`a)` Thay `x=2` vào `B` có: `B=[-10]/[2-4]=5`

`b)` Với `x ne -1;x ne -5` có:

`A=[(x+2)(x+1)-5x-1-(x+5)]/[(x+1)(x+5)]`

`A=[x^2+x+2x+2-5x-1-x-5]/[(x+1)(x+5)]`

`A=[x^2-3x-4]/[(x+1)(x+5)]`

`A=[(x+1)(x-4)]/[(x+1)(x+5)]`

`A=[x-4]/[x+5]`

`c)` Với `x ne -5; x ne -1; x ne 4` có:

`P=A.B=[x-4]/[x+5].[-10]/[x-4]`

           `=[-10]/[x+5]`

Để `P` nguyên `<=>[-10]/[x+5] in ZZ`

    `=>x+5 in Ư_{-10}`

Mà `Ư_{-10}={+-1;+-2;+-5;+-10}`

`=>x={-4;-6;-3;-7;0;-10;5;-15}` (t/m đk)

a) A= ^{_{\(\dfrac{\sqrt{x}}{\sqrt{x-2}}-\dfrac{4}{x-2\sqrt{x}}=\dfrac{\sqrt{x}\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}-\dfrac{4}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{x-4}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}+2}{\sqrt{x}}=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\sqrt{x}\sqrt{x}}=\dfrac{x+2\sqrt{x}}{x}\)}}

b) Ta có x >0  nên \(\sqrt{x}\) >0 

 <=>  \(2\sqrt{x}\)  > 0 

 <=>  \(x+2\sqrt{x}\)  > x 

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

 hay A > 1

c) 

a: \(A=\dfrac{x^2-8x+16-x^2+16}{\left(x-4\right)\left(x+4\right)}\cdot\dfrac{x}{2\left(x-1\right)}\)

\(=\dfrac{-8\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}\cdot\dfrac{x}{2\left(x-1\right)}\)

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

a: A=−4x/(x+4)(x−1)

Bài 1 : 

a, \(A=\frac{2x^2-4x+8}{x^3+8}=\frac{2\left(x^2-2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}=\frac{2}{x+2}\)

b, Ta có : \(\left|x\right|=2\Rightarrow\orbr{\begin{cases}x=2\\x=-2\end{cases}}\)

TH1 : Thay x = 2 vào biểu thức trên ta được : 

\(\frac{2}{2+2}=\frac{2}{4}=\frac{1}{2}\)

TH2 : Thay x = -2 vào biểu thức trên ta được : 

\(\frac{2}{-2+2}=\frac{2}{0}\)vô lí 

c, ta có A = 2 hay \(\frac{2}{x+2}=2\)ĐK : \(x\ne-2\)

\(\Rightarrow2x+4=2\Leftrightarrow2x=-2\Leftrightarrow x=-1\)

Vậy với x = -1 thì A = 2 

d, Ta có A < 0 hay \(\frac{2}{x+2}< 0\)

\(\Rightarrow x+2< 0\)do 2 > 0 

\(\Leftrightarrow x< -2\)

Vậy với A < 0 thì x < -2 

e, Để A nhận giá trị nguyên khi \(x+2\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)

2.

ĐKXĐ : \(x\ne\pm2\)

a. \(B=\frac{x^2-4x+4}{x^2-4}=\frac{\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}=\frac{x-2}{x+2}\)

b. | x - 1 | = 2 <=>\(\hept{\begin{cases}x-1=2\\x-1=-2\end{cases}}\)<=>\(\hept{\begin{cases}x=3\\x=-1\end{cases}}\)

Với x = 3 thì \(B=\frac{3-2}{3+2}=\frac{1}{5}\)

Với x = - 1 thì \(B=\frac{-1-2}{-1+2}=-3\)

Vậy với | x - 1 | = 2 thì B đạt được 2 giá trị là B = 1/5 hoặc B = - 3

c. \(B=\frac{x-2}{x+2}=-1\)<=>\(-\left(x-2\right)=x+2\)

<=> \(-x+2=x+2\)<=>\(-x=x\)<=>\(x=0\)

d. \(B=\frac{x-2}{x+2}< 1\)<=>\(x-2< x+2\)luôn đúng \(\forall\)x\(\ne\pm2\)

e. \(B=\frac{x-2}{x+2}=\frac{x+2-4}{x+2}=1-\frac{4}{x+2}\)

Để B nguyên thì 4/x+2 nguyên => x + 2\(\in\){ - 4 ; - 2 ; - 1 ; 1 ; 2 ; 4 }

=> x \(\in\){ - 6 ; - 4 ; - 3 ; - 1 ; 0 ; 2 }

a)\(A=\frac{x^2}{5x+25}+\frac{2x-10}{x}+\frac{50+5x}{x^2+5x}\left(ĐK:x\ne0;-5\right)\)

\(\Leftrightarrow A=\frac{x^2}{5\left(x+5\right)}+\frac{2\left(x-5\right)}{x}+\frac{5\left(x+10\right)}{x\left(x+5\right)}\)

\(\Leftrightarrow A=\frac{x^3+10\left(x^2-25\right)+25x+250}{5x\left(x+5\right)}\)

\(\Leftrightarrow A=\frac{x^3+10x^2+25x}{5x\left(x+5\right)}\)

\(\Leftrightarrow A=\frac{x\left(x+5\right)^2}{5x\left(x+5\right)}\)

\(\Leftrightarrow A=\frac{x+5}{5}\)

b)Để A=-4 \(\Leftrightarrow\frac{x+5}{5}=-4\)

                  \(\Leftrightarrow x+5=-20\)

                   \(\Leftrightarrow x=-25\)

a).....

