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\(K=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{3\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{6\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
\(K\le\frac{1}{2}\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+1}\le\frac{1}{2}\)
\(\Leftrightarrow2\sqrt{x}-2\le\sqrt{x}+1\) (do \(\sqrt{x}+1>0;\forall x\in D\))
\(\Leftrightarrow\sqrt{x}\le3\Rightarrow x\le9\)
\(\Rightarrow x=\left\{0;2;3;4;5;6;7;8;9\right\}\Rightarrow\sum x=44\)
\(K=\left(\frac{a}{\sqrt{a}\left(\sqrt{a}-1\right)}-\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{\sqrt{a}-1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right)\)
\(=\left(\frac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right)\)
\(=\left(\frac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}\right).\left(\sqrt{a}-1\right)\)
\(=\frac{a-1}{\sqrt{a}}\Rightarrow\left\{{}\begin{matrix}m=1\\n=-1\end{matrix}\right.\Rightarrow m^2+n^2=2\)
\(A=\frac{x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{x+2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{\sqrt{x}}{\sqrt{x}-2}\Rightarrow\left\{{}\begin{matrix}m=0\\n=-2\end{matrix}\right.\Rightarrow m-n=2\)
\( a)A = \dfrac{{a - \sqrt a - 6}}{{4 - a}} - \dfrac{1}{{\sqrt a - 2}}\\ A = \dfrac{{a + 2\sqrt a - 3\sqrt a - 6}}{{\left( {2 - \sqrt a } \right)\left( {2 + \sqrt a } \right)}} - \dfrac{1}{{\sqrt a - 2}}\\ A = \dfrac{{\left( {\sqrt a + 2} \right)\left( {\sqrt a - 3} \right)}}{{\left( {2 - \sqrt a } \right)\left( {2 + \sqrt a } \right)}} - \dfrac{1}{{\sqrt a - 2}}\\ A = - \dfrac{{\sqrt a - 3}}{{\sqrt a - 2}} - \dfrac{1}{{\sqrt a - 2}}\\ A = - \dfrac{{\sqrt a - 2}}{{\sqrt a - 2}} = - 1 \)
\( b)B = \dfrac{1}{{\sqrt x - 1}} + \dfrac{1}{{\sqrt x + 1}} - \dfrac{2}{{x - 1}}\\ B = \dfrac{1}{{\sqrt x - 1}} + \dfrac{1}{{\sqrt x + 1}} - \dfrac{2}{{\left( {\sqrt x - 1} \right)\left( {\sqrt x + 1} \right)}}\\ B = \dfrac{{\sqrt x + 1 + \sqrt x - 1 - 2}}{{\left( {\sqrt x - 1} \right)\left( {\sqrt x + 1} \right)}}\\ B = \dfrac{{2\sqrt x - 2}}{{\left( {\sqrt x - 1} \right)\left( {\sqrt x + 1} \right)}}\\ B = \dfrac{{2\left( {\sqrt x - 1} \right)}}{{\left( {\sqrt x - 1} \right)\left( {\sqrt x + 1} \right)}} = \dfrac{2}{{\sqrt x + 1}} \)
a, x = \(\frac{4\left(\sqrt{3}+1\right)}{3-1}-\frac{4\left(\sqrt{3}-1\right)}{3-1}\)
x = \(\left(2\sqrt{3}+2\right)-\left(2\sqrt{3}-2\right)\)
x = \(2\sqrt{3}+2-2\sqrt{3}+2\)
x = 4 (TMĐK)
=> A = \(\frac{2\sqrt{4}+1}{3\sqrt{4}+1}\)
=> A = \(\frac{5}{7}\)
Vậy x = \(\frac{4}{\sqrt{3}-1}-\frac{4}{\sqrt{3}+1}\) thì A = \(\frac{5}{7}\)
b, B = \(\left(\frac{1}{\sqrt{x}-1}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}-1}\)
B = \(\frac{\sqrt{x}+1+\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}:\frac{1}{\sqrt{x}-1}\)
B = \(\frac{2\sqrt{x}+1}{\sqrt{x}+1}\)
c, \(\frac{B}{A}>2\) <=> \(\frac{2\sqrt{x}+1}{\sqrt{x}+1}:\frac{2\sqrt{x}+1}{3\sqrt{x}+1}\) > 2
<=> \(\frac{3\sqrt{x}+1}{\sqrt{x}+1}>2\)
<=> \(\frac{3\sqrt{x}+1}{\sqrt{x}+1}-2>0\)
<=> \(\frac{3\sqrt{x}+1-2\sqrt{x}-2}{\sqrt{x}+1}>0\)
<=> \(\frac{\sqrt{x}-1}{\sqrt{x}+1}>0\)
mà \(\sqrt{x}+1>0\) \(\forall\) \(x\in\) ĐKXĐ
=> \(\sqrt{x}-1>0\)
<=> \(\sqrt{x}>1\)
<=> \(x>1\)
Kết hợp ĐKXĐ : x \(\ge0\) ; x \(\ne\) 1
=> x > 1 thì \(\frac{B}{A}>2\)
a. \(A=\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3\left(x+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}=\frac{3\sqrt{x}-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\frac{3}{\sqrt{x}+3}\)
. \(x=2.\left(4+\sqrt{15}\right).\left(\sqrt{10}-\sqrt{6}\right).\sqrt{4-\sqrt{15}}\)
\(\Rightarrow x=\left(\sqrt{5}+\sqrt{3}\right)^2.\sqrt{2}\left(\sqrt{5}-\sqrt{3}\right).\frac{\left(\sqrt{5}-\sqrt{3}\right)^2}{\sqrt{2}}\)
\(=\left(\sqrt{5}+\sqrt{3}\right)^2.\left(\sqrt{5}-\sqrt{3}\right)^3\)\(=4\left(\sqrt{5}-\sqrt{3}\right)\)
Thay \(x=4\left(\sqrt{5}-\sqrt{3}\right)\Rightarrow A=\frac{3}{\sqrt{4\left(\sqrt{5}-\sqrt{3}\right)}+3}\)
\(=\frac{3}{2\sqrt{\left(\sqrt{5}-\sqrt{3}\right)}+3}\)
\(K=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{3\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{6\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
\(K\le\frac{1}{2}\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+1}\le\frac{1}{2}\Leftrightarrow2\sqrt{x}-2\le\sqrt{x}+1\) (do \(\sqrt{x}+1>0;\forall x\))
\(\Leftrightarrow\sqrt{x}\le3\Rightarrow x\le9\)
\(\Rightarrow x=\left\{2;3;4;5;6;7;8;9\right\}\Rightarrow T=44\)