\(M=\dfrac{\sqrt{a}+2}{\sqrt{a}-2}=\dfrac{\sqrt{a}-2}{\sqrt{a}-2}+\dfrac{4}{\sqrt{a}-2}=1+\dfrac{4}{\sqrt{a}-2}\in Z\)

\(\Rightarrow\sqrt{a}-2\inƯ\left(4\right)=\left\{1;-1;2;-2;4;-4\right\}\)

Do \(\sqrt{a}\ge0\)

\(\Leftrightarrow\sqrt{a}\in\left\{3;1;4;0;6\right\}\)

\(\Rightarrow a\in\left\{9;1;16;0;36\right\}\)

Đề yêu cầu tìm a nguyên thì đúng hơn.

Vì yêu cầu tìm a hữu tỉ bài này sẽ có vô số số hữu tỉ thỏa mãn

Ta có \(M=\dfrac{\sqrt{a}+2}{\sqrt{a}-2}=1+\dfrac{4}{\sqrt{a}-2}\)

Để \(M\) nhận giá trị nguyên thì \(4⋮\left(\sqrt{a}-2\right)\Rightarrow\left(\sqrt{a}-2\right)\inƯ\left(4\right)\)

\(\Rightarrow\sqrt{a}-2\in\left\{1;-1;2;-2;4;-4\right\}\)

\(\Rightarrow\sqrt{a}\in\left\{3;1;4;0;6\right\}\)

\(\Rightarrow a\in\left\{\sqrt{3};1;2;0;\sqrt{6}\right\}\)

mà a là số hữu tỉ nên \(a\in\left\{1;2;0\right\}\)

Lời giải:
a.

Áp dụng BĐT Bunhiacopxky:

$A^2=(\sqrt{x-1}+\sqrt{9-x})^2\leq (x-1+9-x)(1+1)=16$

$\Rightarrow A\leq 4$

Vậy $A_{\max}=4$. Giá trị này đạt tại $x=5$

b.

$A=\frac{3(\sqrt{x}+2)+5}{\sqrt{x}+2}=3+\frac{5}{\sqrt{x}+2}$

Để $A$ nguyên thì $\frac{5}{\sqrt{x}+2}=m$ với $m$ nguyên dương

$\Leftrightarrow \sqrt{x}+2=\frac{5}{m}$

$\sqrt{x}=\frac{5-2m}{m}$

Vì $\sqrt{x}\geq 0$ nên $\frac{5-2m}{m}\geq 0$

Mà $m$ nguyên dương nên $5-2m\geq 0$

$\Leftrightarrow m\leq 2,5$. 

$\Rightarrow m=1; 2$

$\Rightarrow x=9; x=\frac{1}{4}$

Ta có : \(M=\frac{\sqrt{x}+6}{\sqrt{x}+1}=\frac{\sqrt{x}+1+5}{\sqrt{x}+1}=\frac{\sqrt{x}+1}{\sqrt{x}+1}+\frac{5}{\sqrt{x}+1}=1+\frac{5}{\sqrt{x}+1}\)

Để M nguyên thì 5 chia hết cho \(\sqrt{x}+1\)

Nên : \(\sqrt{x}+1\inƯ\left(5\right)=\left\{-5;-1;1;5\right\}\)

Ta có bảng : 

bài có nhầm đề không bạn? vì tử = mẫu thì M=1 rồi kìa

Nhầm đề bài :p

1: Ta có: \(A=\left(\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{x-1}\right):\dfrac{\sqrt{x}}{\sqrt{x}+1}\)

\(=\left(\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}-\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\right):\dfrac{\sqrt{x}}{\sqrt{x}+1}\)

\(=\dfrac{x-\sqrt{x}+2\sqrt{x}-2-\left(x+\sqrt{x}-2\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}:\dfrac{\sqrt{x}}{\sqrt{x}+1}\)

\(=\dfrac{x+\sqrt{x}-2-x+\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\cdot\dfrac{\sqrt{x}+1}{\sqrt{x}}\)

\(=\dfrac{2\sqrt{x}}{\sqrt{x}\left(x-1\right)}\)

\(=\dfrac{2}{x-1}\)

2: ĐKXĐ: \(\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\)

Để A là số nguyên thì \(2⋮x-1\)

\(\Leftrightarrow x-1\inƯ\left(2\right)\)

\(\Leftrightarrow x-1\in\left\{1;-1;2;-2\right\}\)

\(\Leftrightarrow x\in\left\{2;0;3;-1\right\}\)

