Question

Zadania w załączniku z matematyki. Bardzo prosze o rozwiazanie wszystkich zadan na tej stronie daje naj

Answer 1

1.

a)

(2y-3)^2 -(3y-2)(3y+2) = 4y^2 -12y +9 -(9y^2 -4) = 4y^2 -12y +9 -9y^2 +4 =

= -5y^2 -12y +13

b)

(5y+1)*1-5y)-(1+5y)^2 = (1+5y)(1-5y) -(1=10y+25y^2) = 1-25y^2 -1-10y -25y^2 =

= -50y^2 -10y

c)

(x^2 -3)^2 + (3+x^2)^2 -(x^2 -1)^2 = (x^2 -3)^2 +(x^2 +3)^2 -9x^2 -1)^2 =

= x^4 -6x^2 +9 +x^4 +6x^2 +9 -(x^4 -2x^2 +1) = 2x^4 +18-x^4 +2x^2 -1 =

= x^4 +2x^2 =17

d)

(x^2 -2)(x^2 +2) -(4-x^2)(x^2 +4) = x^4 -4 -(16-x^4 = x^4 -4-16+x^2 = 2x^4 -20

 

2.

a)

(2x-1)^2 -(2x_1)(1+2x) -(2x+1)^2 = 4x^4 -4x+1 -(4x^2 -1) -(4x^2 +4x +1) =

= 4x^2 -4x+1 -4x^2 +1 -4x^2 -4x-1 = -4x^2 -8x-1

dla x = V2

= -4(V2)^2 -8V2 -1 = -8-8V2+1 = -7-8V2

b)

(V5+x)^2 -(V5-x)^2 +(x+V5)(V5-x) = 5 +2V5 x +x^2 -(5-2V5 x +x^2) +5 -x^2 =

= 5-2V5x +x^2 -5+2V5x -x^2 +5-x^2 = -x^2 + 4V5x +5

dla x = V5 -1

= -(V5-1)^2 + 4V5(V5-1) + 5 = -(5-2V5+1) +20-4V5 +5 = -5+2V5-1=20-4V5+5 =

= 19 - 2V5 

 

3.

a)

(x-y)(x+y) +(x+2y)(x-2y) = x^2 -y^2 + x^2 - 4y^2 = 2x^2 - 5y^2

dla x = V2,  y = 1-V3

= 2(V2)^2 -5(1-V3)^2 = 4-5(1-2V3 +3) = 4-20+10V3 = 10V3 -16

b)

(3x+y)(3x-y) -(x-5y)(x+5y) = 9x^2 -y^2 -(x^2 -25y^2) = 9x^2 -y^2 - x^2 =25y^2 =

= 8x^2 +24y^2

dla x = 1 +V3/2;  y = 1+V2

= 8(1+V3/2)^2 + 24(1+V2)^2 = 8(1+V3+3/4) + 24(1+2V2 +2) = 

= 8(1,75+V3) +24(3+2V2) = 14+8V3+72+48V2 = 86+8V3+48V2

c)

(y-3x)^2 -(y-4x)^2 -6xy = y^2 -6xy +9x^2 -(y^2 -8xy +16x^2)-6xy = 

= y^2 -6xy +9x^2 -y^2 + 8xy -16x^2 -6xy = -7x^2 -4xy

dla x = 1/(V2-1); y = V2

= -7[1/(V2-1)]^2 - 4 *1/(V2-1) *V2 = -7[1/(V2-1) *(V2+1)/(V2+1)]^2 = 

= 4 *1/(V2-1) *(V2+1)/(V2+1)*V2 =

 -7[(V2+1)/(2-1)]^2 - 4[(V2+1)/(2-1)] *V2 =

= -7(2+2V2+1) -4(2+V2) = -7(3+2V2)-(8+4V2) = -21-14V2-8-4V2 = -18V2 - 29

d)

