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2010-03-20T22:01:21+01:00
Zadanie 5
a)
(√3 + √2)/(√3 - √2) - (√3 - √2)/(√3 + √2) + (1 - 16√6)/4 = [(√3 + √2)² - (√3 - √2)²]/[(√3 - √2)(√3 + √2)] + (1 - 16√6)/4 = [(√3 + √2 + √3 - √2)(√3 + √2 - √3 + √2)]/[3 - 2] + (1 - 16√6)/4 = 2√3*2√2 + 1/4 - 4√6 = 1/4

b)
(√7 - √6)/(√7 + √6) - (√7 + √6)/(√7 - √6) + (8√42 + 3)/2 = [(√7 - √6)² - (√7 + √6)²]/[(√7 - √6)(√7 + √6)] + (8√42 + 3)/2 = [(√7 - √6 + √7 + √6)(√7 - √6 - √3 - √2)]/[7 - 6] + (8√42 + 3)/2 = - 2√7*2√6 + 4√42 + 3/2 = 3/2

c)
2/(√3 - 1) - 1/(√3 + 1) + (5 - 2√3)/4 = [2(√3 + 1) - (√3 - 1)]/[(√3 + 1)(√3 - 1)] + (5 - 2√3)/4 = [√3 + 3]/[3 - 1] + 5/4 - √3/2 = 3/2 + 5/4 = 11/4

d)
4/(√7 - 2) - 2/(√7 + 2) + (1 - 2√7)/3 = [4(√7 + 2) - 2(√7 - 2)]/[(√7 + 2)(√7 - 2)] + (1 - 2√7)/3 = [2√7 + 12]/[7 - 4] + (1 - 2√7)/3 = 13/3

zadanie 6
a)
(√(2 - √3) - √(2 + √3))² = |2 - √3| + |2 + √3| - 2√[(2 - √3)(2 + √3)] = 4 - 2√(4 - 3) = 4 - 2 = 2

b)
(√(√2 - 1) + √(√2 + 1))² = |√2 - 1| + |√2 + 1| + 2√[(√2 - 1)(√2 + 1)] = 2√2 + 2√(2 - 1) = 2√2 + 2 = 2(√2 + 1)

zadanie 7
5¹² - 1 = (5⁶ - 1)(5⁶ + 1) = (5³ - 1)(5³ + 1)(5⁶ + 1) = 124(5³ + 1)(5⁶ + 1) = 31*4(5³ + 1)(5⁶ + 1)
4(5³ + 1)(5⁶ + 1) ∈ N

zadanie 8
3¹⁸ - 2¹⁸ = (3⁹ - 2⁹)(3⁹ + 2⁹) = (3³ - 2³)(3⁶ + 3³*2³ + 2⁶)(3⁹ + 2⁹) = 19(3⁶ + 3³*2³ + 2⁶)(3⁹ + 2⁹)
(3⁶ + 3³*2³ + 2⁶)(3⁹ + 2⁹) ∈ N

zadanie 9
a)
2(x - 3y)² - 3(x - 2y)(x + 2y) + 12xy = 2x² - 12xy + 18y² - 3x² + 12y² + 12xy = - x² + 30y²
- 3² + 30*(- 1)² = 21

b)
(2a - 3b)² - (2a + b)(2a - b) - 4b(2b - 3a) + a = 4a² - 12ab + 9b² - 4a² + b² - 8b² + 12ab + a = 2b² + a
2(√2)² + 2 = 6

c)
(2a - b)² - (a/2 - 1)(a/2 + 1) - (4a² + b²) = 4a² - 4ab + b² - a²/4 + 1 - 4a² - b² = - 4ab + 1 - a²/4
- 4* 1/4 * 1/5 + 1 - 1/4*25 = 4/5 - 1/100 = 79/100

d)
2(5x² + 4) + (2x - 3)(2x + 3) - 2(3x - 1)² + 3(1 - 2x)² = 10x² + 8 + 4x² - 9 - 18x² + 12x - 2 + 3 - 12x + 12x² = 8x²
8*(- 1/2)² = 8* 1/4 = 2

zadanie 10
a)
(x - 1)³ - 4(x + 1)² + (2 - x)(4 + 2x + x²) = (x - 1)³ - 4(x + 1)² + 8 - x³ = x³ - 3x² + 3x - 1 - 4(x + 1)² + 8 - x³ = - 3x² + 3x + 7 - 4x² - 8x - 4 = - 7x² - 5x + 3

b)
(2 + x)³ - (4x - 2)²/4 + (2 - x)³ = (2 + x + 2 - x)(4 + 4x + x² - 4 + x² + 4 - 4x + x²) - (4x - 2)²/4 = 4(3x² + 4) - (4x - 2)²/4 = 12x² + 16 - 4x² + 4x - 1 = 8x² + 4x + 15

c)
2(3x - 5)² - (x + 3)³ + 3(x + 1)(x² - x + 1) = 18x² - 60x + 50 - x³ - 9x² - 27x - 27 + 3x³ + 3 = 2x³ + 9x² - 87x + 26

d)
2(3x + √2)(3x - √2) + (√2*x + √8)² = 18x² - 4 + 2x² + 8x + 8 = 20x² + 8x + 4 = 4(5x² + 2x + 1) = 20x² + 8x + 4

zadanie 11
a)
12 - 2(x - 1)² = 4(x - 2) - (x - 3)(2x - 5)
12 - 2x² + 4x - 2 = 4x - 8 - 2x² + 11x - 15
11x = 33
x = 3

b)
(3x - 1)² - 5(2x + 1)² + (6x - 3)(2x + 1) = (x - 1)²
9x² - 6x + 1 - 20x² - 20x - 5 + 12x² - 3 = x² - 2x + 1
24x = - 8
x = - 1/3

c)
5(x - 1)² - 2(x + 3)² = 3(x + 2)² - 7(6x - 1)
5x² - 10x + 5 - 2x² - 12x - 18 = 3x² + 12x + 12 - 42x + 7
8x = 32
x = 4

d)
2x² + (x + 5)² - 2(x + 7)² = 2(3x - 72,5) + (x - 6)²
2x² + x² + 10x + 25 - 2x² - 28x - 98 = 6x - 145 + x² - 12x + 36
12x = 36
x = 3

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