已知3a+2b+c=24且(a^2+b^2+c^2)=ab+bc+ca,求a^3+b^2+c的值.求你啦,

Question

已知3a+2b+c=24且(a^2+b^2+c^2)=ab+bc+ca,求a^3+b^2+c的值.求你啦,

Answer 1

(a^2+b^2+c^2)=ab+bc+ca
2(a^2+b^2+c^2)=2ab+2bc+2ca
a^2-2ab+b^2+a^2-2ac+c^2+b^2-2bc+c^2=0
(a-b)^2+(b-c)^2+(a-c)^2=0
因为(a-b)^2≥0
(b-c)^2≥0
(a-c)^2≥0
所以a=b=c
3a+2b+c=24
6a=24
a=4
a^3+b^2+c
=a^3+a^2+a
=4^3+4^2+4
=64+16+4
=84