Kết hợp ĐKXĐ, ta được: \(x\in\left\{2;3\right\}\)

Vậy: Để A là số nguyên thì \(x\in\left\{2;3\right\}\)

\(M=\dfrac{7\sqrt{a}-2}{2\sqrt{a}+1}\left(đk:a\ge0\right)=\dfrac{3\left(2\sqrt[]{a}+1\right)+\sqrt{a}-5}{2\sqrt{a}+1}=3+\dfrac{\sqrt{a}-5}{2\sqrt{a}+1}\)

Để \(M\in Z,M>0\) thì \(\sqrt{a}-5\ge0\Leftrightarrow a\ge25\) và:

\(\left\{{}\begin{matrix}\sqrt{a}-5⋮2\sqrt{a}+1\\2\sqrt{a}+1⋮2\sqrt{a}+1\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}2\sqrt{a}-10⋮2\sqrt{a}+1\\2\sqrt{a}+1⋮2\sqrt{a}+1\end{matrix}\right.\)

\(\Rightarrow\left(2\sqrt{a}+1\right)-\left(2\sqrt{a}-10\right)⋮2\sqrt{a}+1\)

\(\Rightarrow11⋮2\sqrt{a}+1\Rightarrow2\sqrt{a}+1\inƯ\left(11\right)=\left\{-11;-1;1;11\right\}\)

Do \(\sqrt{a}\ge0\forall a\)

\(\Rightarrow\sqrt{a}\in\left\{0;5\right\}\)

\(\Rightarrow a\in\left\{0\left(loại\right);25\left(nhận\right)\right\}\)

a) \(M=\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{6\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\left(x\ge0,x\ne1\right)\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)-6\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\dfrac{x-4\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\dfrac{\sqrt{x}-3}{\sqrt{x}+2}\)

b) \(M=\dfrac{\sqrt{x}-3}{\sqrt{x}+2}=1-\dfrac{5}{\sqrt{x}+2}\in Z\)

\(\Rightarrow\sqrt{x}+2\inƯ\left(5\right)=\left\{-5;-1;1;5\right\}\)

Do \(\sqrt{x}\ge0\forall x\)

\(\Rightarrow\sqrt{x}\in\left\{3\right\}\Rightarrow x=9\left(tm\right)\)

a: ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x\notin\left\{4;1\right\}\end{matrix}\right.\)

Ta có: \(A=\dfrac{x-4\sqrt{x}+3-\left(2x-4\sqrt{x}-\sqrt{x}+2\right)+x+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{2x-4\sqrt{x}+5-2x+5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{\sqrt{x}+3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

a: ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x\notin\left\{1;4\right\}\end{matrix}\right.\)

\(A=\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}-1}{\sqrt{x}-1}+\dfrac{x-2}{x-3\sqrt{x}+2}\)

\(=\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}-1}{\sqrt{x}-1}+\dfrac{x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)-\left(2\sqrt{x}-1\right)\left(\sqrt{x}-2\right)+x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{x-4\sqrt{x}+3-2x+5\sqrt{x}-2+x-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}=\dfrac{1}{\sqrt{x}-2}\)

b: Để A>2 thì A-2>0

=>\(\dfrac{1-2\left(\sqrt{x}-2\right)}{\sqrt{x}-2}>0\)

=>\(\dfrac{5-2\sqrt{x}}{\sqrt{x}-2}>0\)

=>\(\dfrac{2\sqrt{x}-5}{\sqrt{x}-2}< 0\)

TH1: \(\left\{{}\begin{matrix}2\sqrt{x}-5>0\\\sqrt{x}-2< 0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\sqrt{x}>\dfrac{5}{2}\\\sqrt{x}< 2\end{matrix}\right.\)

=>\(x\in\varnothing\)

TH2: \(\left\{{}\begin{matrix}2\sqrt{x}-5< 0\\\sqrt{x}-2>0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\sqrt{x}< \dfrac{5}{2}\\\sqrt{x}>2\end{matrix}\right.\)

=>\(2< \sqrt{x}< \dfrac{5}{2}\)

=>4<x<25/4

c: Để A là số nguyên thì \(1⋮\sqrt{x}-2\)

=>\(\sqrt{x}-2\in\left\{1;-1\right\}\)

=>\(\sqrt{x}\in\left\{3;1\right\}\)

=>\(x\in\left\{1;9\right\}\)

kết hợp ĐKXĐ, ta được: x=9

em tham khảo