Question

已知a^2=b+5,b^2=a+5(a≠b),求a^3-2ab+b^3的值

Answer 1

a²=b+5

b²=a+5

两式相减

(a+b)(a-b)=-(a-b)

a+b=-1

两式相乘

(ab)²=ab+5(a+b)+25

(ab)²-ab-20=0

(ab-5)(ab+4)=0

解得ab=5或ab=-4

a³-2ab+b³

=(a+b)(a²-ab+b²)-2ab

=(a+b)[(a+b)²-3ab]-2ab

当ab=5时a³-2ab+b³=4

当ab=-4时a³-2ab+b³=-4