已知x,y,z均不为0,并且x^2+y^2+16z^2=2x^3+y^3+12z^3=4x^4+y^4+9z^4,则(2x-1)^2+(2y-2)^2+(2z-3)^2=?x^2表示x的平方a^b表示a的b次方

Question

已知x,y,z均不为0,并且x^2+y^2+16z^2=2x^3+y^3+12z^3=4x^4+y^4+9z^4,则 (2x-1)^2+(2y-2)^2+(2z-3)^2=?
x^2表示x的平方
a^b表示a的b次方

Answer 1

x^2+y^2+16z^2=4x^4+y^4+9z^4,有(4x^2-1)x^2+(y^2-1)y^2+(9z^2-16)z^2=0[(1)式],
x^2+y^2+16z^2=2x^3+y^3+12z^3,有（2x-1）x^2+（y-1）y^2+（12z-16）z^2=0[(2)式],
（1）式－（2）式*2
（4x^2－4x+1)x^2+(y^2-2y+1)y^2+(9z^2-24z+16)z^2=0
[(2x-1)*x]^2+[(y-1)*y]^2+[(3z-4)*z]^2=0,
所以(2x-1)*x＝0,（y-1)*y＝0,（3z-4)*z＝0,因为x,y,z均不为0,
x=1/2,y=1,z=4/3,
(2x-1)^2+(2y-2)^2+(2z-3)^2=(2*4/3-3)^2=1/9