a) P = \(\dfrac{\sqrt{a}+2}{\sqrt{a}+3}-\dfrac{5}{a+\sqrt{a}-6}+\dfrac{1}{2-\sqrt{a}}\left(ĐKXĐ:a\ge0;a\ne4\right)\)

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

P = \(\dfrac{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)-5-\left(\sqrt{a}+3\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+3\right)}\)

P = \(\dfrac{a-4-5-\sqrt{a}-3}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+3\right)}=\dfrac{a-\sqrt{a}-12}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+3\right)}\)

P  = \(\dfrac{\left(\sqrt{a}-4\right)\left(\sqrt{a}+3\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+3\right)}\)

P = \(\dfrac{\sqrt{a}-4}{\sqrt{a}-2}\) 

b) Ta có: P = \(\dfrac{\sqrt{a}-4}{\sqrt{a}-2}\) = 1 - \(\dfrac{2}{\sqrt{a}-2}\)

Để \(P\in Z\) <=> 1 - \(\dfrac{2}{\sqrt{a}-2}\) \(\in Z\) <=> \(\sqrt{a}-2\inƯ\left(2\right)=\left\{1;-1;2;-2\right\}\)

Ta có bảng sau: 

Vậy để \(P\in Z\) thì  \(a\in\left\{0;1;9;16\right\}\)

Lời giải:ĐK: $a\geq 0; a\neq 9; a\neq 4$

a) 

\(A=\frac{2\sqrt{a}-9}{(\sqrt{a}-2)(\sqrt{a}-3)}-\frac{\sqrt{a}+3}{\sqrt{a}-2}+\frac{2\sqrt{a}+1}{\sqrt{a}-3}\)

\(\frac{2\sqrt{a}-9}{(\sqrt{a}-2)(\sqrt{a}-3)}-\frac{(\sqrt{a}+3)(\sqrt{a}-3)}{(\sqrt{a}-2)(\sqrt{a}-3)}+\frac{(2\sqrt{a}+1)(\ \sqrt{a}-2)}{(\sqrt{a}-3)(\sqrt{a}-2)}\)

\(=\frac{2\sqrt{a}-9-(a-9)+(2a-3\sqrt{a}-2)}{(\sqrt{a}-3)(\sqrt{a}-2)}=\frac{a-\sqrt{a}-2}{(\sqrt{a}-3)(\sqrt{a}-2)}=\frac{(\sqrt{a}-2)(\sqrt{a}+1)}{(\sqrt{a}-3)(\sqrt{a}-2)}=\frac{\sqrt{a}+1}{\sqrt{a}-3}\)

b) Để \(A< 1\Leftrightarrow \frac{\sqrt{a}+1}{\sqrt{a}-3}<1\Leftrightarrow 1+\frac{4}{\sqrt{a}-3}<1\)

\(\Leftrightarrow \frac{4}{\sqrt{a}-3}< 0\Leftrightarrow \sqrt{a}-3< 0\Leftrightarrow 0\leq a< 9\)

Kết hợp ĐKXĐ: suy ra $0\leq a< 9; a\neq 4$

c) Với $a$ nguyên,  \(A=1+\frac{4}{\sqrt{a}-3}\in\mathbb{Z}\Leftrightarrow 4\vdots \sqrt{a}-3\)

$\Rightarrow \sqrt{a}-3\in\left\{\pm 1; \pm 2;\pm 4\right\}$

$\Rightarrow a\in\left\{4;16; 1;25; 49\right\}$

Kết hợp ĐKXĐ suy ra $a\in\left\{16;1;25;49\right\}$

ĐKXĐ: \(\left\{{}\begin{matrix}a\ge0\\a\notin\left\{4;9\right\}\end{matrix}\right.\)

a) Ta có: \(A=\dfrac{2\sqrt{a}-9}{a-5\sqrt{a}+6}-\dfrac{\sqrt{a}+3}{\sqrt{a}-2}-\dfrac{2\sqrt{a}+1}{3-\sqrt{a}}\)

\(=\dfrac{\left(2\sqrt{a}-9\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-3\right)}-\dfrac{\left(\sqrt{a}+3\right)\left(\sqrt{a}-3\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-3\right)}+\dfrac{\left(2\sqrt{a}+1\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-3\right)\left(\sqrt{a}-2\right)}\)

