Question

已知x的平方加根号2y等于根号3，y的平方加根号2x等于根号3，x≠y，求xy的值

Answer 1

由已知可知
x*x+√2
*
y=√3，y*y+√2
*
x=√3
两式相减，则有 x*x+√2
*
y
=y*y+√2
*
x

==>
(x*x-y*y)=
√2
*
(x-y)

==>
(x-y)*(x+y)=√2
*
(x-y)
[因为x≠y]

==>
x+y
= √2
两式相加，则有 x*x+√2
*
y
+
y*y+√2
*
x
=
2√3

==>
(x*x
+
y*y)+(√2
*
x+√2
*
y)
=
2√3
==>
[(x
+
y)^2
-
2xy]
+
√2
*
(x+y)
=
2√3

==>
(√2)^2
-2xy
+
√2
*
√2
=
2√3

==>
4
-
2xy
=
2√3

==>
xy
=
2
-
√3
所以
xy
=
2
-
√3
。

Answer 2

解，x^2+根号2y=根号3；y^2+根号2x=根号3。因为x≠y，且x,y都为正数，所以x^2不等于y^2，即x^2=根号2x，y^2=根号2y，解得x,y一个等于0另一个等于2开三次方根，所以xy
=0