Question

已知x+根号2y＝根号3,y+根号2x＝根号3,x不等于y,求根号x分之1+根号y分之1的值

Answer 1

a=根号x
b=根号y
x!=y ==> a!=b
a^2 + (根号2)b = 根号3...(1)
b^2 + (根号2)a = 根号3...(2)
(1)-(2)
(a-b)(a+b-根号2)=0
a!=b ==> a-b!=0 ==> (a+b-根号2)
a+b=根号2...(3)
(1)+(2)
a^2+b^2+(根号2)(a+b)=2(根号3)
(a+b)^2-2ab + (根号2)(a+b)=2(根号3)...(4)
(3)代入(4)
2-2ab+2=2(根号3)
ab=2-根号3...(5)
根号x分之1+根号y分之1
=1/a+1/b
=(a+b)/ab...(6)
(3)(5)代入(6)
根号x分之1+根号y分之1
=(根号2)/(2-根号3)
=2根号2+根号6