\(a,A=\frac{2}{\sqrt{x}-3}+\frac{2\sqrt{x}}{x-4\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-1}\)

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

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

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

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

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

\(b,A=\frac{\sqrt{x}-3+5}{\sqrt{x}-3}=1+\frac{5}{\sqrt{x}-3}\)

để A nguyên \(5⋮\sqrt{x}-3\)

lập bảng ra đc 

\(x=\left\{2\right\}\)

a) \(A=\dfrac{\sqrt{x}+2}{\sqrt{x}-2}=1+\dfrac{4}{\sqrt{x}-2}\)

Để A nguyên thì 4 ⋮ √x - 2

\(\Rightarrow\sqrt{x}-2\inƯ\left(4\right)\)

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

\(\Rightarrow\sqrt{x}\in\left\{3;1;4;0;6;-2\right\}\)

Mà x \(\sqrt{x}\ge0\)

=> x thuộc {9; 1; 16; 0; 36}

b) 

Thuy Duong Nguyen đánh đề cẩn thận hơn bạn nhé

Lời giải :

a) ĐKXĐ : \(x\ne1\)

 \(A=\frac{15\sqrt{x}-11}{x+2\sqrt{x}-3}+\frac{3\sqrt{x}-2}{1-\sqrt{x}}-\frac{2\sqrt{x}+3}{\sqrt{x}+3}\)

\(A=\frac{15\sqrt{x}-11}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}+\frac{\left(\sqrt{x}+3\right)\left(2-3\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}-\frac{\left(2\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)

\(A=\frac{15\sqrt{x}-11-3x+6-7\sqrt{x}-2x-\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)

\(A=\frac{-5x+7\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)

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

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

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

\(\Leftrightarrow\sqrt{x}=\sqrt{2}-1\)

Khi đó \(A=\frac{2-5\left(\sqrt{2}-1\right)}{\sqrt{2}-1+3}\)

\(A=\frac{2-5\sqrt{2}+5}{\sqrt{2}+2}=\frac{7-5\sqrt{2}}{\sqrt{2}+2}\)

c) \(A=\frac{1}{2}\)

\(\Leftrightarrow\frac{2-5\sqrt{x}}{\sqrt{x}+3}=\frac{1}{2}\)

\(\Leftrightarrow2\left(2-5\sqrt{x}\right)=\sqrt{x}+3\)

\(\Leftrightarrow4-10\sqrt{x}-\sqrt{x}-3=0\)

\(\Leftrightarrow1-11\sqrt{x}=0\)

\(\Leftrightarrow11\sqrt{x}=1\)

\(\Leftrightarrow\sqrt{x}=\frac{1}{11}\)

\(\Leftrightarrow x=\frac{1}{121}\)( thỏa )

d) A nguyên \(\Leftrightarrow2-5\sqrt{x}⋮\sqrt{x}+3\)

\(\Leftrightarrow-5\left(\sqrt{x}+3\right)+17⋮\sqrt{x}+3\)

Vì \(-5\left(\sqrt{x}+3\right)⋮\sqrt{x}+3\)

\(\Rightarrow17⋮\sqrt{x}+3\)

\(\Rightarrow\sqrt{x}+3\inƯ\left(17\right)=\left\{17\right\}\)( vì \(\sqrt{x}+3\ge3\))

\(\Leftrightarrow\sqrt{x}=14\)

\(\Leftrightarrow x=196\)( thỏa )

Vậy....

\(a,ĐKXĐ:\orbr{\begin{cases}x+2\sqrt{x}+3\ne0\\\sqrt{x}+3\ne0\end{cases}}\)

\(\Leftrightarrow\orbr{ }\sqrt{x}\ne-3\)

Rút gọn: p/s: sau phân số thứ 2 ở mẫu ko có x à? Bạn chép đề sai?

Hình như đề phần rút gọn sai nhé! 

\(x+2\sqrt{x}+3\) không thể tách được 

Và đa số mình làm mẫu sẽ không như này :\(1-\sqrt{2}\) ,phải có x nữa .

