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【另类解一元三次方程】通过化掉一次项直接求解

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发表于 2017-5-7 00:02 | 显示全部楼层 |阅读模式
本帖最后由 风花飘飘 于 2018-8-10 18:49 编辑

x^3+ax^2+bx+c=0化为:y^3+dY^2+e=0?

设 : x=y+k

则 :(y+k)^3+a(y+k)^2+b(y+k)+c=0

即: y^3+(3*k+a)*y^2+(3*k^2+2*a*k+b)*y+k^3+a*k^2+b*k+c=0

令:3*k^2+2*a*k+b=0

取: k = (((a^2-3*b)^(1/2)-a)/3)

由:(y+k)^3+a(y+k)^2+b(y+k)+c=0

得:(27*y^3+sqrt(a^2-3*b)*(27*y^2+6*b-2*a^2)+27*c-9*a*b+2*a^3)/27=0

取:y = (sqrt(729*c^2+((324*sqrt(a^2-3*b)-486*a)*b+108*(a^2-3*b)^(3/2)-108*a^2*sqrt(a^2-3*b)+108*a^3)*c+108*b^3+(36*(a^2-3*b)-108*a*sqrt(a^2-3*b)-27*a^2)*b^2+((-24*a^2*(a^2-3*b))+24*(a^2-3*b)^2-36*a*(a^2-3*b)^(3/2)+60*a^3*sqrt(a^2-3*b))*b+4*a^4*(a^2-3*b)+4*(a^2-3*b)^3-8*a^2*(a^2-3*b)^2+8*a^3*(a^2-3*b)^(3/2)-8*a^5*sqrt(a^2-3*b))/54-(27*c+6*sqrt(a^2-3*b)*b-9*a*b+2*(a^2-3*b)^(3/2)-2*a^2*sqrt(a^2-3*b)+2*a^3)/54)^(1/3)+(a^2-3*b)/(9*(sqrt(729*c^2+((324*sqrt(a^2-3*b)-486*a)*b+108*(a^2-3*b)^(3/2)-108*a^2*sqrt(a^2-3*b)+108*a^3)*c+108*b^3+(36*(a^2-3*b)-108*a*sqrt(a^2-3*b)-27*a^2)*b^2+((-24*a^2*(a^2-3*b))+24*(a^2-3*b)^2-36*a*(a^2-3*b)^(3/2)+60*a^3*sqrt(a^2-3*b))*b+4*a^4*(a^2-3*b)+4*(a^2-3*b)^3-8*a^2*(a^2-3*b)^2+8*a^3*(a^2-3*b)^(3/2)-8*a^5*sqrt(a^2-3*b))/54-(27*c+6*sqrt(a^2-3*b)*b-9*a*b+2*(a^2-3*b)^(3/2)-2*a^2*sqrt(a^2-3*b)+2*a^3)/54)^(1/3))-sqrt(a^2-3*b)/3

则:
x=y+k
=(sqrt(729*c^2+((324*sqrt(a^2-3*b)-486*a)*b+108*(a^2-3*b)^(3/2)-108*a^2*sqrt(a^2-3*b)+108*a^3)*c+108*b^3+(36*(a^2-3*b)-108*a*sqrt(a^2-3*b)-27*a^2)*b^2+((-24*a^2*(a^2-3*b))+24*(a^2-3*b)^2-36*a*(a^2-3*b)^(3/2)+60*a^3*sqrt(a^2-3*b))*b+4*a^4*(a^2-3*b)+4*(a^2-3*b)^3-8*a^2*(a^2-3*b)^2+8*a^3*(a^2-3*b)^(3/2)-8*a^5*sqrt(a^2-3*b))/54-(27*c+6*sqrt(a^2-3*b)*b-9*a*b+2*(a^2-3*b)^(3/2)-2*a^2*sqrt(a^2-3*b)+2*a^3)/54)^(1/3)+(a^2-3*b)/(9*(sqrt(729*c^2+((324*sqrt(a^2-3*b)-486*a)*b+108*(a^2-3*b)^(3/2)-108*a^2*sqrt(a^2-3*b)+108*a^3)*c+108*b^3+(36*(a^2-3*b)-108*a*sqrt(a^2-3*b)-27*a^2)*b^2+((-24*a^2*(a^2-3*b))+24*(a^2-3*b)^2-36*a*(a^2-3*b)^(3/2)+60*a^3*sqrt(a^2-3*b))*b+4*a^4*(a^2-3*b)+4*(a^2-3*b)^3-8*a^2*(a^2-3*b)^2+8*a^3*(a^2-3*b)^(3/2)-8*a^5*sqrt(a^2-3*b))/54-(27*c+6*sqrt(a^2-3*b)*b-9*a*b+2*(a^2-3*b)^(3/2)-2*a^2*sqrt(a^2-3*b)+2*a^3)/54)^(1/3))-sqrt(a^2-3*b)/3+(((a^2-3*b)^(1/2)-a)/3)

