Метод Феррари:
уравнение вида
![(1)\ x^4+ax^3+bx^2+cx+d=0](https://tex.z-dn.net/?f=%281%29%5C%20x%5E4%2Bax%5E3%2Bbx%5E2%2Bcx%2Bd%3D0)
с помощью замены ![x=y-\frac{a}{4}](https://tex.z-dn.net/?f=x%3Dy-%5Cfrac%7Ba%7D%7B4%7D)
приводим к виду
![(2)\ y^4+p*y^2+qy+r=0](https://tex.z-dn.net/?f=%282%29%5C%20y%5E4%2Bp%2Ay%5E2%2Bqy%2Br%3D0)
где:
![p=b-\frac{3a^2}{8}\\q=\frac{a^3}{8}-\frac{a*b}{2}+c\\r=-\frac{3a^4}{256}+\frac{a^2b}{16}-\frac{a*c}{4}+d](https://tex.z-dn.net/?f=p%3Db-%5Cfrac%7B3a%5E2%7D%7B8%7D%5C%5Cq%3D%5Cfrac%7Ba%5E3%7D%7B8%7D-%5Cfrac%7Ba%2Ab%7D%7B2%7D%2Bc%5C%5Cr%3D-%5Cfrac%7B3a%5E4%7D%7B256%7D%2B%5Cfrac%7Ba%5E2b%7D%7B16%7D-%5Cfrac%7Ba%2Ac%7D%7B4%7D%2Bd)
добавим и вычтем из левой части уравнения 2 выражение
, где s - некоторое число:![y^4+p*y^2+qy+r=y^2+py^2+2sy^2+qy+r+s^2-2sy^2-s^2=\\=y^4+2sy^2+s^2+y^2(p-2s)+qy+r-s^2=\\=(y^4+2s*y^2+s^2)+(p-2s)(y^2+\frac{2*qy}{2*(p-2s)})+r-s^2=\\=(y^2+s)^2+(p-2s)(y^2+2(\frac{qy}{2(p-2s)}+\frac{q^2}{4(p-2s)^2})-\frac{\frac{q^2}{4(p-2s)^2}}{p-2s}+r-s^2=\\=(y^2+s)^2+(p-2s)(y+\frac{q}{2(p-2s)})^2+r^2-s^2-\frac{q^2}{4(p-2s)}](https://tex.z-dn.net/?f=y%5E4%2Bp%2Ay%5E2%2Bqy%2Br%3Dy%5E2%2Bpy%5E2%2B2sy%5E2%2Bqy%2Br%2Bs%5E2-2sy%5E2-s%5E2%3D%5C%5C%3Dy%5E4%2B2sy%5E2%2Bs%5E2%2By%5E2%28p-2s%29%2Bqy%2Br-s%5E2%3D%5C%5C%3D%28y%5E4%2B2s%2Ay%5E2%2Bs%5E2%29%2B%28p-2s%29%28y%5E2%2B%5Cfrac%7B2%2Aqy%7D%7B2%2A%28p-2s%29%7D%29%2Br-s%5E2%3D%5C%5C%3D%28y%5E2%2Bs%29%5E2%2B%28p-2s%29%28y%5E2%2B2%28%5Cfrac%7Bqy%7D%7B2%28p-2s%29%7D%2B%5Cfrac%7Bq%5E2%7D%7B4%28p-2s%29%5E2%7D%29-%5Cfrac%7B%5Cfrac%7Bq%5E2%7D%7B4%28p-2s%29%5E2%7D%7D%7Bp-2s%7D%2Br-s%5E2%3D%5C%5C%3D%28y%5E2%2Bs%29%5E2%2B%28p-2s%29%28y%2B%5Cfrac%7Bq%7D%7B2%28p-2s%29%7D%29%5E2%2Br%5E2-s%5E2-%5Cfrac%7Bq%5E2%7D%7B4%28p-2s%29%7D)
получим:
![(3)\ (y^2+s)^2+(p-2s)(y+\frac{q}{2(p-2s)})^2+r^2-s^2-\frac{q^2}{4(p-2s)}=0](https://tex.z-dn.net/?f=%283%29%5C%20%28y%5E2%2Bs%29%5E2%2B%28p-2s%29%28y%2B%5Cfrac%7Bq%7D%7B2%28p-2s%29%7D%29%5E2%2Br%5E2-s%5E2-%5Cfrac%7Bq%5E2%7D%7B4%28p-2s%29%7D%3D0)
Пусть s - корень уравнения
![(4)\ r^2-s^2-\frac{q^2}{4(p-2s)}=0](https://tex.z-dn.net/?f=%284%29%5C%20r%5E2-s%5E2-%5Cfrac%7Bq%5E2%7D%7B4%28p-2s%29%7D%3D0)
Тогда уравнение 3 примет вид:
![(5)(y^2+s)^2+(p-2s)(y+\frac{q}{2(p-2s)})^2=0](https://tex.z-dn.net/?f=%285%29%28y%5E2%2Bs%29%5E2%2B%28p-2s%29%28y%2B%5Cfrac%7Bq%7D%7B2%28p-2s%29%7D%29%5E2%3D0)
Избавляемся в уравнении 4 от знаменателя:
![r(p-2s)-s^2(p-2s)-\frac{q^2}{4}=0](https://tex.z-dn.net/?f=r%28p-2s%29-s%5E2%28p-2s%29-%5Cfrac%7Bq%5E2%7D%7B4%7D%3D0)
Раскроем скобки и получим:
![(6)\ 2s^3-ps^2-2rs+rp-\frac{q^2}{4}=0](https://tex.z-dn.net/?f=%286%29%5C%202s%5E3-ps%5E2-2rs%2Brp-%5Cfrac%7Bq%5E2%7D%7B4%7D%3D0)
Уравнение 6 называется кубической резольвентой уравнения 4 степени.
