1、[x]表示不大于x的最大整数,则方程 1/2×[x2+x]=19x+99的实数解x是 .

出处:游戏玩家inone    更新日期:2018-05-22
[x^2+x]=38x+198
38x+198≤x^2+x<38x+198+1=38x+199
(37+√2161)/2≤x<(37+√2165)/2且(37-√2165)/2<x≤(37-√2161)/2
41.74≤x<41.75或-4.75<x≤-4.74
[x1]=41,[x2]=-5
设x1=[x1]+a=41+a,0≤a<1
x2=[x2]+b=-5+b,0≤b<1,
1.当x=[x]+a=41+a,0≤a<1时,
x^2+x=41^2+82a+41+a^2+a=41^2+83a+41+a^2
[x^2+x]=[41^2+83a+41+a^2]
=41^2+41+[83a+a^2]
=38*41+38a+198
[83a+a^2]=38a+34
设a=m/38,m∈Z,0<m<38
38a+34≤83a+a^2<38a+34+1=38a+35
34≤a^2+45a<35
a^2+45a-34≥0且a^2+45a-35<0
(-45-√2165)/2<a<(-45+√2165)/2且
a≥(-45+√2161)/2
(-45+√2161)/2≤a<(-45+√2165)/2
1.486/2≤m/38<1.529/2
28.23≤m<29.06
m=29,
a=29/38
x=41+29/38=1587/38;
2.当x=[x]+b=-5+b,0≤b<1时,
x^2+x=5^2-10b+b^2-5+b=b^2-9b+20
[x^2+x]=[b^2-9b+20]
=20+[b^2-9b]
=38(-5)+38b+198
[b^2-9b]=38b-12
设b=n/38,n∈Z,0<n<38
38b-12≤b^2-9b<38b-12+1=38b-11
-12≤b^2-47b<-11
(47-√2165)/2<b<(47+√2165)/2且b≥(47+√2161)/2或b≤(47-√2161)/2
(47-√2165)/2<b≤(47-√2161)/2或
(47+√2161)/2≤b<(47+√2165)/2
0.47/2<b≤0.514/2或
(47+46.487)/2≤b<(47+46.529)/2(因b<舍去)
0.47/2<n/38≤0.514/2
8.93<n≤9.766
n=9
b=n/38=9/38
x=-5+9/38=-181/38,
综上所述x1=1587/38,x2=-181/38。