a = ∛(3x + √(1 + x²))
b = ∛(3x - √(1 + x²))
ab = ∛(3x + √(1 + x²))∛(3x - √(1 + x²)) = ∛(9x² - 1 - x²) = ∛(8x² - 1)
∛(3x + √(1 + x²))² + 3∛(8x² - 1 ) + 2∛(3x - √(1 + x²))²= 0
a² + 3ab + 2b² = 0
D = 9b² - 8b² = b²
a12 = (-3b +- b)/2 = -2b -b
1. a = -b
∛(3x + √(1 + x²)) = -∛(3x - √(1 + x²))
3x + √(1 + x²) = - (3x - √(1 + x²))
6x = 0
x = 0
2. a = -2b
∛(3x + √(1 + x²)) = -2∛(3x - √(1 + x²))
3x + √(1 + x²) = -8 (3x - √(1 + x²))
3x + √(1 + x²) = -24x + 8√(1 + x²)
27x = 7√(1 + x²)
x > 0
729x² = 49 + 49x²
680x² = 49
x² = 49/680
x = - √49/680 < 0 нет
x = √49/680 = 7/2√170 = 7√170/340
ответ 7√170/340, 0
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Verified answer
a = ∛(3x + √(1 + x²))
b = ∛(3x - √(1 + x²))
ab = ∛(3x + √(1 + x²))∛(3x - √(1 + x²)) = ∛(9x² - 1 - x²) = ∛(8x² - 1)
∛(3x + √(1 + x²))² + 3∛(8x² - 1 ) + 2∛(3x - √(1 + x²))²= 0
a² + 3ab + 2b² = 0
D = 9b² - 8b² = b²
a12 = (-3b +- b)/2 = -2b -b
1. a = -b
∛(3x + √(1 + x²)) = -∛(3x - √(1 + x²))
3x + √(1 + x²) = - (3x - √(1 + x²))
6x = 0
x = 0
2. a = -2b
∛(3x + √(1 + x²)) = -2∛(3x - √(1 + x²))
3x + √(1 + x²) = -8 (3x - √(1 + x²))
3x + √(1 + x²) = -24x + 8√(1 + x²)
27x = 7√(1 + x²)
x > 0
729x² = 49 + 49x²
680x² = 49
x² = 49/680
x = - √49/680 < 0 нет
x = √49/680 = 7/2√170 = 7√170/340
ответ 7√170/340, 0