Question

已知m^2=n+2，n^2=m+2，求m^3-2mn+n^3

m ≠ n

Answer 1

m^2=n+2，n^2=m+2
m²-n²=(n-m)
(m-n)(m+n)=(n-m)
(m-n)(m+n)+(m-n)=0
(m-n)(m+n+1)=0
① 当m-n=0时，即m=n，解得m=n=2（或者 -1）
② 当m+n+1=0时，m+n = -1

m³=m×m²=m（n+2）=mn+2m
n³=n×n²=n（m+2）=mn+2n

m^3-2mn+n^3
=(mn+2m)-2mn+(mn+2n)
=2(m+n)
=8或者-4或者-1

Answer 2

m^3-2mn+n^3= 2（m+n）
已知m^2=n+2，n^2=m+2，
m^2-n^2=n+2-m-2

(m+n)(m-n)=-(m-n)
当m-n= 0时 m=n=2或者-1 则 值为8 或者-4
当m-n不等于0时，m+n=-1 则值为-2

Answer 3

m^2=n+2,n^2=m+2(m≠0)
m^2-n^2=n+2-m-2
(m+n)(m-n)=-(m-n)
m+n=-1
m^3-2mn+n^3
=m(n+2)-2mn+n(m+2)
=mn+2m-2mn+mn+2n
=2m+2n
=2(m+n)
=2*(-1)
=-2