2026/065) If x and y are positive real numbers that satisfy the equation $x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y =3$ evaluate $\frac{\sqrt{x} + 2\sqrt{y} + 2014}{ 4- \sqrt{x} - 2\sqrt{y}}$

Solution 

We are given

$x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y =3$ 

Adding 1 to both sides

$x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y + 1=4$

Or $ (\sqrt{x} +  2\sqrt{y} -1)^2 = 4$ or $ (\sqrt{x} +  2\sqrt{y} -1) = 2$ as both square roots are positive

So  $ (\sqrt{x} +  2\sqrt{y}) = 3$

So  $\frac{\sqrt{x} + 2\sqrt{y} + 2014}{ 4- \sqrt{x} -  2\sqrt{y}}=  \frac{3 + 2014}{ 4- 3} = 2017$