MHB Can You Solve This Advanced Algebra Puzzle?

[anemone]

Here is this week's POTW:

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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 $\dfrac{\sqrt{x}+2\sqrt{y}+2014}{4-\sqrt{x}-2\sqrt{y}}$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution!(Cool)

1. kaliprasad
2. castor28
3. MegaMoh
4. lfdahl
5. Greg

Solution from Greg:

Spoiler

Note that the L.H.S. of the given equation factors as $(\sqrt x+2\sqrt y)(\sqrt x+2\sqrt y-2)$.

Setting $u=\sqrt x+2\sqrt y$ we obtain the quadratic $u^2-2u-3=(u-3)(u+1)=0$ and this implies that $\sqrt x+2\sqrt y=3$.

Thus we may conclude that $\frac{\sqrt{x}+2\sqrt{y}+2014}{4-\sqrt{x}-2\sqrt{y}}=2017$