Question

If $x$ and $y$ are positive real numbers for which $(x + y)^2 + (x - y)^2 = 10$ and $(x + y)^4 + (x - y)^4 = 98$, what is the value of $xy$? Express your answer in simplest radical form.

Answer 1

$(x+y)^2+(x-y)^2=10$
$\implies 2x^2+2y^2=10$
$\implies x^2+y^2=5$
$\implies (x^2+y^2)^2=x^4+2x^2y^2+y^4=25$

$(x+y)^4+(x-y)^4=98$
$\implies 2x^4+12x^2y^2+2y^4=98$
$\implies x^4+6x^2y^2+y^4=49$
$\implies 25+4x^2y^2=49$

$4x^2y^2=24$
$\implies x^2y^2=6$
$\implies xy=\sqrt6$