Question

设m＞n＞0，m^2+n^2=4mn，则m^2-n^2/mn的值等于

Answer 1

解：m^2+n^2=4mn得m/n+n/m=4 设m/n=x 则x+1/x=4 (x>1)
则（x-1/x)^2=(x+1/x)^2-4=16-4=12
故x-1/x=2√3
则m^2-n^2/mn=m/n-n/m=x-1/x=2√3

Answer 2

(m+n)²=m²+n²+2mn=4mn+2mn=6mn
(m-n)²=m²+n²-2mn=4mn-2mn=2mn
(m²-n²)²=(m+n)²(m-n)²=6mn×2mn=12m²n²又∵m＞n＞0
∴m²-n²=2√3mn
∴（m²-n²)/mn=2√3

Answer 3

(m+n)^2=m^2+n^2+2mn=4mn+2mn=6mn
(m-n)^2=m^2+n^2-2mn=4mn-2mn=2mn
(m^2-n^2)^2=(m+n)^2(m-n)^2=6mn*2mn=12m^2n^2
m^2-n^2=2√3mn
m^2-n^2/mn=2√3