Cho P = ( √ $\frac{15√x -11}{x+2√x -3}$ + $\frac{3√x -2}{1-√x}$ – $\frac{2√x +3}{√x+3}$ )
Tìm điều kiện để P có nghĩa và rút gọn P.
Cho P = ( √ $\frac{15√x -11}{x+2√x -3}$ + $\frac{3√x -2}{1-√x}$ – $\frac{2√x +3}{√x+3}$ )
Tìm điều kiện để P có nghĩa và rút gọn P.
Đáp án:
$\begin{array}{l}
Đkxđ:\left\{ \begin{array}{l}
x + 2\sqrt x – 3 \ne 0\\
1 – \sqrt x \ne 0\\
\sqrt x + 3 \ne 0\\
x \ge 0
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right) \ne 0\\
x \ne 1\\
x \ge 0
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
x \ge 0\\
x \ne 1
\end{array} \right.\\
P = \left( {\frac{{15\sqrt x – 11}}{{x + 2\sqrt x – 3}} + \frac{{3\sqrt x – 2}}{{1 – \sqrt x }} – \frac{{2\sqrt x + 3}}{{\sqrt x + 3}}} \right)\\
= \frac{{15\sqrt x – 11}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}} – \frac{{3\sqrt x – 2}}{{\sqrt x – 1}} – \frac{{2\sqrt x + 3}}{{\sqrt x + 3}}\\
= \frac{{15\sqrt x – 11 – \left( {3\sqrt x – 2} \right)\left( {\sqrt x + 3} \right) – \left( {2\sqrt x + 3} \right)\left( {\sqrt x – 1} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \frac{{15\sqrt x – 11 – 3x – 9\sqrt x + 2\sqrt x + 6 – 2x + 2\sqrt x – 3\sqrt x + 3}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \frac{{ – 5x + 7\sqrt x – 2}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \frac{{\left( { – 5\sqrt x + 2} \right)\left( {\sqrt x – 1} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \frac{{ – 5\sqrt x + 2}}{{\sqrt x + 3}}
\end{array}$