Cho biểu thức: \(P=\left(\frac{3x+3}{x-9}-\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{3-\sqrt{x}}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)
với x≥0và x≠9
a) Rút gọn P
b) Tính giá trị của P khi x=20−6√11
c) Tìm x để P > \(\frac{1}{2}\)
d) Tìm giá trị nguyên của x để biếu thức Q=\(\frac{2P\sqrt{x}}{3}\) nhận giá trị nguyên
\(P=\left(\frac{3x+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\frac{2\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\frac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\left(\frac{2\sqrt{x}-2-\left(\sqrt{x}-3\right)}{\sqrt{x}-3}\right)\)
\(=\left(\frac{3x+3-2x+6\sqrt{x}-x-3\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-3}\right)\)
\(=\frac{3\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)\left(\sqrt{x}+1\right)}=\frac{3}{\sqrt{x}+3}\)
\(x=20-6\sqrt{11}=\left(\sqrt{11}-3\right)^2\Rightarrow\sqrt{x}=\sqrt{11}-3\)
\(\Rightarrow P=\frac{3}{\sqrt{11}-3+3}=\frac{3\sqrt{11}}{11}\)
\(P>\frac{1}{2}\Rightarrow\frac{3}{\sqrt{x}+3}>\frac{1}{2}\Rightarrow\sqrt{x}+3< 6\Rightarrow x< 9\)
Kết hợp ĐKXD \(\Rightarrow0\le x< 9\)
\(Q=\frac{2\sqrt{x}}{3}.\frac{3}{\left(\sqrt{x}+3\right)}=\frac{2\sqrt{x}}{\sqrt{x}+3}=2-\frac{6}{\sqrt{x}+3}\)
Để Q nguyên \(\Rightarrow\sqrt{x}+3=Ư\left(6\right)\)
Mà \(\sqrt{x}+3\ge3\Rightarrow\sqrt{x}+3=\left\{3;6\right\}\)
\(\Rightarrow\left[{}\begin{matrix}\sqrt{x}+3=3\\\sqrt{x}+3=6\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=0\\x=9\left(l\right)\end{matrix}\right.\)
