x+y+z=3 x的平方+y的平方+z的平方=29 x的三次方+y的三次方+z的三次方=45 求xyz的值

x+y+z=3 x的平方+y的平方+z的平方=29 x的三次方+y的三次方+z的三次方=45 求xyz的值
x+y+z=3 x的平方+y的平方+z的平方=29 x的三次方+y的三次方+z的三次方=45 求xyz的值

x+y+z=3 x的平方+y的平方+z的平方=29 x的三次方+y的三次方+z的三次方=45 求xyz的值
X+Y+Z=3,
X^2+Y^2+Z^2=29
X^3+Y^3+Z^3=45
思路:轮称式,三元三次方程,代入法,过程比较复杂.
答案为
X=2,Y=4,Z=-3
X=2,Y=-3,Z=4
X=4,Y=2,Z=-3
X=4,Y=-3,Z=2
X=-3,Y=2,Z=4
X=-3,Y=4,Z=2

[（一式的三次方-三式）-3（一式*二式-三式）]/2 结果为-72

所有的解，包括复数域
x =

...

全部展开

所有的解，包括复数域
x =

1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (55 + 26/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) + 6*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1)^2)^(1/2)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (55 + 26/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) + 6*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1)^2)^(1/2)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 - (55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2
(55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 + 1
1 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 - (55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2
(55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 + 1


y=

1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (55 + 26/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) + 6*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1)^2)^(1/2)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (55 + 26/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) + 6*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1)^2)^(1/2)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
(55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 + 1
1 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 - (55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) - 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2
(55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 + 1
1 + 13/(12*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/4 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/4 - (55 - 13/(27^(1/2)*((13*i)/27) - 26/3)^(1/3) - 3*((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*3*i - 3*(13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 1)^2)^(1/2)/2


z =

13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1
13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) + ((27^(1/2)*13*i)/27 - 26/3)^(1/3) + 1
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 - (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))
1 - ((27^(1/2)*13*i)/27 - 26/3)^(1/3)/2 + (3^(1/2)*(13/(3*(27^(1/2)*((13*i)/27) - 26/3)^(1/3)) - ((27^(1/2)*13*i)/27 - 26/3)^(1/3))*i)/2 - 13/(6*(27^(1/2)*((13*i)/27) - 26/3)^(1/3))

收起