Cho biểu thức
[tex]P=\frac{x}{(\sqrt{x}+\sqrt{y})(1-\sqrt{y})}-\frac{y}{(\sqrt{x}+\sqrt{y})(1+\sqrt{x})}-\frac{xy}{(\sqrt{x}+1)(1-\sqrt{y})}[/tex]
a) Rút gọn P
b) Tìm x, y nguyên sao cho P = 2
a) ĐK: $x\ge 0;y\ge 0;y\ne 1$
$P=\dfrac{x(1+\sqrt x)-y(1-\sqrt y)-xy(\sqrt x+\sqrt y)}{(\sqrt x+\sqrt y)(1+\sqrt x)(1-\sqrt y)}$
Biến đổi tử thức của $P$ ta được:
$x+x\sqrt x-y+y\sqrt y-xy(\sqrt x+\sqrt y)
\\=(x-y)+(x\sqrt x+y\sqrt y)-xy(\sqrt x+\sqrt y)
\\=(\sqrt x+\sqrt y)(\sqrt x-\sqrt y)+(\sqrt x+\sqrt y)(x-\sqrt{xy}+y)-xy(\sqrt x+\sqrt y)
\\=(\sqrt x+\sqrt y)(\sqrt x-\sqrt y+x-\sqrt{xy}+y-xy)
\\=(\sqrt x+\sqrt y)[(\sqrt x+x)-(\sqrt y+\sqrt{xy})+(y-xy)]
\\=(\sqrt x+\sqrt y)[\sqrt x(1+\sqrt x)-\sqrt y(1+\sqrt{x})+y(1-\sqrt x)(1+\sqrt x)]
\\=(\sqrt x+\sqrt y)(1+\sqrt x)(\sqrt x-\sqrt y+y-y\sqrt x)
\\=(\sqrt x+\sqrt y)(1+\sqrt x)[(\sqrt x-y\sqrt x)-(\sqrt y-y)]
\\=(\sqrt x+\sqrt y)(1+\sqrt x)[\sqrt x(1+\sqrt y)(1-\sqrt y)-\sqrt y(1-\sqrt y)]
\\=(\sqrt x+\sqrt y)(1+\sqrt x)(1-\sqrt y)(\sqrt x+\sqrt{xy}-\sqrt y)$
$\Rightarrow P=\sqrt x+\sqrt{xy}-\sqrt y$
b) $P=2\Leftrightarrow \sqrt x+\sqrt{xy}-\sqrt y=2$
$\Leftrightarrow \sqrt{xy}-\sqrt y+\sqrt x-1=1$
$\Leftrightarrow \sqrt y(\sqrt x-1)+(\sqrt x-1)=1$
$\Leftrightarrow (\sqrt x-1)(\sqrt y+1)=1$
Mà $x,y\in \mathbb{Z}\Rightarrow \left\{\begin{matrix}\sqrt x-1=1\\ \sqrt y+1=1\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix}x=4\\ y=0\end{matrix}\right.$ (TM)
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