\(=\frac{x^2}{5\left(x+5\right)}+\frac{2x-10}{x}+\frac{50+5x}{x\left(x+5\right)}\)                                MTC= 5x (x+5)                 ĐK\(\hept{\begin{cases}x\ne0\\x\ne-5\end{cases}}\)

\(=\frac{x^2.x}{5x\left(x+5\right)}+\frac{5.\left(2x-10\right).\left(x+5\right)}{5x\left(x+5\right)}+\frac{5.\left(50+5x\right)}{5x\left(x+5\right)}\)

\(=\frac{x^3+\left(10x-50\right).\left(x+5\right)+250+25x}{5x\left(x+5\right)}\)

\(=\frac{x^3+10x^2+50x-50x-250+250+25x}{5x\left(x+5\right)}\)

\(=\frac{x^3+10x^2+25x}{5x\left(x+5\right)}\)

\(=\frac{x\left(x^2+10x+25\right)}{5x\left(x+5\right)}\)

\(=\frac{x\left(x+5\right)^2}{5x\left(x+5\right)}=\frac{x+5}{5}\)

b) A=-4

=>\(\frac{x+5}{5}=-4\)

=> x = -25

c)

d) Để A đạt gt nguyên thì 5\(⋮\)x+5

=> \(\left(x+5\right)\inƯ\left(5\right)=\left\{1;-1;5;-5\right\}\)

*x+5=1 => x=-4 \(\in Z\)

*x+5=-1 => x=-6\(\in Z\)

*x+5=5  => x=0\(\in Z\)

*x+5=-5  => x=-10\(\in Z\)

Vậy...........

a) Ta có: A = \(\frac{x+1}{x-2}+\frac{x-1}{x+2}+\frac{x^2+4x}{4-x^2}\)

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

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

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

A = \(\frac{\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}=\frac{x-2}{x+2}\)

b) Với x = 4 => A = \(\frac{4-2}{4+2}=\frac{2}{8}=\frac{1}{4}\)

c) ĐKXĐ: \(\hept{\begin{cases}x-2\ne0\\x+2\ne0\\4-x^2\ne0\end{cases}}\) <=> \(\hept{\begin{cases}x\ne2\\x\ne-2\\x\ne\pm2\end{cases}}\) <=> \(x\ne\pm2\)

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

Để A  nhận giá trị nguyên dương <=> \(1-\frac{4}{x+2}\) nguyên dương

<=> \(-\frac{4}{x+2}\) nguyên dương <=> -4 \(⋮\)x + 2

 <=> x + 2 \(\in\)Ư(-4) = {1; -1; 2; -2; 4; -4}

Lập bảng: 

Vậy ....

Bạn nên viết đề bằng công thức toán để được hỗ trợ tốt hơn (biểu tượng $\sum$ góc trái khung soạn thảo). 

\(A=\left(\dfrac{x}{x-2}+\dfrac{12}{x^2-4}-\dfrac{x}{x+2}\right):\dfrac{4}{x-2}\left(x\ne2;x\ne-2\right)\)

\(a,A=\left(\dfrac{x}{x-2}+\dfrac{12}{x^2-4}-\dfrac{x}{x+2}\right):\dfrac{4}{x-2}\)

\(=\left[\dfrac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{12}{\left(x-2\right)\left(x+2\right)}-\dfrac{x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\right]:\dfrac{4}{x-2}\)

\(=\left[\dfrac{x^2+2x+12-x^2+2x}{\left(x-2\right)\left(x+2\right)}\right]:\dfrac{4}{x-2}\)

\(=\dfrac{4x+12}{\left(x-2\right)\left(x+2\right)}:\dfrac{4}{x-2}\)

\(=\dfrac{4\left(x+3\right)}{\left(x-2\right)\left(x+2\right)}.\dfrac{x-2}{4}\)

\(=\dfrac{x+3}{x+2}\)

\(b,x=-1\Rightarrow A=\dfrac{\left(-1\right)+3}{\left(-1\right)+2}=2\)

\(c,A=\dfrac{x+3}{x+2}=\dfrac{x+2+1}{x+2}=1+\dfrac{1}{x+2}\)

\(A\in Z\Leftrightarrow x+2\inƯ\left(1\right)=\left\{1;-1\right\}\)

\(\Rightarrow x\in\left\{-1;-3\right\}\) (thỏa mãn điều kiện)

a)

\(ĐKXĐ:\left\{{}\begin{matrix}x-2\ne0\\x+2\ne0\\x^2-4\ne0\end{matrix}\right.< =>\left\{{}\begin{matrix}x\ne2\\x\ne-2\end{matrix}\right.\)

b)

\(\dfrac{1}{x-2}-\dfrac{1}{x+2}+\dfrac{x^2+4x}{x^2-4}\)

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

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

\(=\dfrac{x+2-x+2+x^2+4x}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{x^2+4x+4}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{\left(x+2\right)^2}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{x+2}{x-2}\)

c)

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

vậy M nhận giá trị nguyên thì 4⋮x-2

=> x-2 thuộc ước của 4

\(Ư\left(4\right)\in\left\{-1;1;-2;2;;4;-4\right\}\)

ta có bảng sau