Разложим уравнение 5 на множители:
![(y^2+s)^2+(p-2s)(y+\frac{q}{2(p-2s)})^2=0\\(y^2+s)^2-(2s-p)(y-\frac{q}{2(2s-p)})^2=0\\(y^2+s^2)^2-(y*\sqrt{2s-p}-\frac{q}{2\sqrt{2s-p}})^2=0\\(y^2-y\sqrt{2s-p}+\frac{q}{2\sqrt{2s-p}}+s)(y^2+y\sqrt{2s-p}-\frac{q}{2\sqrt{2s-p}}+s)=0](https://tex.z-dn.net/?f=%28y%5E2%2Bs%29%5E2%2B%28p-2s%29%28y%2B%5Cfrac%7Bq%7D%7B2%28p-2s%29%7D%29%5E2%3D0%5C%5C%28y%5E2%2Bs%29%5E2-%282s-p%29%28y-%5Cfrac%7Bq%7D%7B2%282s-p%29%7D%29%5E2%3D0%5C%5C%28y%5E2%2Bs%5E2%29%5E2-%28y%2A%5Csqrt%7B2s-p%7D-%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%29%5E2%3D0%5C%5C%28y%5E2-y%5Csqrt%7B2s-p%7D%2B%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%2Bs%29%28y%5E2%2By%5Csqrt%7B2s-p%7D-%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%2Bs%29%3D0)
Получим два квадратных уравнения:
![(7)\ y^2-y\sqrt{2s-p}+\frac{q}{2\sqrt{2s-p}}+s=0\\(8)\ y^2+y\sqrt{2s-p}-\frac{q}{2\sqrt{2s-p}}+s=0](https://tex.z-dn.net/?f=%287%29%5C%20y%5E2-y%5Csqrt%7B2s-p%7D%2B%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%2Bs%3D0%5C%5C%288%29%5C%20y%5E2%2By%5Csqrt%7B2s-p%7D-%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%2Bs%3D0)
Применяем этот метод для решения уравнения:
![x^{4} -4x^{3} -51x^{2} +306x-432=0](https://tex.z-dn.net/?f=x%5E%7B4%7D%20-4x%5E%7B3%7D%20-51x%5E%7B2%7D%20%2B306x-432%3D0)
коэффициенты:
a=-4
b=-51
c=306
d=-432
Определяем p,q и r:
![p=b-\frac{3a^2}{8}=-51-\frac{3*4^2}{8}=-57 \\q=\frac{a^3}{8}-\frac{a*b}{2}+c=\frac{(-4)^3}{8}-\frac{-4*(-51)}{2}+306=196 \\r=-\frac{3a^4}{256}+\frac{a^2b}{16}-\frac{a*c}{4}+d=-\frac{3*2^8}{256} +\frac{16*(-51)}{16} -\frac{(-4)*306}{4} -432=-180](https://tex.z-dn.net/?f=p%3Db-%5Cfrac%7B3a%5E2%7D%7B8%7D%3D-51-%5Cfrac%7B3%2A4%5E2%7D%7B8%7D%3D-57%20%5C%5Cq%3D%5Cfrac%7Ba%5E3%7D%7B8%7D-%5Cfrac%7Ba%2Ab%7D%7B2%7D%2Bc%3D%5Cfrac%7B%28-4%29%5E3%7D%7B8%7D-%5Cfrac%7B-4%2A%28-51%29%7D%7B2%7D%2B306%3D196%20%20%5C%5Cr%3D-%5Cfrac%7B3a%5E4%7D%7B256%7D%2B%5Cfrac%7Ba%5E2b%7D%7B16%7D-%5Cfrac%7Ba%2Ac%7D%7B4%7D%2Bd%3D-%5Cfrac%7B3%2A2%5E8%7D%7B256%7D%20%2B%5Cfrac%7B16%2A%28-51%29%7D%7B16%7D%20-%5Cfrac%7B%28-4%29%2A306%7D%7B4%7D%20-432%3D-180)
