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+

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+

Answer 1

x²+y²+16z² = 4x⁴+y⁴+9z⁴
x²(4x² - 1) + y²(y² - 1) + z²(9z² - 16) = 0 ……①
x²+y²+16z² = 2x³+y³+12z³
x²(2x - 1) +y²(y-1) + z²(12z - 16) = 0 ……②
x²(4x² - 4x+1) + y²(y² - 2y+1) +z²(9z² - 24z +16) = 0
x²(2x-1)² +y²(y-1)² + z²(3z - 4)² = 0
∵x,y,z均不为零
x = 1/2,y = 1,z= 4/3