Algebra

x - y = sqrt (3)

x + y = 2              add these

2x  = 2 + sqrt (3)

x =   [ 2 + sqrt (3) ]  /2

y =     2 - x

y =  2 - [2 + sqrt (3) ] / 2  = ( 4 - [2 + sqrt 3] )  / 2  =     [ 2 - sqrt (3)] / 2

xy  =   [ 2 + sqrt (3) ]  /  2   *    [2 -sqrt (3) ] / 2  =     (4 - 3) / 4 =   1/4

Square the first equation,

$\displaystyle x^{2}+y^{2}-2xy=3,\dots(1)$

Square the  second equation,

$\displaystyle x^{2}+y^{2}+2xy =4,\dots(2)$

Subtract (1) from (2),

$\displaystyle 4xy=1, \\ xy=1/4.$