1. xét biểu thức: P=(\(\dfrac{x+3\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\dfrac{x+\sqrt{x}}{x-1}\)) : ( \(\dfrac{1}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\))
a) Rút gọn P b)Tìm x để \(\dfrac{1}{P}-\dfrac{\sqrt{x}+1}{8}\ge1\)
2. cho biểu thức:P=\(\dfrac{\sqrt{x}}{\sqrt{x}-1}+\dfrac{3}{\sqrt{x}+1}-\dfrac{6\sqrt{x}-4}{x-1}\)
a) rút gọn P b)tìm x để P <\(\dfrac{1}{2}\)
3. cho biểu thức: P=\(\dfrac{\sqrt{x}+1}{x-1}-\dfrac{x-2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
a)rút gọn P b)tìm giá trị nhỏ nhất của biểu thức \(\dfrac{2}{p}+\sqrt{x}\)
4.cho biểu thức: Q=\(\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\)
a)rút gọn Q b)tìm giá trị nhỏ nhất của Q
c)tìm các số nguyên x để \(\dfrac{3Q}{\sqrt{x}}\) nhận giá trị nguyên
\(P=\left[\dfrac{x+3\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\dfrac{x+\sqrt{x}}{x-1}\right]\div\left(\dfrac{1}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right)\)
\(=\left[\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right]\)\(\div\left[\dfrac{\left(\sqrt{x}-1\right)-\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right]\)
\(=\dfrac{\sqrt{x}+1-\sqrt{x}}{\sqrt{x}-1}\times\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{-2}\)
\(=-\dfrac{\sqrt{x}+1}{2}\)
~ ~ ~
\(-\dfrac{2}{\sqrt{x}+1}-\dfrac{\sqrt{x}+1}{8}\ge1\)
\(\Leftrightarrow\dfrac{2}{\sqrt{x}+1}+\dfrac{\sqrt{x}+1}{8}\le-1\)
\(\Leftrightarrow16+x+2\sqrt{x}+1\le-8\sqrt{x}-8\)
\(\Leftrightarrow x+10\sqrt{x}+25\le0\)
\(\Leftrightarrow\left(\sqrt{x}+5\right)^2\le0\)
\(\Leftrightarrow\sqrt{x}\le-5\) (vô lý)
Vậy không có giá trị nào của x thoả mãn yêu cầu.
\(Q=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{x+\sqrt{x}+1}-\dfrac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+\dfrac{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\)
\(=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2\)
\(=x-\sqrt{x}+1\)
\(=\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)
Dấu "=" xảy ra khi x = 0,25
~ ~ ~
\(Q_1=\dfrac{3\left(x-\sqrt{x}+1\right)}{\sqrt{x}}=\dfrac{3\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}}+\dfrac{3}{\sqrt{x}}=3\sqrt{x}-3+\dfrac{3}{\sqrt{x}}\)
Q1 đạt giá trị nguyên
<=> \(\sqrt{x}\in\text{Ư}\left(3\right)=\left\{1;3\right\}\Leftrightarrow x\in\left\{1;9\right\}\)
\(P=\dfrac{\sqrt{x}+1}{x-1}-\dfrac{x-1}{x\sqrt{x}-1}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(=\dfrac{1}{\sqrt{x}-1}-\dfrac{x-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(=\dfrac{\left(x+\sqrt{x}+1\right)-\left(x-1\right)+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{x+\sqrt{x}+1-x+1+x-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{1}{\sqrt{x}-1}\)
~ ~ ~
\(2\div\dfrac{1}{\sqrt{x}-1}+\sqrt{x}\)
\(=2\sqrt{x}+\sqrt{x}-2\)
\(=3\sqrt{x}-2\ge-2\)
Dấu "=" xảy ra khi x = 0