2) \(ĐKXĐ:x\ge0;x\ne1\)
1. \(P=\dfrac{15\sqrt{x}-11}{x+2\sqrt{x}-3}+\dfrac{3\sqrt{x}-2}{1-\sqrt{x}}+\dfrac{2\sqrt{x}+3}{3+\sqrt{x}}\)
\(=\dfrac{15\sqrt{x}-11}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}-\dfrac{3\sqrt{x}-2}{\sqrt{x}-1}+\dfrac{2\sqrt{x}+3}{3+\sqrt{x}}\)
\(=\dfrac{15\sqrt{x}-11-\left(3\sqrt{x}-2\right)\left(\sqrt{x}+3\right)+\left(2\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}\)
\(=\dfrac{15\sqrt{x}-11-\left(3x+7\sqrt{x}-6\right)+\left(2x+\sqrt{x}-3\right)}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}\)
\(=\dfrac{15\sqrt{x}-11-3x-7\sqrt{x}+6+2x+\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}\)
\(=\dfrac{-x+9\sqrt{x}-8}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}\)
\(=\dfrac{\left(1-\sqrt{x}\right)\left(\sqrt{x}-8\right)}{\left(\sqrt{x}-1\right)\left(3+\sqrt{x}\right)}\)
\(=\dfrac{8-\sqrt{x}}{\sqrt{x}+3}\)
2. \(P=\dfrac{8-\sqrt{x}}{\sqrt{x}+3}=\dfrac{-\left(\sqrt{x}+3\right)+11}{\sqrt{x}+3}=-1+\dfrac{11}{\sqrt{x}+3}\)
Để P là số nguyên thì:
\(11⋮\left(\sqrt{x}+3\right)\)
\(\Rightarrow\left(\sqrt{x}+3\right)\inƯ\left(11\right)\)
\(\Rightarrow\left(\sqrt{x}+3\right)\in\left\{1;11;-1;-11\right\}\)
| \(\sqrt{x}+3\) | 1 | 11 | -1 | -11 |
| \(x\) | 64 |
Vậy \(x=64\) thì P là 1 số nguyên.
