【四元消法】之精要……

风花飘飘

风花飘飘

解3元3次方程组：

x^3+5*x-y=0----------------------------(1)
4*x^3-y*x^2+y+2=0------------------(2)
5*x^3+5*x+z=0------------------------(3)

x^5-5*x-2=0的根式解，也就在里面啦。

风花飘飘

风花飘飘

【碟】【恋】【花】【公】【式】


(x^3+b*x+c=0)乘以(x^2+f=0)等于零
【则】
(x^3+b*x+c)*(x^2+f)= x^5+(f+b)*x^3+c*x^2+b*f*x+c*f=0

m=(-108*c)

n = (4*3^(3/2)*sqrt(27*c^2+4*b^3))

m^2-n^2=((-12*b)^3)

p=(2^(2/3)*3^(1/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3))

【当p不等于0时】
q=(-2^(4/3)*3^(2/3)*b/(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3))

【当p=0时】
q=(-2^(2/3)*3^(1/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)+9*c)^(1/3))

x1
=((3^(2/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)-3*2^(2/3)*b)/(2^(1/3)*3^(4/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3)))

X2,3=(……)

X4,5=(……)        


【其中：】

b=
(-(q^5+5*p*q^4+10*p^2*q^3+10*p^3*q^2+5*p^4*q+p^5+7776*c)/((q+p)^(2/3)*(2^(10/3)*3^(10/3)*q+2^(10/3)*3^(10/3)*p)))

c= (-(q^5+5*p*q^4+10*p^2*q^3+10*p^3*q^2+(q+p)^(2/3)*(2^(10/3)*3^(10/3)*b*q+2^(10/3)*3^(10/3)*b*p)+5*p^4*q+p^5)/7776)

f=
(-(q+p)^(1/3)*(q^3+3*p*q^2+3*p^2*q+p^3)/(2^(10/3)*3^(10/3)))

***************************************************************

x1 = -(q+p)^(5/3)/(2^(5/3)*3^(5/3))

x2 = (q+p)^(5/3)/(2^(5/3)*3^(5/3))

