已知m^3+n^3=10,m^2+n^2=7,求m+n

Question

已知m^3+n^3=10,m^2+n^2=7,求m+n

Answer 1

设m+n=u,mn=v,
则m^2+n^2=u^2-2v=7,
v=(u^2-7)/2,
m^3+n^3=u(u^2-3v)=u[u^2-3(u^2-7)/2]
=u(21-u^2)/2=10,
整理得u^3-21u+20=0,
分解因式得(u-1)(u-4)(u+5)=0,
解得m+n=u=1,4,-5.

Answer 2

已知m^3+n^3=10,m^2+n^2=7,求m+n
m+n=t
m³+n³=(m+n)[(m+n) ²-3mn]=t(t ²-3mn)=10
m ²+n ²=(m+n) ²-2mn=t ²-2mn=7 mn=(t ²-7)/2
t[t ²-3(t ²-7)/2]=10
t(-t ²+21)=20
t³-21t+20=0
t³-1-20t+20=0
(t-1)(t²+t+1-20)
t=1 or t²+t-19=0 t=(-1+√77)/2 or t=(-1-√77)/2
∵m+n=1 or m+n=(-1+√77)/2 or m+n=(-1-√77)/2