二次方程与三次方程同解的关系式。

风花飘飘

风花飘飘

【同解关系式：】
若：
x^3+a*x^2+b*x+c=0与x^2-a*x+d=0有同解
则有：
(-a*x+d)*(a*x+b)^2-2*(-a*x+d)^2*(a*x+b)+c^2+(-a*x+d)^3=0
即：
4*a^3*x^3+(-8*a^2*d+4*a^2*b)*x^2+((a*b^2)-6*a*b*d+5*a*d^2)*x-d^3+2*b*d^2-b^2*d-c^2=0
解：
a1= (-(b-2*d)/(3*x))+(d^2+2*b*d+b^2)/(36*((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)*x^2)+((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)
a2= (-(b-2*d)/(3*x))+(((sqrt(3)*%i)/2-1/2)*(d^2+2*b*d+b^2))/(36*((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)*x^2)+((-1/2)-(sqrt(3)*%i)/2)*((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)
a3= (-(b-2*d)/(3*x))+(((-(sqrt(3)*%i)/2)-1/2)*(d^2+2*b*d+b^2))/(36*((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)*x^2)+((sqrt(3)*%i)/2-1/2)*((d^3+3*b*d^2+3*b^2*d+27*c^2+b^3)/(216*x^3)+(c*sqrt(2*d^3+6*b*d^2+6*b^2*d+27*c^2+2*b^3))/(8*3^(3/2)*x^3))^(1/3)

b有唯一解 = (((c*sqrt(a*x-d)-2*a^2*x^2+3*a*d*x-d^2)/(a*x-d)))

c1 = (sqrt(a*x-d)*(2*a*x-d+b))
c 2= (((-2*a*x)+d-b)*sqrt(a*x-d))
d1= ((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(a^2*x^2+2*a*b*x+b^2)/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-((-5*a*x)-2*b)/3

d2= ((sqrt(3)*%i)/2-1/2)*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(((-(sqrt(3)*%i)/2)-1/2)*(a^2*x^2+2*a*b*x+b^2))/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-((-5*a*x)-2*b)/3
d3= ((-1/2)-(sqrt(3)*%i)/2)*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3)+(((sqrt(3)*%i)/2-1/2)*(a^2*x^2+2*a*b*x+b^2))/(9*((c*sqrt(4*a^3*x^3+12*a^2*b*x^2+12*a*b^2*x+27*c^2+4*b^3))/(2*3^(3/2))-(2*a^3*x^3+6*a^2*b*x^2+6*a*b^2*x+27*c^2+2*b^3)/54)^(1/3))-((-5*a*x)-2*b)/3

风花飘飘

【同解关系式：】
  若：
    x^3+a*x^2+b*x+c=0 与 x^2-a*x+d=0 有同解
  则有：
    (-a*x+d)*(a*x+b)^2-2*(-a*x+d)^2*(a*x+b)+c^2+(-a*x+d)^3=0
  即：
   4*a^3*x^3+(-8*a^2*d+4*a^2*b)*x^2
+((a*b^2)-6*a*b*d+5*a*d^2)*x-d^3+2*b*d^2-b^2*d-c^2=0