已知m,n满足不等式m平方=5n-2n平方=5m-2,且m≠n,试求m立方-5mn-2n的值

Question

已知m,n满足不等式m平方=5n-2
n平方=5m-2,且m≠n,试求m立方-5mn-2n的值

Answer 1

m^2=5n-2
n^2=5m-2
相减
m^2-n^2=5n-5m
(m+n)(m-n)+5(m-n)=0
(m-n)(m+n+5)=0
m不等于n,所以m-n不等于0
所以m+n+5=0
m+n=-5
m^2=5n-2
m^3=m^2*m=m(5n-2)=5mn-2m
所以
m^3-5mn-2n
=5mn-2m-5mn-2n
=-2(m+n)
=-2*(-5)
=10