Ищем s:
![2s^3+57s^2+360s+57*180-\frac{196^2}{4}=0\\2s^3+57s^2+360s+656=0\\P(s)=2s^3+57s^2+360s+656\\s=-1\Rightarrow P(-1)=-2+57-360+656\neq 0\\s=-2\Rightarrow P(-2)=-2*8+57*4-360*2+656=148\neq 0\\s=-4\Rightarrow P(-4)=-2*4^3+57*16-360*4+656=0 \Rightarrow s_1=-4](https://tex.z-dn.net/?f=2s%5E3%2B57s%5E2%2B360s%2B57%2A180-%5Cfrac%7B196%5E2%7D%7B4%7D%3D0%5C%5C2s%5E3%2B57s%5E2%2B360s%2B656%3D0%5C%5CP%28s%29%3D2s%5E3%2B57s%5E2%2B360s%2B656%5C%5Cs%3D-1%5CRightarrow%20P%28-1%29%3D-2%2B57-360%2B656%5Cneq%200%5C%5Cs%3D-2%5CRightarrow%20P%28-2%29%3D-2%2A8%2B57%2A4-360%2A2%2B656%3D148%5Cneq%200%5C%5Cs%3D-4%5CRightarrow%20P%28-4%29%3D-2%2A4%5E3%2B57%2A16-360%2A4%2B656%3D0%20%5CRightarrow%20s_1%3D-4)
Возможно, у этого уравнения третьей степени есть и другие действительные корни. Но для данной задачи находить их все не обязательно. Достаточно одного корня, т.е числа, при котором выражение обращается в ноль.
Подставляем p,q,r и s в квадратные уравнения 7 и 8:
![y^2-y\sqrt{2s-p}+\frac{q}{2\sqrt{2s-p}}+s=0\\y^2-y\sqrt{-8+57}+\frac{196}{2\sqrt{-8+57}} -4=0\\y^2-7y+10=0\\D=49-40=9=3^2\\y_1=\frac{7+3}{2}=5\\y_2=\frac{7-3}{2}=2](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%5C%5Cy%5E2-y%5Csqrt%7B-8%2B57%7D%2B%5Cfrac%7B196%7D%7B2%5Csqrt%7B-8%2B57%7D%7D%20-4%3D0%5C%5Cy%5E2-7y%2B10%3D0%5C%5CD%3D49-40%3D9%3D3%5E2%5C%5Cy_1%3D%5Cfrac%7B7%2B3%7D%7B2%7D%3D5%5C%5Cy_2%3D%5Cfrac%7B7-3%7D%7B2%7D%3D2)
![y^2+y\sqrt{2s-p}-\frac{q}{2\sqrt{2s-p}}+s=0\\y^2+y\sqrt{-8+57}-\frac{196}{2\sqrt{-8+57}} -4=0\\y^2+7y-18=0\\D=49+72=121=11^2\\y_3=\frac{-7+11}{2}=2\\y_4=\frac{-7-11}{2}=-9](https://tex.z-dn.net/?f=y%5E2%2By%5Csqrt%7B2s-p%7D-%5Cfrac%7Bq%7D%7B2%5Csqrt%7B2s-p%7D%7D%2Bs%3D0%5C%5Cy%5E2%2By%5Csqrt%7B-8%2B57%7D-%5Cfrac%7B196%7D%7B2%5Csqrt%7B-8%2B57%7D%7D%20-4%3D0%5C%5Cy%5E2%2B7y-18%3D0%5C%5CD%3D49%2B72%3D121%3D11%5E2%5C%5Cy_3%3D%5Cfrac%7B-7%2B11%7D%7B2%7D%3D2%5C%5Cy_4%3D%5Cfrac%7B-7-11%7D%7B2%7D%3D-9)
Находим x:
![x=y-\frac{a}{4} \\x_1=5+1=6\\x_2=2+1=3\\x_3=-9+1=-8](https://tex.z-dn.net/?f=x%3Dy-%5Cfrac%7Ba%7D%7B4%7D%20%5C%5Cx_1%3D5%2B1%3D6%5C%5Cx_2%3D2%2B1%3D3%5C%5Cx_3%3D-9%2B1%3D-8)
Ответ: -8; 3; 6