∛(3x + 6) - ∛(x + 1) = 1
∛(3x + 6) = a
∛(x + 1) = b
a - b = 1
a³ - 3b³ = 3
a = b + 1
(b + 1)³ - 3b³ = 3
b³ + 3b² + 3b + 1 - 3b³ - 3 = 0
-2b³ - 2 + 3b² + 3b = 0
-2(b³ + 1) + 3b(b + 1) = 0
-2(b + 1)(b² - b + 1) + 3b(b + 1) = 0
(b + 1)(-2b² + 2b - 2 + 3b) = 0
(b + 1)(2b² -5b + 2) = 0
b = -1
D = 25 - 4*4 = 9
b23 = (5 +- 3)/4 = 2 1/2
1. b = -1
∛(x + 1) = -1
x + 1 = -1
x = -2
проверяем
∛(-6 + 6) - ∛(-2 + 1) = 0 - (-1) = 1 да
2. b = 2
∛(x + 1) = 2
x + 1 = 8
x = 7
∛(21 + 6) - ∛(7 + 1) = 3 - 2 = 1 да
3. b = 1/2
∛(x + 1) = 1/2
x + 1 = 1/8
x = -7/8
∛(-21/8 + 6) - ∛(-7/8 + 1) =∛(-21/8 + 48/8) - ∛ 1/8 = 3/2 - 1/2 = 1 да
Ответ x = {-2,-7/8,7}
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Answers & Comments
∛(3x + 6) - ∛(x + 1) = 1
∛(3x + 6) = a
∛(x + 1) = b
a - b = 1
a³ - 3b³ = 3
a = b + 1
(b + 1)³ - 3b³ = 3
b³ + 3b² + 3b + 1 - 3b³ - 3 = 0
-2b³ - 2 + 3b² + 3b = 0
-2(b³ + 1) + 3b(b + 1) = 0
-2(b + 1)(b² - b + 1) + 3b(b + 1) = 0
(b + 1)(-2b² + 2b - 2 + 3b) = 0
(b + 1)(2b² -5b + 2) = 0
b = -1
D = 25 - 4*4 = 9
b23 = (5 +- 3)/4 = 2 1/2
1. b = -1
∛(x + 1) = -1
x + 1 = -1
x = -2
проверяем
∛(-6 + 6) - ∛(-2 + 1) = 0 - (-1) = 1 да
2. b = 2
∛(x + 1) = 2
x + 1 = 8
x = 7
проверяем
∛(21 + 6) - ∛(7 + 1) = 3 - 2 = 1 да
3. b = 1/2
∛(x + 1) = 1/2
x + 1 = 1/8
x = -7/8
проверяем
∛(-21/8 + 6) - ∛(-7/8 + 1) =∛(-21/8 + 48/8) - ∛ 1/8 = 3/2 - 1/2 = 1 да
Ответ x = {-2,-7/8,7}
Verified answer
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