If x - y = sqrt(3) and x + y = 2, then xy = ?
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),
\(\displaystyle 4xy=1, \\ xy=1/4.\)