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2010-01-24T22:01:03+01:00
Zadanie 10
x = 2 - √5
y = 1 + 3√5

a)
mnożymy "każdy z każdym"
x * y = (2 - √5)(1 + 3√5) = 2 + 2*3√5 - √5 - √5 * 3√5 = 2 + 6√5 - √5 - 15 = - 13 + 5√5

b)
3x + y = 3(2 - √5) + 1 + 3√5 = 3*2 - 3√5 + 1 + 3√5 = 7 + 0*√5 = 7

c)
y : x = (1 + 3√5)/(2 - √5) = (2 + √5)(1 + 3√5)/(2 - √5)(2 + √5) = (2 + 6√5 + √5 + 3√5 * √5)/(4 - 5) = - (2 + 15 + 7√5) = - 17 - 7√5

d)
(2y - x)/(2x + y) = [2(1 + 3√5) - (2 - √5)]/[2(2 - √5) + (1 + 3√5)] = [2 + 6√5 - 2 + √5]/[4 - 2√5 + 1 + 3√5] = 7√5/[5 + √5] = 7√5(5 - √5)/[5 + √5](5 - √5) = 7√5(5 - √5)/(25 - 5) = (35√5 - 35)/20 = - 35/20 + 35√5/20 = - 7/4 + 7/4 * √5

zadanie 13
a) (2√3 - x)² - 2(2√3 - x)(2√3 + x) + (2√3 + x)² = [(2√3 - x) - (2√3 + x)]² = (2√3 - x - 2√3 - x)² = (- 2x)² = 4x²

x = √3/2
4*(√3/2)² = 4*3/4 = 3

b) (x + 1)⁴ - 2(x² - 1)² + (x - 1)⁴ = (x + 1)⁴ - 2(x - 1)²(x + 1)² + (x - 1)⁴ = [(x + 1)² - (x - 1)²]² = [(x + 1 - x + 1)(x + 1 + x - 1)]² = [2*2x]² = 16x²

x = √3/2
16*(√3/2)² = 16*3/4 = 12

c) (x + √2)⁴ - (x - √2)⁴ = [(x + √2)²]² - [(x - √2)²]² = [(x + √2)² - (x - √2)²]*[(x + √2)² + (x - √2)²] = [(x + √2 + x - √2)(x + √2 - x + √2)]*[x² + x*2√2 + 2 + x² - x*2√2 + 2] = [2x*2√2]*[2x² + 4] = x*8√2*(x² + 1)

x = √3/2
(√3/2)*8√2*((√3/2)² + 1) = 4√6*(3/4 + 1) = 4√6 * 7/4 = 7√6

d) (x + √3)³ + 2x(x + 3)(x - 3) + (x - √3)³ = x³ + 3x³*√3 + 9x + 3√3 + 2x(x² - 9) + x³ - 3x³*√3 + 9x - 3√3 = 2x³ + 18x + 2x(x² - 9) = 4x³

x = √3/2
4(√3/2)³ = 4*3√3/8 = 3√3/2

e) (x - √3)(x² + x√3 + 3) + (x + √3)(x² - x√3 + 3) = x³ - (√3)³ + x³ + (√3)³ = 2x³

x = √3/2
2(√3/2)³ = 2*3√3/8 = 3√3/4

f) (x² - 1)(x⁴ + x² + 1) + (x - 1)(x² + x + 1)(x + 1)(x² - x + 1) = (x²)³ - 1³ + (x³ - 1)(x³ + 1) = x⁶ - 1 + x⁶ - 1 = 2x⁶ - 2

x = √3/2
2(√3/2)⁶ - 2 = 2*27/64 - 2 = 54/64 - 2 = - 1 10/64

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