已知实数mn满足m+1/2的绝对值与n^2-n+1/4互为相反数,化简并求值：（3m-2n）^2+(3m+2n)(3m-2n)-9(m+n)^2

题目

已知实数mn满足m+1/2的绝对值与n^2-n+1/4互为相反数,化简并求值：（3m-2n）^2+(3m+2n)(3m-2n)-9(m+n)^2

答案

∵m+1/2的绝对值非负数,n^2-n+1/4=（n-1/2)^2也非负数
又：m+1/2的绝对值与n^2-n+1/4互为相反数
∴m+1/2的绝对值=0,n^2-n+1/4=（n-1/2)^2=0
∴m=-1/2,n=1/2
∴m+n=0
（3m-2n）^2+(3m+2n)(3m-2n)-9(m+n)^2
= (3m-2n)(3m-2n+3m+2n)-9(m+n)^2
= 6m(3m-2n)-9(m+n)^2
= 6*(-1/2)*{3*(-1/2)-2*1/2}-9*0
= -3*(-5/2)-0
= 15/2