Применим метод Феррари.
Пусть
![x=y+ \frac{3}{4}](https://tex.z-dn.net/?f=x%3Dy%2B+%5Cfrac%7B3%7D%7B4%7D+)
. Подставив в исходное уравнение, получим
![y^4-11.375y^2-18.375y+8.30078125=0](https://tex.z-dn.net/?f=y%5E4-11.375y%5E2-18.375y%2B8.30078125%3D0)
(*)
![p=-11.375;\,\,\,\,\, q=-18.375;\,\,\, r=8.30078125](https://tex.z-dn.net/?f=p%3D-11.375%3B%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C+q%3D-18.375%3B%5C%2C%5C%2C%5C%2C+r%3D8.30078125)
Для кубической резольвентой уравнения (*) есть уравнение
![2s^3+11.375s^2-16.6015625s-178.83154296875=0\\ 4096s^3+23296s^2-34000s-366247=0](https://tex.z-dn.net/?f=2s%5E3%2B11.375s%5E2-16.6015625s-178.83154296875%3D0%5C%5C+4096s%5E3%2B23296s%5E2-34000s-366247%3D0)
s - некоторое число
Здесь же применим метод Виета-Кардано
Q=(a²-3b)/9 ≈ 6.361
R=(2a³-9ab+27c)/54 ≈ -30.025
S=Q³-R² = -644.134 <0
Поскольку S<0, то кубическое уравнение имеет один единственный корень.
α = arccos(|R|/√Q³)/3 ≈ 0.413
s = -2sgn(R)√Q*chα - (a/3) ≈ 3.585 - наш корень
Подставляя наши значения в уравнение
![y^2-y \sqrt{2S-p} + \frac{q}{2 \sqrt{2S-p} } +S=0](https://tex.z-dn.net/?f=y%5E2-y+%5Csqrt%7B2S-p%7D+%2B+%5Cfrac%7Bq%7D%7B2+%5Csqrt%7B2S-p%7D+%7D+%2BS%3D0)
, получим
![y^2-y \sqrt{2\times3.585+11.375} - \frac{18.375}{2 \sqrt{2\times3.85+11.375} } +3.585=0\\ \\ \\ y^2-(0.05 \sqrt{7418} )y+ \frac{106374.12-735 \sqrt{7418} }{29672} =0](https://tex.z-dn.net/?f=y%5E2-y+%5Csqrt%7B2%5Ctimes3.585%2B11.375%7D+-+%5Cfrac%7B18.375%7D%7B2+%5Csqrt%7B2%5Ctimes3.85%2B11.375%7D+%7D+%2B3.585%3D0%5C%5C+%5C%5C+%5C%5C+y%5E2-%280.05+%5Csqrt%7B7418%7D+%29y%2B+%5Cfrac%7B106374.12-735+%5Csqrt%7B7418%7D+%7D%7B29672%7D+%3D0)
![D= \frac{31192.69+735 \sqrt{7418} }{7418}](https://tex.z-dn.net/?f=D%3D+%5Cfrac%7B31192.69%2B735+%5Csqrt%7B7418%7D+%7D%7B7418%7D+)
![y_1= \frac{370.9 \sqrt{7418}- \sqrt{231387374.42+5452230 \sqrt{7418} } }{14836} \approx 0.3686\\ \\ \\ y_2= \frac{370.9 \sqrt{7418}+ \sqrt{231387374.42+5452230 \sqrt{7418} } }{14836} \approx3.9378](https://tex.z-dn.net/?f=y_1%3D+%5Cfrac%7B370.9+%5Csqrt%7B7418%7D-+%5Csqrt%7B231387374.42%2B5452230+%5Csqrt%7B7418%7D+%7D++%7D%7B14836%7D+%5Capprox+0.3686%5C%5C+%5C%5C+%5C%5C+y_2%3D+%5Cfrac%7B370.9+%5Csqrt%7B7418%7D%2B+%5Csqrt%7B231387374.42%2B5452230+%5Csqrt%7B7418%7D+%7D++%7D%7B14836%7D+%5Capprox3.9378)
Возвращаемся к обратной замене
![x_1= \frac{370.9 \sqrt{7418}- \sqrt{231387374.42+5452230 \sqrt{7418} } }{14836} +0.75\approx1.1186\\ \\\\ x_2= \frac{370.9 \sqrt{7418}+ \sqrt{231387374.42+5452230 \sqrt{7418} } }{14836} +0.75\approx4.6878](https://tex.z-dn.net/?f=x_1%3D+%5Cfrac%7B370.9+%5Csqrt%7B7418%7D-+%5Csqrt%7B231387374.42%2B5452230+%5Csqrt%7B7418%7D+%7D++%7D%7B14836%7D+%2B0.75%5Capprox1.1186%5C%5C+%5C%5C%5C%5C+x_2%3D+%5Cfrac%7B370.9+%5Csqrt%7B7418%7D%2B+%5Csqrt%7B231387374.42%2B5452230+%5Csqrt%7B7418%7D+%7D++%7D%7B14836%7D+%2B0.75%5Capprox4.6878)
Ответ:
![x_1\approx1.1186;\,\,\,\,\, x_2\approx4.6878](https://tex.z-dn.net/?f=x_1%5Capprox1.1186%3B%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C+x_2%5Capprox4.6878)