)< ¼^(x-1)

 

2^(3x+6) ≤ 2^(-2x+2)

(3x+6) ≤ (-2x+2)

5x ≤ -4

X ≤ -4/5

X € (- ∞ ; -4/5]

 

15.

9^x-1 < 9^(-2x+8)

 

x-1 ≤ -2x+8

x ≤ 3

x€ (-∞; 3]

 

16.

log3²⁷ =log ½  4* log 7 ³√49

 

17.

log 1/3 9* log 2 3корень2/8:7² log 7 2

 

18.

log 4 х= log 4 2+ log 47

 

log₄ x = log₄14

x=14

 

19.

log 23 (2х-1)= log 25х

23 (2x-1) = 25x

64x-23=25x

 

20.

log 2 (х²+7х+5)= log 7 (4х-1)

 

log 2 (х²+7х+5)= (log₂(4x-1)/log₂7)

 

(х²+7х+5)= (4x-1)/7

7x²+45x+36=0

D =2025-1008= 1017; √D= 3√ 113

X₁= (-45-√113)/14

X₂= (-45+√113)/14

 

преобразовать и упростить:

1. sin 180⁰ - sin 495⁰ + cos 945⁰ = sin 180⁰ – sin (450⁰+45⁰)+cos (900⁰+45⁰)= sin 180⁰- (sin450⁰cos45⁰+cos450⁰sin45⁰) + cos900⁰cos45⁰-sin900⁰sin45⁰ =

= 0- (1*√2/2+0*√2/2)+(-1*√2/2-0*√2/2) = -√2/2-√2/2 = -√2

 

2. sin (90⁰ -t)+ cos (180⁰ +t)+tg(270⁰ +t)+сtg(360⁰ +t) = cos t + (-cost) – ctg t + ctgt = 0

3. arcsin (√3) /2+ arccos (√ 3) /2 = π/3 + π/6 = 3π/6 = π/2
4. arc cos (-1/2)- arc sin (-1) = 2π/3 – (-π/2) = 7π/6
5. sin х-1/2=0

sin x = ½

arcsin x = arcsin ½

x = π/6

6. сos х+(√3)/2=0

cosx=-√3/2

arcos x = arcos (-√3/2)

x= 5π/6