x3 = (-sqrt(3)*%i/2-1/2)*(sqrt((-4*b^3*q^15-60*b^3*p*q^14+(q+p)^2*(27*q^13+351*p*q^12+2106*p^2*q^11+7722*p^3*q^10+19305*p^4*q^9+34749*p^5*q^8+46332*p^6*q^7+46332*p^7*q^6+34749*p^8*q^5+19305*p^9*q^4+7722*p^10*q^3+2106*p^11*q^2+351*p^12*q+27*p^13)-420*b^3*p^2*q^13-1820*b^3*p^3*q^12-5460*b^3*p^4*q^11+(-12012*b^3*p^5-93312*b^3*c)*q^10+(q+p)^(8/3)*(2^(13/3)*3^(19/3)*b*q^9+2^(13/3)*3^(25/3)*b*p*q^8+2^(19/3)*3^(25/3)*b*p^2*q^7+7*2^(19/3)*3^(22/3)*b*p^3*q^6+7*2^(16/3)*3^(25/3)*b*p^4*q^5+7*2^(16/3)*3^(25/3)*b*p^5*q^4+7*2^(19/3)*3^(22/3)*b*p^6*q^3+2^(19/3)*3^(25/3)*b*p^7*q^2+2^(13/3)*3^(25/3)*b*p^8*q+2^(13/3)*3^(19/3)*b*p^9)+(-20020*b^3*p^6-933120*b^3*c*p)*q^9+(-25740*b^3*p^7-4199040*b^3*c*p^2)*q^8+(-25740*b^3*p^8-11197440*b^3*c*p^3)*q^7+(-20020*b^3*p^9-19595520*b^3*c*p^4)*q^6+(q+p)^(10/3)*(2^(20/3)*3^(29/3)*b^2*q^5+5*2^(20/3)*3^(29/3)*b^2*p*q^4+5*2^(23/3)*3^(29/3)*b^2*p^2*q^3+5*2^(23/3)*3^(29/3)*b^2*p^3*q^2+5*2^(20/3)*3^(29/3)*b^2*p^4*q+2^(20/3)*3^(29/3)*b^2*p^5)+(-12012*b^3*p^10-23514624*b^3*c*p^5-725594112*b^3*c^2)*q^5+(-5460*b^3*p^11-19595520*b^3*c*p^6-3627970560*b^3*c^2*p)*q^4+(-1820*b^3*p^12-11197440*b^3*c*p^7-7255941120*b^3*c^2*p^2)*q^3+(-420*b^3*p^13-4199040*b^3*c*p^8-7255941120*b^3*c^2*p^3)*q^2+(-60*b^3*p^14-933120*b^3*c*p^9-3627970560*b^3*c^2*p^4)*q-4*b^3*p^15-93312*b^3*c*p^10-725594112*b^3*c^2*p^5-1880739938304*b^3*c^3)/(q+p))/(64*3^(13/2)*(q+p)^2)+(2^(10/3)*3^(10/3)*b*(q^2*(q+p)^(2/3)+p^2*(q+p)^(2/3))+2^(13/3)*3^(10/3)*b*p*q*(q+p)^(2/3)+q^6+6*(p*q^5+p^5*q)+3*(5*p^2*q^4+5*p^4*q^2)+20*p^3*q^3+p^6)/(7776*(2*q+2*p)))^(1/3)+(sqrt(3)*%i/2-1/2)*b*(2592*q^5+12960*p*q^4+25920*p^2*q^3+25920*p^3*q^2+12960*p^4*q+2592*p^5+20155392*c)/(2^(25/3)*3^(25/3)*(q*(q+p)^(2/3)+p*(q+p)^(2/3))*(sqrt((-4*b^3*q^15-60*b^3*p*q^14+(q+p)^2*(27*q^13+351*p*q^12+2106*p^2*q^11+7722*p^3*q^10+19305*p^4*q^9+34749*p^5*q^8+46332*p^6*q^7+46332*p^7*q^6+34749*p^8*q^5+19305*p^9*q^4+7722*p^10*q^3+2106*p^11*q^2+351*p^12*q+27*p^13)-420*b^3*p^2*q^13-1820*b^3*p^3*q^12-5460*b^3*p^4*q^11+(-12012*b^3*p^5-93312*b^3*c)*q^10+(q+p)^(8/3)*(2^(13/3)*3^(19/3)*b*q^9+2^(13/3)*3^(25/3)*b*p*q^8+2^(19/3)*3^(25/3)*b*p^2*q^7+7*2^(19/3)*3^(22/3)*b*p^3*q^6+7*2^(16/3)*3^(25/3)*b*p^4*q^5+7*2^(16/3)*3^(25/3)*b*p^5*q^4+7*2^(19/3)*3^(22/3)*b*p^6*q^3+2^(19/3)*3^(25/3)*b*p^7*q^2+2^(13/3)*3^(25/3)*b*p^8*q+2^(13/3)*3^(19/3)*b*p^9)+(-20020*b^3*p^6-933120*b^3*c*p)*q^9+(-25740*b^3*p^7-4199040*b^3*c*p^2)*q^8+(-25740*b^3*p^8-11197440*b^3*c*p^3)*q^7+(-20020*b^3*p^9-19595520*b^3*c*p^4)*q^6+(q+p)^(10/3)*(2^(20/3)*3^(29/3)*b^2*q^5+5*2^(20/3)*3^(29/3)*b^2*p*q^4+5*2^(23/3)*3^(29/3)*b^2*p^2*q^3+5*2^(23/3)*3^(29/3)*b^2*p^3*q^2+5*2^(20/3)*3^(29/3)*b^2*p^4*q+2^(20/3)*3^(29/3)*b^2*p^5)+(-12012*b^3*p^10-23514624*b^3*c*p^5-725594112*b^3*c^2)*q^5+(-5460*b^3*p^11-19595520*b^3*c*p^6-3627970560*b^3*c^2*p)*q^4+(-1820*b^3*p^12-11197440*b^3*c*p^7-7255941120*b^3*c^2*p^2)*q^3+(-420*b^3*p^13-4199040*b^3*c*p^8-7255941120*b^3*c^2*p^3)*q^2+(-60*b^3*p^14-933120*b^3*c*p^9-3627970560*b^3*c^2*p^4)*q-4*b^3*p^15-93312*b^3*c*p^10-725594112*b^3*c^2*p^5-1880739938304*b^3*c^3)/(q+p))/(64*3^(13/2)*(q+p)^2)+(2^(10/3)*3^(10/3)*b*(q^2*(q+p)^(2/3)+p^2*(q+p)^(2/3))+2^(13/3)*3^(10/3)*b*p*q*(q+p)^(2/3)+q^6+6*(p*q^5+p^5*q)+3*(5*p^2*q^4+5*p^4*q^2)+20*p^3*q^3+p^6)/(7776*(2*q+2*p)))^(1/3))

……………………
第4、5两个根算不出来了，直接把电脑累“休克”了……

风花飘飘

【碟】【恋】【花】【公】【式】


(x^3+b*x+c=0)乘以(x^2+f=0)等于零
【则】
(x^3+b*x+c)*(x^2+f)= x^5+(f+b)*x^3+c*x^2+b*f*x+c*f=0

m=(-108*c)

n = (4*3^(3/2)*sqrt(27*c^2+4*b^3))

m^2-n^2=((-12*b)^3)

p=(2^(2/3)*3^(1/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3))

【当p不等于0时】
q=(-2^(4/3)*3^(2/3)*b/(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3))

