设√5+1/(√5-1)的整数部分为a，小数部分为b，求a^2+1/2*ab+b^2的值

[5^(1/2) + 1]/[5^(1/2) - 1]

= [5^(1/2) + 1]^2/[5 - 1]

= [6 + 2*5^(1/2)]/4

= [3 + 5^(1/2)]/2

= 2 + [5^(1/2) - 1]/2

5^(1/2) - 1 > 0.

0 < [5^(1/2) - 1]/2

= [5 - 1]/{2[5^(1/2) + 1]}

= 2/[5^(1/2) + 1]

< 2/[2 + 1]

= 2/3 < 1

所以，
a = 2,

b = [5^(1/2) - 1]/2

a^2 + 1/2*ab + b^2

= 4 + 1/2*2*[5^(1/2) - 1]/2 + {[5^(1/2) - 1]/2}^2

= 4 + [5^(1/2) - 1]/2 + [6 - 2*5^(1/2)]/4

= 4 + [5^(1/2) - 1]/2 + [3 - 5^(1/2)]/2

= 4 - 1/2 + 3/2

= 5

a+b=√5+1/(√5-1);

a^2+1/2*ab+b^2 = (a+b)^2 =[√5+1/(√5-1)]^2 = (3+√5)/(3-√5);

看错了，呵呵，作废