Question

设x=根号n+1 - 根号n / 根号n+1 + 根号n，y=根号n+1 +根号n / 根号n+1 - 根号n，

设x=根号n+1 - 根号n / 根号n+1 + 根号n，y=根号n+1 +根号n / 根号n+1 - 根号n，若10x^2+10y^2-12xy+80=2008,n为何值？？？？？？？

Answer 1

x=(√n+1 -√n)/(√n+1 +√n) 分母有理化x=2n+1+2√(n^2+n)
y=(√n+1 +√n)/(√n+1 -√n) 分母有理化y=2n+1-2√(n^2+n)
10x^2+10y^2-12xy+80=2008
10(x-y)^2+8xy+80=2008
x,y带入其中10(x-y)^2=10(2n+1+2√(n^2+n)-(2n+1-2√(n^2+n))=160n^2+160n
x,y带入其中8xy=8(2n+1+2√(n^2+n))(2n+1-2√(n^2+n))由平方差公式
=8（2n+1）^2-8(2√(n^2+n))^2=8
160n^2+160n+8+80=2008
n^2+n-12=0
n1=3 n2=-4