Cho biểu thức: A= $\frac{15√x -11}{x+2√x -3}$ + $\frac{3√x -2}{1-√x}$ – $\frac{2√x+3}{3+√x}$
a) Tìm x để A có nghĩa.
b) Rút gọn biểu thức A.
CỨUuuuuuuu~
Cho biểu thức: A= $\frac{15√x -11}{x+2√x -3}$ + $\frac{3√x -2}{1-√x}$ – $\frac{2√x+3}{3+√x}$ a) Tìm x để A có nghĩa. b) Rút gọn biểu thức A. CỨU
By Remi
Đáp án:
$A = \dfrac{2 – 5\sqrt x}{\sqrt x + 3} \qquad (x \geq 0; \, x \ne 1)$
Giải thích các bước giải:
$\begin{array}{l}A = \dfrac{15\sqrt x – 11}{x +2\sqrt x – 3} + \dfrac{3\sqrt x – 2}{1 – \sqrt x} – \dfrac{2\sqrt x +3}{3 + \sqrt x}\\ a)\,\,\text{A có nghĩa}\,\,\Leftrightarrow \begin{cases}x \geq 0\\x + 2\sqrt x – 3 \ne 0\end{cases} \Leftrightarrow \begin{cases} x \geq 0\\x \ne 1\end{cases}\\ b) A = \dfrac{15\sqrt x – 11}{(\sqrt x +3)(\sqrt x -1)} – \dfrac{(3\sqrt x – 2)(\sqrt x + 3)}{(\sqrt x +3)(\sqrt x -1)} – \dfrac{(2\sqrt x + 3)(\sqrt x – 1)}{(\sqrt x +3)(\sqrt x -1)}\\ \to A = \dfrac{15\sqrt x – 11 – (3x + 9\sqrt x – 2\sqrt x – 6) – (2x – 2\sqrt x + 3\sqrt x – 3)}{(\sqrt x +3)(\sqrt x -1)}\\ \to A = \dfrac{-5x + 7\sqrt x – 2}{(\sqrt x +3)(\sqrt x -1)}\\ \to A = \dfrac{-(\sqrt x – 1)(5\sqrt x – 2)}{(\sqrt x +3)(\sqrt x -1)}\\ \to A = \dfrac{2 – 5\sqrt x}{\sqrt x + 3}\end{array}$
Đáp án:
$\begin{array}{l}
a)Dkxd:\left\{ \begin{array}{l}
x \ge 0\\
1 – \sqrt x \ne 0
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
x \ge 0\\
x \ne 1
\end{array} \right.\\
b)x \ge 0;x \ne 1\\
A = \dfrac{{15\sqrt x – 11}}{{x + 2\sqrt x – 3}} + \dfrac{{3\sqrt x – 2}}{{1 – \sqrt x }} – \dfrac{{2\sqrt x + 3}}{{3 + \sqrt x }}\\
= \dfrac{{15\sqrt x – 11}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}} + \dfrac{{2 – 3\sqrt x }}{{\sqrt x – 1}} – \dfrac{{2\sqrt x + 3}}{{\sqrt x + 3}}\\
= \dfrac{{15\sqrt x – 11 + \left( {2 – 3\sqrt x } \right)\left( {\sqrt x + 3} \right) – \left( {2\sqrt x + 3} \right)\left( {\sqrt x – 1} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \dfrac{{15\sqrt x – 11 – 3x – 7\sqrt x + 6 – \left( {2x + \sqrt x – 3} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \dfrac{{ – 5x + 7\sqrt x – 2}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= \dfrac{{ – \left( {5\sqrt x – 7\sqrt x + 2} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= – \dfrac{{\left( {\sqrt x – 1} \right)\left( {5\sqrt x – 2} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 3} \right)}}\\
= – \dfrac{{5\sqrt x – 2}}{{\sqrt x + 3}}\\
= \dfrac{{2 – 5\sqrt x }}{{\sqrt x + 3}}
\end{array}$