Odpowiedź:
a)
(3x - 2)³ - 64 = 0
(3x - 2)³- 4³ = 0
Korzystamy ze wzoru a³ - b³ = (a - b)(a² + a + b)
(3x - 2)³ - 4³ = 27x³-54x² +36x - 72
27x³-54x² +36x - 72 =0
9(3x³- 6x² + 4x - 8) = 0
(3x³- 6x² + 4x - 8) = 0
3x²(x - 2) + 4(x -2) = 0
(x - 2)(3x²+ 4) = 0
Ponieważ 3x²+ 4 > 0 dla x ∈ R ,wiec :
x - 2 = 0
x = 2
b)
x²(x - 2) = x(8 - 4x)
x²(x - 2) - x(8-4x) = 0
x²(x - 2) - 4x(2 - x) = 0
x²(x - 2) + 4x(x - 2) = 0
(x - 2)(x² + 4x) = 0
x(x - 2)(x + 2) = 0
x = 0 ∨ x - 2 = 0 ∨ x + 2 = 0
x = 0 ∨ x = 2 ∨ x = - 2
x = { - 2 , 0 , 2 }
c)
(x - 1)(x² + x + 1) < (x - 1)³
(x - 1)(x² + x + 1) - (x - 1)³ < 0
(x - 1)[x² + x + 1 - (x - 1)²] < 0
(x - 1)(x² +x + 1 - x² +2x - 1) < 0
(x - 1)3x < 0
3x(x - 1) < 0
3x > 0∧ x - 1 < 0 ∨ 3x < 0 ∧ x - 1 > 0
x > 0 ∧ x < 1 ∨ x < 0 ∧ x > 1
x > 0 ∧ x < 1
x ∈ ( 0 , 1 )
d)
√8x³- 6√2 = 4√18x - √2x²
2√2x³ - 6√2 = 4 * 3√2x - √2x² | : √2
2x³ - 6 = 12x - x²
2x³ + x²- 12x - 6 = 0
x² (2x + 1) - 6(2x + 1) = 0
(2x+1)(x² - 6) = 0
(2x + 1)(x - √6)(x + √6) = 0
2x+1 =0 ∨ x - √6= 0 ∨ x + √6 = 0
x= - 1/2 ∨ x = √6 ∨ x = - √6
x = { - √6 , - 1/2 , √6 }
