ĐKXĐ \(x\ge0,x\ne4\)
a) \(B=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)-\left(\sqrt{x}+3\right)\cdot\left(\sqrt{x}+1\right)-\left(x+5\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{x-2\sqrt{x}-\sqrt{x}+2-\left(x+\sqrt{x}+3\sqrt{x}+3\right)-x-5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{-x-7\sqrt{x}-6}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}=\dfrac{-\left(\sqrt{x}+1\right)\cdot\left(\sqrt{x}+6\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}+6}{2-\sqrt{x}}\)
b) B > -1 <=> B + 1 > 0.
\(\Leftrightarrow\dfrac{\sqrt{x}+6}{2-\sqrt{x}}+1>0\Leftrightarrow\dfrac{8}{2-\sqrt{x}}>0\)
=> \(2-\sqrt{x}>0\Leftrightarrow\sqrt{x}< 2\Rightarrow x< 4\)
Vậy \(0\le x< 4\) thì B > -1.
c) \(B=\dfrac{\sqrt{x}+6}{2-\sqrt{x}}=-1-\dfrac{8}{2-\sqrt{x}}\in Z\)
\(\Rightarrow2-\sqrt{x}\inƯ_{\left(8\right)}=\left\{\pm1;\pm2;\pm4;\pm8\right\}\)
\(\Rightarrow\sqrt{x}\in\left\{1;3;0;4;-2;6;-6;10\right\}\)
\(\Rightarrow x\in\left\{1;9;0;16;36;100\right\}\)thì \(B\in Z\)
a) đk : \(x\ne4;x\ge0\)
B = \(\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{\sqrt{x}+3}{\sqrt{x}-2}-\dfrac{x+5}{x-\sqrt{x}-2}\)
B = \(\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)-\left(\sqrt{x}+3\right)\left(\sqrt{x}+1\right)-\left(x+5\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
B = \(\dfrac{x-2\sqrt{x}-\sqrt{x}+2-\left(x+\sqrt{x}+3\sqrt{x}+3\right)-x-5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
B = \(\dfrac{x-2\sqrt{x}-\sqrt{x}+2-x-\sqrt{x}-3\sqrt{x}-3-x-5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
B = \(\dfrac{-x-7\sqrt{x}-6}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\) = \(\dfrac{\left(-\sqrt{x}-6\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
B = \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}\)
a) B = \(\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{\sqrt{x}+3}{\sqrt{x}-2}-\dfrac{x+5}{x-\sqrt{x}+2}\) ( đk: x \(\ge\) 0; x\(\ne\)4)
<=> B = \(\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)-\left(\sqrt{x}+3\right)\left(\sqrt{x}+1\right)-x-5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
<=> B = \(\dfrac{x-3\sqrt{x}+2-x-4\sqrt{x}-3-x-5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
<=> B = \(\dfrac{x+7\sqrt{x}+6}{-\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}=\dfrac{-\left(\sqrt{x}+1\right)\left(\sqrt{x}+6\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
=> B = \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}\)
b) Để B > -1 => \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}>-1\)
=> \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}+1>0\)
<=> \(\dfrac{-8}{\sqrt{x}-2}\) > 0 => \(\sqrt{x}-2< 0\) => \(x< 4\)
Đối chiếu với điều kiện ta được: \(0\le x< 4\)
c) Để B \(\in\) Z => \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}\) \(\in\) Z
Mà \(\dfrac{-\sqrt{x}-6}{\sqrt{x}-2}\) = \(\dfrac{-\left(\sqrt{x}-2\right)-8}{\sqrt{x}-2}=-1-\dfrac{8}{\sqrt{x}-2}\)
=> \(\dfrac{8}{\sqrt{x}-2}\in Z\) => 8 \(⋮\) \(\sqrt{x}-2\)
=> \(\sqrt{x}\inƯ\left(8\right)=\left\{\pm1;\pm2;\pm4;\pm8\right\}\)
=> \(\sqrt{x}\in\left\{3;1;4;0;6;-4;10;-6\right\}\)
Mà \(x\ge0\) => \(x\in\left\{9;0;1;16;36;100\right\}\)
Vậy .......................................................