\(=\dfrac{2\sqrt{a}-9-\left(a-9\right)+2a-4\sqrt{a}+\sqrt{a}-2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-3\right)}\)

\(=\dfrac{2a-\sqrt{a}-11-a+9}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-3\right)}\)

\(=\dfrac{a-\sqrt{a}-2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-3\right)}\)

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

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

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

\(=\dfrac{\sqrt{a}+1}{\sqrt{a}-3}\)

b) Để A<1 thì A-1<0

\(\Leftrightarrow\dfrac{\sqrt{a}+1}{\sqrt{a}-3}-1< 0\)

\(\Leftrightarrow\dfrac{\sqrt{a}+1}{\sqrt{a}-3}-\dfrac{\sqrt{a}-3}{\sqrt{a}-3}< 0\)

\(\Leftrightarrow\dfrac{\sqrt{a}+1-\sqrt{a}+3}{\sqrt{a}-3}< 0\)

\(\Leftrightarrow\dfrac{4}{\sqrt{a}-3}< 0\)

mà 4>0

nên \(\sqrt{a}-3< 0\)

\(\Leftrightarrow\sqrt{a}< 3\)

hay a<9

Kết hợp ĐKXĐ, ta được:

\(\left\{{}\begin{matrix}0\le a< 9\\a\ne4\end{matrix}\right.\)

Vậy: Để A<1 thì \(\left\{{}\begin{matrix}0\le a< 9\\a\ne4\end{matrix}\right.\)

c) Để A nguyên thì \(\sqrt{a}+1⋮\sqrt{a}-3\)

\(\Leftrightarrow\sqrt{a}-3+4⋮\sqrt{a}-3\)

mà \(\sqrt{a}-3⋮\sqrt{a}-3\)

nên \(4⋮\sqrt{a}-3\)

\(\Leftrightarrow\sqrt{a}-3\inƯ\left(4\right)\)

\(\Leftrightarrow\sqrt{a}-3\in\left\{1;-1;2;-2;4;-4\right\}\)

mà \(\sqrt{a}-3\ge-3\forall a\) thỏa mãn ĐKXĐ

nên \(\sqrt{a}-3\in\left\{1;-1;2;-2;4\right\}\)

\(\Leftrightarrow\sqrt{a}\in\left\{4;2;5;1;7\right\}\)

\(\Leftrightarrow a\in\left\{16;4;25;1;49\right\}\)

Kết hợp ĐKXĐ, ta được: \(a\in\left\{1;16;25;49\right\}\)

Vậy: Để A nguyên thì \(a\in\left\{1;16;25;49\right\}\)

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

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

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

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

a) Ta có: \(A=\left(\dfrac{x-5\sqrt{x}}{x-25}-1\right):\left(\dfrac{25-x}{x+2\sqrt{x}-15}-\dfrac{\sqrt{x}+3}{\sqrt{x}+5}+\dfrac{\sqrt{x}-5}{\sqrt{x}-3}\right)\)

\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}-5\right)}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}-1\right):\left(\dfrac{25-x}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}-\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}+\dfrac{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)

\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}+5}-1\right):\left(\dfrac{25-x-\left(x-9\right)+x-25}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)

\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}+5}-\dfrac{\sqrt{x}+5}{\sqrt{x}+5}\right):\left(\dfrac{25-x-x+9+x-25}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)

\(=\dfrac{\sqrt{x}-\sqrt{x}-5}{\sqrt{x}+5}:\dfrac{x+9}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\)

\(=\dfrac{-5}{\sqrt{x}+5}\cdot\dfrac{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}{x+9}\)

\(=\dfrac{-5\left(\sqrt{x}-3\right)}{x+9}\)

ĐK: \(a\ge0;a\ne4\)

a) ⇔ \(P=\dfrac{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}-\dfrac{5}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}-\dfrac{\sqrt{a}+3}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}\)

⇔ \(P=\dfrac{a-4-5-\sqrt{a}-3}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}=\dfrac{a-\sqrt{a}-12}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}=\dfrac{\left(\sqrt{a}+3\right)\left(\sqrt{a}-4\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-2\right)}=\dfrac{\sqrt{a}-4}{\sqrt{a}-2}\)

b) \(P< 1\Leftrightarrow\dfrac{\sqrt{a}-4}{\sqrt{a}-2}< 1\)

