Question

Sabendo que x². y²= 0,333... e x² + y² = 3. calcule (x+y)²

Answer 1

$(x+y)^2\\= x^2 + 2xy + y^2\\= (x^2 + y^2) + 2(xy)\:\:\:(i)$

Conforme o enunciado:
$x^2y^2 = 0,333...\\\\x^2y^2 = \cfrac{1}{3} \\\\\sqrt{x^2y^2} = \sqrt{\cfrac{1}{3}} \\\\xy = \cfrac{1}{\sqrt{3} } \\\\xy= \cfrac{\sqrt{3} \cdot 1 }{\sqrt{3} \cdot \sqrt{3} } \\\\xy = \cfrac{\sqrt{3} }{3}$

E também:
$x^2 + y^2 = 3$

Substituindo em (i):

$(3) + 2(\cfrac{\sqrt{3} }{3} )\\\\= \cfrac{9}{3} + \cfrac{2\sqrt{3} }{3}\\\\= \cfrac{2\sqrt{3} + 9}{3}$

$\boxed{(x+y)^2 = \cfrac{2\sqrt{3} + 9}{3}}$