Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\ge0\\\sqrt{y}=b\ge0\end{matrix}\right.\)
\(P=a^2-2ab+3b^2-2a+2009,5\)
\(P=\frac{1}{3}\left(9b^2-6ab+a^2\right)+\frac{2}{3}\left(a^2-3a+\frac{9}{4}\right)+2008\)
\(P=\frac{1}{3}\left(3b-a\right)^2+\frac{2}{3}\left(a-\frac{3}{2}\right)^2+2008\ge2008\)
\(P_{min}=2008\) khi \(\left\{{}\begin{matrix}a-\frac{3}{2}=0\\3b-a=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=\frac{3}{2}\\b=\frac{1}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=\frac{9}{4}\\y=\frac{1}{4}\end{matrix}\right.\)