1/(2x^2 -6),  dla x = 1+V2

= 1/[2(1+V2)^2 -6] = 1/[2(3+2V2) -6] = 1/(6+4V2-6) = 1/8V2 *V2/V2 = V2/8

e)

(x+1)/(x^2 -4),  dla x = -1+V3

= (-1+V3+1)/[(V3-1)^2 -4] = V3/(3-2V3+1-4) = V3/(-2V3) = -1/2

 

4.

a) 

(3-x)^2 -(x+1/3)^2 = 2/9

9-6x+x^2 -(x^2 +2/3x +1/9) = 2/9

9-6x+x^2 - x^2 -2/3x -1/9 = 2/9

-6x-2/3x = 2/9 +1/9 -9

-6x-2/3x = 3/9 -9

-6x -2/3x = 1/3 -9  I*3

-18x -2x = 1-27

-20x = -26

x = 26/20

x = 13/10

b)

4(1/2x -3)^2 -(2-x)^2 = 0

4(1/4x^2 -3x +9)-(4-4x+x^2) = 0

x^2 -12x +36 -4 +4x -x^2 = 0

-8x = -32

x = 4

c)

(6+1/3x)(-1/3x +6) +(1/3x -4)^2 = 4

(6x+1/3x)(6-1/3x) +1/9x^2 -8/3 +16 = 4

 36 -1/9x^2 +1/9x^2 -8/3x +16 = 4

-8/3x = -48 I*(-3)

8x = 144

x = 18

d)

(-4x-3)(4x-3) + 8(1-V2x)^2 = 1

-(4x+3)(4x-3) + 8(1-2V2+2x^2) = 1

-16x^2 + 9 + 8 -16V2x +16x^2 = 1

-16V2x = 1-17

-16Vx = -16  I:(-16)

V2x = 1

x = 1/V2 *V2/V2

x = V2/2

 

5.

a)

4(x-3)^2 -(2x-5)^2 >= 2

4(x^2 -6x+9) -(4x^2 -20x +25) >= 2

4x^2 -24x +36 -4x^2 +20x -25 >= 2

-4x >= -9  I:(-4)

x <= 9/4

b)

9(4/9x^2 -4x +9) < 1-4x +4x^2

4x^2 -36x +81 < 1-4x+4x^2

-36x + 4x < 1-81

-32x < -80  I:(-32)

x > 80/32

x > 5/2

c)

-9(2-x)^2 -(1-3x)(3x+1) <= 11

-9(4 - 4x + x^2) -(1- 9x^2) <= 11

-36+36-9x^2 -1+9x^2 <= 11

36x <= 11+37

36x <= 48

x <= 48/36

x <= 4/3

d)

(1/4x +2)^2 + 1/4(1-1/2x)(1+1/2x) >= 0

1/16x^2 + x +4 +1/4(1-1/4x^2) >= 0

1/16x^2 +x +4 +1/4 -1/16x^2 >= 0

x >= -17/4

 

6.

a)   a = 3+V2

V = a^3 = (3+V2)^3 = 3^3 + 3*3^2 *V2 +3*3*(V2)^2 = 27 +27V2 +18 = 45 + 27V2

Pc = 6a^2 = 6(3+V2)^2 = 6(9-6V2+2) = 6(11-6V2)

b)   a = 2V3-1

V = a^3 = (2V3-1)^3 = (2V3)^3 -3(2V2)^2 +3*2V3 -1 = 24V3-36+ 6V3 -1 =30V3-37

Pc = 6a^2 = 6(2V3-1)^2 = 6(12-4V3-1) = 6(13-4V3)

c)   a = V6+V2

V = a^3 = (V6+V2)^3 = (V6)^3  +3(v6)^2 *V2 +3V6 *(V2)^2 +(V2)^3 =

= 18V2 +12V6 +2V2 = 12V6 +20V2 = 4(3V6 +5V2)

Pc = 6a^2 = 6(V6+V2) = 6(6+2V12+2) = 6(8+2*2V3) = 48 +24V3 = 24(2+V3)