【当p=0时】
q=(-2^(2/3)*3^(1/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)+9*c)^(1/3))

x1
=((3^(2/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)-3*2^(2/3)*b)/(2^(1/3)*3^(4/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3)))

X2,3=(……)

X4,5=(……)        


【其中：】

b=
(-(q^5+5*p*q^4+10*p^2*q^3+10*p^3*q^2+5*p^4*q+p^5+7776*c)/((q+p)^(2/3)*(2^(10/3)*3^(10/3)*q+2^(10/3)*3^(10/3)*p)))

c= (-(q^5+5*p*q^4+10*p^2*q^3+10*p^3*q^2+(q+p)^(2/3)*(2^(10/3)*3^(10/3)*b*q+2^(10/3)*3^(10/3)*b*p)+5*p^4*q+p^5)/7776)

f=
(-(q+p)^(1/3)*(q^3+3*p*q^2+3*p^2*q+p^3)/(2^(10/3)*3^(10/3)))

f = ((3^(4/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(4/3)-2^(5/3)*3^(5/3)*b*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)+9*2^(4/3)*b^2)/(2^(2/3)*3^(8/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)) )
=( -(2^(5/3)*b*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)+(3^(5/3)*c-3^(1/6)*sqrt(27*c^2+4*b^3))*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3)-2^(4/3)*3^(1/3)*b^2)/(3*2^(2/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)))

风花飘飘

表达式
(x^2-(-(2^(5/3)*b*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)+(3^(5/3)*c-3^(1/6)*sqrt(27*c^2+4*b^3))*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3)-2^(4/3)*3^(1/3)*b^2)/(3*2^(2/3)*(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3))))*(x^3+b*x+c)

= -(sqrt(27*c^2+4*b^3)*((-2*3^(19/3)*c^3-4*3^(10/3)*b^3*c)*x^5+(-10*3^(16/3)*b*c^3-20*3^(7/3)*b^4*c)*x^3+(-2*3^(19/3)*c^4-4*3^(10/3)*b^3*c^2)*x^2+(-4*3^(16/3)*b^2*c^3-8*3^(7/3)*b^5*c)*x-4*3^(16/3)*b*c^4-8*3^(7/3)*b^4*c^2)+(2*3^(47/6)*c^4+8*3^(29/6)*b^3*c^2+4*3^(11/6)*b^6)*x^5+(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(2/3)*(sqrt(27*c^2+4*b^3)*((243*2^(1/3)*c^3+9*2^(4/3)*b^3*c)*x^3+(243*2^(1/3)*b*c^3+9*2^(4/3)*b^4*c)*x+243*2^(1/3)*c^4+9*2^(4/3)*b^3*c^2)+(-2^(1/3)*3^(13/2)*c^4-2^(7/3)*3^(7/2)*b^3*c^2-2^(4/3)*sqrt(3)*b^6)*x^3+(-2^(1/3)*3^(13/2)*b*c^4-2^(7/3)*3^(7/2)*b^4*c^2-2^(4/3)*sqrt(3)*b^7)*x-2^(1/3)*3^(13/2)*c^5-2^(7/3)*3^(7/2)*b^3*c^3-2^(4/3)*sqrt(3)*b^6*c)+(sqrt(3)*sqrt(27*c^2+4*b^3)-9*c)^(1/3)*(sqrt(27*c^2+4*b^3)*((-2^(2/3)*3^(11/3)*b^2*c^2-2^(2/3)*3^(2/3)*b^5)*x^3+(-2^(2/3)*3^(11/3)*b^3*c^2-2^(2/3)*3^(2/3)*b^6)*x-2^(2/3)*3^(11/3)*b^2*c^3-2^(2/3)*3^(2/3)*b^5*c)+(2^(2/3)*3^(31/6)*b^2*c^3+2^(2/3)*3^(19/6)*b^5*c)*x^3+(2^(2/3)*3^(31/6)*b^3*c^3+2^(2/3)*3^(19/6)*b^6*c)*x+2^(2/3)*3^(31/6)*b^2*c^4+2^(2/3)*3^(19/6)*b^5*c^2)+(10*3^(41/6)*b*c^4+40*3^(23/6)*b^4*c^2+20*3^(5/6)*b^7)*x^3+(2*3^(47/6)*c^5+8*3^(29/6)*b^3*c^3+4*3^(11/6)*b^6*c)*x^2+(4*3^(41/6)*b^2*c^4+16*3^(23/6)*b^5*c^2+8*3^(5/6)*b^8)*x+4*3^(41/6)*b*c^5+16*3^(23/6)*b^4*c^3+8*3^(5/6)*b^7*c)/(-2*3^(47/6)*c^4+sqrt(27*c^2+4*b^3)*(2*3^(19/3)*c^3+4*3^(10/3)*b^3*c)-8*3^(29/6)*b^3*c^2-4*3^(11/6)*b^6)

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