Bạn xem lại đề rồi mình sẽ làm tiếp

\(dkxd\Leftrightarrow\hept{\begin{cases}x\ge0\\\sqrt{x}-2\ne0\end{cases}\Rightarrow\hept{\begin{cases}x\ge0\\x\ne4\end{cases}}}\)

\(A=\left(\frac{\sqrt{x}}{x-4}-\frac{2}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\right):\frac{1}{\sqrt{x}+2}.\)

\(=\left(\frac{\sqrt{x}}{x-4}-\frac{2\left(\sqrt{x}+2\right)}{x-4}+\frac{\sqrt{x}-2}{x-4}\right):\frac{1}{\sqrt{x}+2}\)

\(=\frac{\sqrt{x}-2\sqrt{x}-4+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}.\frac{\sqrt{x}+2}{1}\)

\(=\frac{-6\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=-\frac{6}{\sqrt{x}-2}\)

\(A=\)\(\left(\frac{\sqrt{x}}{x-4}+\frac{2}{2-\sqrt{x}}+\frac{1}{\sqrt{x}+2}\right)\)\(:\frac{1}{\sqrt{x}+2}\)

a,ĐKXĐ:\(\hept{\begin{cases}x\ge0\\2-\sqrt{x}\\x-4\ne0\end{cases}\ne0}\)\(\Rightarrow\)\(\hept{\begin{cases}x\ge0\\x\ne4\end{cases}}\)

\(A=\)\(\left(\frac{\sqrt{x}}{x-4}+\frac{2}{2-\sqrt{x}}+\frac{1}{\sqrt{x}+2}\right)\)\(:\frac{1}{\sqrt{x}+2}\)

\(A=\)\(\left(\frac{\sqrt{x}}{x-4}-\frac{2}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\right)\)\(.\left(\sqrt{x}+2\right)\)

\(A=\)\(\left(\frac{\sqrt{x}-2\left(\sqrt{x}+2\right)+\sqrt{x}-2}{x-4}\right)\)\(.\left(\sqrt{x}+2\right)\)

\(A=\)\(\left(\frac{\sqrt{x}-2\sqrt{x}-4+\sqrt{x}-2}{x-4}\right)\)\(.\left(\sqrt{x}+2\right)\)

\(A=\)\(\left(\frac{-6}{x-4}\right)\)\(.\left(\sqrt{x}+2\right)\)

\(A=\)\(\frac{-6}{\sqrt{x}-2}\)

b,\(x=9-4\sqrt{5}\)\(\Rightarrow\)\(A=\)\(\frac{-6}{\sqrt{9-4\sqrt{5}}-2}\)\(=\frac{-6}{\sqrt{5-2.2\sqrt{5}+4}-2}\)

\(A=\)\(\frac{-6}{\sqrt{\left(\sqrt{5}-2\right)^2}-2}\)\(=\frac{-6}{\sqrt{5}-2-2}\)\(=\frac{-6}{\sqrt{5}-4}\)

c,\(A>-1\)\(\Rightarrow\)\(\frac{-6}{\sqrt{x}-2}\)\(>-1\)\(\Rightarrow\)\(\frac{-6}{\sqrt{x}-2}+1>0\)

\(\Leftrightarrow\)\(\frac{-6+\sqrt{x}-2}{\sqrt{x}-2}>0\)

\(\Leftrightarrow\)\(\frac{\sqrt{x}-8}{\sqrt{x}-2}>0\)

\(d,\frac{-6}{\sqrt{x}-2}\)nhỏ nhất \(\Leftrightarrow\frac{6}{\sqrt{x}-2}\)lớn nhất

\(\Rightarrow\sqrt{x}-2\)nhỏ nhất \(\Rightarrow\sqrt{x}=0\Leftrightarrow x=0\)

\(\Rightarrow A_{min}=-2\Leftrightarrow x=0\)

\(e,\)\(A\in Z\Leftrightarrow\frac{6}{\sqrt{x}-2}\in Z\)\(\Leftrightarrow\sqrt{x}-2\inƯ_6\)

Mà \(Ư_6=\left\{\pm1;\pm2;\pm3;\pm6\right\}\Rightarrow...\)

a) \(A=\frac{2\sqrt{x}-3}{\sqrt{x}-4}-\frac{\sqrt{x}+2}{\sqrt{x}+1}-\frac{2-3\sqrt{x}}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{2x-\sqrt{x}-3-x+2\sqrt{x}+8-2+3\sqrt{x}}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{x+4\sqrt{x}+3}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+1\right)}\) 

\(=\frac{\sqrt{x}+3}{\sqrt{x}-4}\)

b) Để \(A\in Z\)

\(\Leftrightarrow\frac{\sqrt{x}+3}{\sqrt{x}-4}=\frac{\sqrt{x}-4}{\sqrt{x}-4}+\frac{7}{\sqrt{x}-4}\in Z\)

=>\(\sqrt{x}-4\inƯ\left(7\right)\)

........