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\(A=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2}\left(đk:x\ge0,x\ne1\right)\)
\(=\dfrac{x+2+\sqrt{x}\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2}{\sqrt{x}-1}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2}{\sqrt{x}-1}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2.2}{\left(\sqrt{x}-1\right)^2\left(x+\sqrt{x}+1\right)}=\dfrac{2}{x+\sqrt{x}+1}\)
Để A nguyên thì: \(x+\sqrt{x}+1\inƯ\left(2\right)=\left\{-2;-1;1;2\right\}\)
Mà \(x+\sqrt{x}+1=\left(x+\sqrt{x}+\dfrac{1}{4}\right)+\dfrac{3}{4}=\left(\sqrt{x}+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\)
\(\Rightarrow x+\sqrt{x}+1\in\left\{1;2\right\}\)
+ Với \(x+\sqrt{x}+1=1\)
\(\Leftrightarrow\sqrt[]{x}\left(\sqrt{x}+1\right)=0\)
\(\Leftrightarrow x=0\left(tm\right)\left(do.\sqrt{x}+1\ge1>0\right)\)
+ Với \(x+\sqrt{x}+1=2\)
\(\Leftrightarrow\left(x+\sqrt{x}+\dfrac{1}{4}\right)=\dfrac{5}{4}\)
\(\Leftrightarrow\left(\sqrt{x}+\dfrac{1}{2}\right)^2=\dfrac{5}{4}\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}+\dfrac{1}{2}=\dfrac{\sqrt{5}}{2}\\\sqrt{x}+\dfrac{1}{2}=-\dfrac{\sqrt{5}}{2}\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}=\dfrac{\sqrt{5}-1}{2}\\\sqrt{x}=-\dfrac{\sqrt{5}+1}{2}\left(VLý\right)\end{matrix}\right.\)
\(\Leftrightarrow x=\dfrac{3-\sqrt{5}}{2}\left(tm\right)\)
Vậy \(S=\left\{1;\dfrac{3-\sqrt{5}}{2}\right\}\)
a: \(P=\left(\dfrac{2+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\dfrac{\sqrt{x}+1-\sqrt{x}}{\sqrt{x}+1}\)
\(=\dfrac{1}{\sqrt{x}-1}\cdot\dfrac{\sqrt{x}+1}{1}=\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)
b: Để P nguyên thì \(\sqrt{x}+1⋮\sqrt{x}-1\)
\(\Leftrightarrow\sqrt{x}-1\in\left\{-1;1;2\right\}\)
hay \(x\in\left\{0;4;9\right\}\)
(a) Với \(x\ge0,x\ne4\), ta có:
\(A=\dfrac{2x-3\sqrt{x}-2}{\sqrt{x}-2}=\dfrac{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}=2\sqrt{x}+1\)
Để \(A\le5\Rightarrow2\sqrt{x}+1\le5\)
\(\Leftrightarrow2\sqrt{x}\le4\Leftrightarrow\sqrt{x}\le2\Leftrightarrow0\le x\le4\).