\(\Leftrightarrow\dfrac{\sqrt{a}-4}{\sqrt{a}-2}-1< 0\Leftrightarrow\dfrac{-2}{\sqrt{a}-2}< 0\)

Do \(-2< 0\) ⇔ \(\sqrt{a}-2< 0\Leftrightarrow a< 4\)

Kết hợp điều kiện ban đầu, ta có: \(0< a< 4\)

Vậy khi \(0< a< 4\) thì \(P< 1\)

Sửa đề: \(P=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{2\sqrt{a}}{a-1}+\dfrac{\sqrt{a}}{\sqrt{a}+1}\right):\dfrac{\sqrt{a}}{\sqrt{a}+1}\)

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

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

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

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

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

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

\(=\dfrac{\sqrt{a}-1}{\sqrt{a}}\)

b) Để \(P< \dfrac{1}{2}\) thì \(P-\dfrac{1}{2}< 0\)

\(\Leftrightarrow\dfrac{\sqrt{a}-1}{\sqrt{a}}-\dfrac{1}{2}< 0\)

\(\Leftrightarrow\dfrac{2\left(\sqrt{a}-1\right)}{2\sqrt{a}}-\dfrac{\sqrt{a}}{2\sqrt{a}}< 0\)

\(\Leftrightarrow\dfrac{2\sqrt{a}-2-\sqrt{a}}{2\sqrt{a}}< 0\)

\(\Leftrightarrow\dfrac{\sqrt{a}-2}{2\sqrt{a}}< 0\)

mà \(2\sqrt{a}>0\forall a\) thỏa mãn ĐKXĐ

nên \(\sqrt{a}-2< 0\)

\(\Leftrightarrow\sqrt{a}< 2\)

hay a<4

Kết hợp ĐKXĐ, ta được: \(\left\{{}\begin{matrix}0< a< 4\\a\ne1\end{matrix}\right.\)

Vậy: Để \(P< \dfrac{1}{2}\) thì \(\left\{{}\begin{matrix}0< a< 4\\a\ne1\end{matrix}\right.\)

\(a,P=B:A\)

\(=\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\right):\left(\dfrac{2}{\sqrt{x}-3}+\dfrac{1}{\sqrt{x}+3}\right)\left(ĐKXĐ:x\ge0;x\ne9\right)\)

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

\(=\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\right):\left[\dfrac{3\sqrt{x}+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right]\)

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

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

\(b,\) Để \(P=\dfrac{\sqrt{x}+3}{3}\) có giá trị nguyên

thì \(\sqrt{x}+3⋮3\)

\(\Leftrightarrow\sqrt{x}+3\in B\left(3\right)\)

\(\Leftrightarrow\sqrt{x}\in B\left(3\right)\) 

Kết hợp với điều kiện, ta được:

\(P\) nguyên khi \(x=m^2\left(m\in Z;m⋮3;m\ne3\right)\)

#Toru

a: 

ĐKXĐ: x>=0; x<>9

\(A=\dfrac{2\sqrt{x}+6+\sqrt{x}-3}{\left(x-9\right)}=\dfrac{3\sqrt{x}+3}{x-9}\)

\(P=B:A=\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\cdot\dfrac{x-9}{3\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+3}{3}\)

b: P nguyên khi \(\sqrt{x}+3⋮3\)

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

=>\(x=k^2\left(k\in Z;k⋮3\right)\)

1: \(P=\dfrac{3a+3\sqrt{a}-3-a+1-a+4}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-1\right)}\)

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

2: Để P nguyên thì \(\sqrt{a}-1+2⋮\sqrt{a}-1\)

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

hay \(a\in\left\{4;0;9\right\}\)

a: \(P=\dfrac{a+\sqrt{a}+1}{a+1}:\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{2\sqrt{a}}{\left(a+1\right)\left(\sqrt{a}-1\right)}\right)\)

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

\(=\dfrac{a+\sqrt{a}+1}{\sqrt{a}-1}\)

b: Để P<1 thì P-1<0

\(\Leftrightarrow\dfrac{a+\sqrt{a}+1-\sqrt{a}+1}{\sqrt{a}-1}< 0\)

hay 0<a<1