已知实数a,b满足a-3a^(2/3)+5a^(1/3)=1,b-3b^(2/3)+5b^(1/3)=5,求a^(1/3
2个回答

令a^(1/3)=x, b^(1/3)=y 化简

a-3a^(2/3)+5a^(1/3)=1

x^3-3x^2+5x-1=0

x(x-1)(x-2)+3(x-1)=-2

(x-1)((x-1)^2+2)=-2

(x-1)^3+2(x-1)=-2

b-3b^(2/3)+5b^(1/3)=5

y^3-3y^2+5y-5=0

y(y-1)(y-2)+3(y-1)=2

(y-1)^3+2(y-1)=2

函数f(x')=x'^3+2x'=x'(x'^2+2)

f(-x')=-f(x')

f(x')为原点对称的奇函数

上述方程组可看成 f(x')=2 f(x')=-2

即 y-1=x0

x-1=-x0

x-1+y-1=x0+(-x0)=0

即x+y=2

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