Question

若a+b=m,a^3+b^3=n,且m,n不等于0,求a^2+b^2的值

若a+b=m,a^3+b^3=n,且m,n不等于0,求a^2+b^2的值。

Answer 1

a³+b³=(a+b)(a²-ab+b²)
a³+b³=(a+b)[(a+b)²-3ab]
n=m(m²-3ab)，求得ab=[m²-(n/m)]/3

a²+b²=(a+b)²-2ab=m²-(2/3)[m²-(n/m)]=[m²+2(n/m)]/3=(m³+2n)/(3m)

Answer 2

a^2+b^2=x
2ab=m^2-x
ab=(m^2-x)/2
n=m[x-(m^2-x)/2]
2n=3mx-m^3
x=(2n+m^3)/3m

Answer 3

a^3+b^3=(a+b)(a^2-ab+b^2)=m(a^2+2ab+b^2-3ab)=m[(a+b)^2-3ab]=m(m^2-3ab)=n
ab=(m^3-n)/3m
a^2+b^2=(a+b)^2-2ab=m^2-2(m^3-n)/3m=(m^3+2n)/3m