化简即得方程之“真根”。呵呵……
 楼主| 发表于 2017-5-11 08:47 | 显示全部楼层
本帖最后由 风花飘飘 于 2017-5-14 19:05 编辑

x^3+ax^2+bx+c=0化为:y^3+dY^2+e=0?

设 : x=y+k

则 :(y+k)^3+a(y+k)^2+b(y+k)+c=0

即: y^3+(3*k+a)*y^2+(3*k^2+2*a*k+b)*y+k^3+a*k^2+b*k+c=0

令:3*k^2+2*a*k+b=0

取: k = (-a/3)

取:a=((3*b)^(1/2))

由:(y+k)^3+a(y+k)^2+b(y+k)+c=0

得:(y+(-a/3))^3+a(y+(-a/3))^2+b(y+(-a/3))+c=0

取:y=(((a^3-27*c)^(1/3))/3)

则:x=Y+k=((((a^3-27*c)^(1/3))/3)-a/3)

即:x^3+ax^2+bx+c=0

((((a^3-27*c)^(1/3))/3)-a/3)^3+a*((((a^3-27*c)^(1/3))/3)-a/3)^2+b*((((a^3-27*c)^(1/3))/3)-a/3)+c=0

得:((3*b-a^2)*(a^3-27*c)^(1/3)-3*a*b+a^3)/9=0

得:a1=((a^3-27*c)^(1/3))

       a2=(3*b)^(1/2)
       a2=-(3*b)^(1/2)

       b=(a^2/3)

代入:x^3+ax^2+bx+c=x^3+((a^3-27*c)^(1/3))x^2+(a^2/3)x+c





(待续……)
 楼主| 发表于 2017-5-12 08:19 | 显示全部楼层
(((5*x+2)*2^(3/5)*(5^(3/2)+11)^(2/5)/(5^(1/2)+3))^(1/5))^5-5*(((5*x+2)*2^(3/5)*(5^(3/2)+11)^(2/5)/(5^(1/2)+3))^(1/5))-2=0
 楼主| 发表于 2017-5-14 12:27 | 显示全部楼层
x^5-5*(((5*x+2)*2^(3/5)*(5^(3/2)+11)^(2/5)/(5^(1/2)+3))^(1/5))-2=0
 楼主| 发表于 2017-5-14 18:16 | 显示全部楼层
解:X^5-5*X-3=0
      
把方程分解因式:
      
