Question

卡尔丹公式通过换元消掉二次项那一段怎么运作的?

Answer 1

是的,作用是消除二次项

x^3 + a1*x^2 + a2*x + a3 = 0…………(1)

设y=x+a1/3,则x=y-a1/3,代入(1)式,得：

(y-a1/3)^3 + a1*(y-a1/3)^2 + a2*(y-a1/3) + a3 = 0

y^3 - 3y^2*a1/3 + 3y*a1^2/9 - a1^3/27 + a1(y^2 - 2y*a1/3 + a1^2/9) + a2*(y-a1/3) + a3 = 0

y^3 - a1*y^2 + a1^2*y/3 - a1^3/27 + a1*y^2 - 2y*a1^2/3 + a1^3/9 + a2*y - a2*a1/3 + a3 = 0

y^3 + (-a1+a1)y^2 + (a1^2/3-2a1^2/3+a2)y + (-a1^3/27+a1^3/9-a2*a1/3+a3) = 0

y^3 + (a2-a1^2/3)y + (2a1^3/27-a1*a2/3+a3) = 0…………(2)

令p=a2-a1^2/3,q=2a1^3/27-a1*a2/3+a3,则(2)式可化为

y^3+py+q=0