Kết hợp với điều kiện thì: \(0\le x< 4.\)
(b) \(\dfrac{A}{2}=\dfrac{2\sqrt{x}+1}{2}\) nguyên khi \(\left(2\sqrt{x}+1\right)\in B\left(2\right)=\left\{0;2;4;...;2n\right\}\left(n\in N\right)\)
\(\Leftrightarrow\sqrt{x}\in\left\{-\dfrac{1}{2};\dfrac{1}{2};\dfrac{3}{2};...;\dfrac{2n+1}{2}\right\}\left(n\in N\right)\)
Hay: \(\sqrt{x}\in\left\{\dfrac{1}{2};\dfrac{3}{2};...;\dfrac{2n+1}{2}\right\}\)
\(\Leftrightarrow x\in\left\{\dfrac{1}{4};\dfrac{9}{4};...;\dfrac{\left(2n+1\right)^2}{4}\right\}\)
\(P=A\cdot B\)
\(=\dfrac{2\sqrt{x}}{\sqrt{x}-3}\cdot\dfrac{2\sqrt{x}+6+x-3\sqrt{x}+3-5\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=\dfrac{2\sqrt{x}}{\left(\sqrt{x}-3\right)}\cdot\dfrac{x-6\sqrt{x}+9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=\dfrac{2\sqrt{x}}{\left(\sqrt{x}+3\right)}\cdot\dfrac{\left(\sqrt{x}-3\right)^2}{\left(\sqrt{x}-3\right)^2}=\dfrac{2\sqrt{x}}{\sqrt{x}+3}\)
Để P nguyên thì
\(2\sqrt{x}⋮\sqrt{x}+3\)
\(\Leftrightarrow2\sqrt{x}+6-6⋮\sqrt{x}+3\)
=>\(\sqrt{x}+3\inƯ\left(-6\right)\)
=>\(\sqrt{x}+3\in\left\{3;6\right\}\)
=>\(\sqrt{x}\in\left\{0;3\right\}\)
=>\(x\in\left\{0;9\right\}\)
Kết hợp ĐKXĐ, ta được: x=0
Lời giải:
$M(2\sqrt{x}-3)=\sqrt{x}+2$
$\Leftrightarrow \sqrt{x}(2M-1)=3M-2$
$\Leftrightarrow x=(\frac{3M-2}{2M-1})^2$
Vì $x$ nguyên nên $\frac{3M-2}{2M-1}$ nguyên
$\Rightarrow 3M-2\vdots 2M-1$
$\Leftrightarrow 6M-4\vdots 2M-1$
$\Leftrightarrow 3(2M-1)-1\vdots 2M-1$
$\Leftrightarrow 1\vdots 2M-1$
$\Rightarrow 2M-1\in\left\{\pm 1\right\}$
$\Rightarrow M=0;1$
$\Leftrightarrow x=4; 1$ (đều tm)
\(\sqrt{x}+\sqrt{2-x}\le\sqrt{2\left(x+2-x\right)}=2\)
\(\sqrt{x}+\sqrt{2-x}\ge\sqrt{x+2-x}=\sqrt{2}\)
\(\Rightarrow\dfrac{2}{2}\le P\le\dfrac{2}{\sqrt{2}}\Rightarrow1\le P\le\sqrt{2}\)
Mà \(P\in Z\Rightarrow P=1\)
\(\Rightarrow\sqrt{x}+\sqrt{2-x}=2\Rightarrow x=1\)
\(\dfrac{\sqrt{x}-5}{\sqrt{x-3}}=1-\dfrac{2}{\sqrt{x}-3}=P\)
Để P nguyên thì \(2⋮\sqrt{x}-3\Leftrightarrow\sqrt{x}-3\inƯ\left(2\right)=\left\{\pm1,\pm2\right\}\)
\(\begin{matrix}\sqrt{x}-3&-1&-2&1&2\\\sqrt{x}&-2\left(L\right)&1&4&5\\x&&1\left(tm\right)&16\left(tm\right)&25\left(tm\right)\end{matrix}\)
Mà x nguyên lớn nhất \(\Rightarrow x=25\)
Để P là số nguyên thì
căn x-3-2 chia hết cho căn x-3
=>căn x-3 thuộc Ư(-2)
mà x nguyên lớn nhất
nên căn x-3=2
=>x=25
ĐKXĐ:\(x\ge0\)
Để \(\dfrac{2\sqrt{x}}{\sqrt{x}+3}\) nhận giá trị nguyên thì \(2\sqrt{x}⋮\sqrt{x}+3\)
\(\Leftrightarrow2\left(\sqrt{x}+3\right)-6⋮\sqrt{x}+3\)
\(\Leftrightarrow-6⋮\sqrt{x}+3hay\sqrt{x}+3\inƯ_{\left(-6\right)}\)
Vì \(\sqrt{x}\ge0\Rightarrow\sqrt{x}+3\ge3\)
TH1.\(\sqrt{x}+3=3\Leftrightarrow\sqrt{x}=0\Leftrightarrow x=0\left(tmĐKXĐ\right)\)
TH2.\(\sqrt{x}+3=6\Leftrightarrow\sqrt{x}=3\Leftrightarrow x=9\left(tmĐKXĐ\right)\)
Vậy,x={0;9}