X^5-5*X-3
=
((-((9*sqrt(7)-13*sqrt(3))^(1/3)*((-2^(5/3)*3^(7/6)*5^(1/3)*x^3)+(2^(5/3)*3^(7/6)*5^(5/6)+2^(8/3)*3^(1/6)*5^(1/3))*x^2+((-2^(2/3)*3^(1/6)*5^(5/6))-7*2^(2/3)*3^(1/6)*5^(1/3))*x-2^(2/3)*3^(1/6)*5^(5/6)-2^(2/3)*3^(7/6)*5^(1/3))+(9*sqrt(7)-13*sqrt(3))^(2/3)*(2^(4/3)*5^(2/3)*x^2+((-2^(4/3)*5^(7/6))-2^(4/3)*5^(2/3))*x+2^(1/3)*5^(7/6)+3*2^(1/3)*5^(2/3))-20*3^(1/3)*x^2+(4*3^(1/3)*5^(3/2)+20*3^(1/3))*x-2*3^(1/3)*5^(3/2)-10*3^(4/3))/(2^(5/3)*3^(7/6)*5^(1/3)*(9*sqrt(7)-13*sqrt(3))^(1/3))))
*
(((9*sqrt(7)-13*sqrt(3))^(1/3)*((2^(1/3)*3^(10/3)*5^(1/3)*sqrt(7)-13*2^(1/3)*3^(11/6)*5^(1/3))*x^2+(2^(4/3)*3^(7/3)*5^(1/3)*sqrt(7)-13*2^(4/3)*3^(5/6)*5^(1/3))*x+2^(4/3)*3^(7/3)*5^(1/3)*sqrt(7)-13*2^(4/3)*3^(5/6)*5^(1/3))+(9*sqrt(7)-13*sqrt(3))^(2/3)*((3^(13/6)*5^(2/3)*sqrt(7)-13*3^(2/3)*5^(2/3))*x+3^(7/6)*5^(2/3)*sqrt(7)-3^(2/3)*5^(2/3))+(195*2^(2/3)-5*2^(2/3)*3^(5/2)*sqrt(7))*x-5*2^(5/3)*3^(3/2)*7^(3/2)+15*2^(20/3))/((9*sqrt(7)-13*sqrt(3))^(1/3)*(2^(1/3)*3^(10/3)*5^(1/3)*sqrt(7)-13*2^(1/3)*3^(11/6)*5^(1/3))))

令:
0=((-((9*sqrt(7)-13*sqrt(3))^(1/3)*((-2^(5/3)*3^(7/6)*5^(1/3)*x^3)+(2^(5/3)*3^(7/6)*5^(5/6)+2^(8/3)*3^(1/6)*5^(1/3))*x^2+((-2^(2/3)*3^(1/6)*5^(5/6))-7*2^(2/3)*3^(1/6)*5^(1/3))*x-2^(2/3)*3^(1/6)*5^(5/6)-2^(2/3)*3^(7/6)*5^(1/3))+(9*sqrt(7)-13*sqrt(3))^(2/3)*(2^(4/3)*5^(2/3)*x^2+((-2^(4/3)*5^(7/6))-2^(4/3)*5^(2/3))*x+2^(1/3)*5^(7/6)+3*2^(1/3)*5^(2/3))-20*3^(1/3)*x^2+(4*3^(1/3)*5^(3/2)+20*3^(1/3))*x-2*3^(1/3)*5^(3/2)-10*3^(4/3))/(2^(5/3)*3^(7/6)*5^(1/3)*(9*sqrt(7)-13*sqrt(3))^(1/3))))
      
求根。

再令:
0=(((9*sqrt(7)-13*sqrt(3))^(1/3)*((2^(1/3)*3^(10/3)*5^(1/3)*sqrt(7)-13*2^(1/3)*3^(11/6)*5^(1/3))*x^2+(2^(4/3)*3^(7/3)*5^(1/3)*sqrt(7)-13*2^(4/3)*3^(5/6)*5^(1/3))*x+2^(4/3)*3^(7/3)*5^(1/3)*sqrt(7)-13*2^(4/3)*3^(5/6)*5^(1/3))+(9*sqrt(7)-13*sqrt(3))^(2/3)*((3^(13/6)*5^(2/3)*sqrt(7)-13*3^(2/3)*5^(2/3))*x+3^(7/6)*5^(2/3)*sqrt(7)-3^(2/3)*5^(2/3))+(195*2^(2/3)-5*2^(2/3)*3^(5/2)*sqrt(7))*x-5*2^(5/3)*3^(3/2)*7^(3/2)+15*2^(20/3))/((9*sqrt(7)-13*sqrt(3))^(1/3)*(2^(1/3)*3^(10/3)*5^(1/3)*sqrt(7)-13*2^(1/3)*3^(11/6)*5^(1/3))))
求根。

完毕!
 楼主| 发表于 2022-5-15 10:57 | 显示全部楼层
好东西要顶一个。
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