x⁴ + 10x³ - 125x - 54 = o
x⁴ + 10x³ + 25x² - 25x² - 125x - 54 = 0
(x⁴ + 10x³ + 25x²) - (25x² + 125x) - 54 = 0
(x² + 5x)² - 25(x² + 5x) - 54 = 0
Сделаем замену : x² + 5x = m
m² - 25m - 54 = 0
m₁ = 27 m₂ = - 2
1) x² + 5x = 27
x² + 5x - 27 = 0
D = 5² - 4 * (- 27) = 25 + 108 = 133
![x_{1}=\frac{-5+\sqrt{133} }{2}\\\\x_{2}=\frac{-5-\sqrt{133} }{2}](https://tex.z-dn.net/?f=x_%7B1%7D%3D%5Cfrac%7B-5%2B%5Csqrt%7B133%7D+%7D%7B2%7D%5C%5C%5C%5Cx_%7B2%7D%3D%5Cfrac%7B-5-%5Csqrt%7B133%7D+%7D%7B2%7D)
2)x² + 5x = - 2
x² + 5x + 2 = 0
D = 5² - 4 * 2 = 25 - 8 = 17
![x_{3}=\frac{-5+\sqrt{17} }{2}\\\\x_{4}=\frac{-5-\sqrt{17} }{2}](https://tex.z-dn.net/?f=x_%7B3%7D%3D%5Cfrac%7B-5%2B%5Csqrt%7B17%7D+%7D%7B2%7D%5C%5C%5C%5Cx_%7B4%7D%3D%5Cfrac%7B-5-%5Csqrt%7B17%7D+%7D%7B2%7D)