a: \(P=\dfrac{x+\sqrt{x}+1+11\sqrt{x}-11+34}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=\dfrac{x+12\sqrt{x}+24}{\sqrt{x}+2}\)

b: Thay \(x=3-2\sqrt{2}\) vào P, ta được:

\(P=\dfrac{3-2\sqrt{2}+12\left(\sqrt{2}-1\right)+24}{\sqrt{2}-1+2}\)

\(=\dfrac{27-2\sqrt{2}+12\sqrt{2}-12}{\sqrt{2}+1}=5+5\sqrt{2}\)

1) \(ĐKXĐ:\hept{\begin{cases}x\ge0\\x\ne4\end{cases}}\)

\(P=\frac{2+\sqrt{x}}{2-\sqrt{x}}-\frac{2-\sqrt{x}}{2+\sqrt{x}}-\frac{4x}{x-4}\)

\(\Leftrightarrow P=\frac{\left(2+\sqrt{x}\right)^2-\left(2-\sqrt{x}\right)^2+4x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}\)

\(\Leftrightarrow P=\frac{4+4\sqrt{x}+x-4+4\sqrt{x}-x+4x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}\)

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

\(\Leftrightarrow P=\frac{4\sqrt{x}}{2-\sqrt{x}}\)

2) Để \(P=2\)

\(\Leftrightarrow\frac{4\sqrt{x}}{2-\sqrt{x}}=2\)

\(\Leftrightarrow4\sqrt{x}=4-2\sqrt{x}\)

\(\Leftrightarrow6\sqrt{x}=4\)

\(\Leftrightarrow\sqrt{x}=\frac{2}{3}\)

\(\Leftrightarrow x=\frac{4}{9}\)

Vậy để \(P=2\Leftrightarrow x=\frac{4}{9}\)

3) Khi \(\left(\sqrt{x}-2\right)\left(2\sqrt{x}-1\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x}-2=0\\2\sqrt{x}-1==0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x}=2\\\sqrt{x}=\frac{1}{2}\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=4\left(ktm\right)\\x=\frac{1}{4}\left(tm\right)\end{cases}}\)

Thay \(x=\frac{1}{4}\)vào P, ta được :

\(\Leftrightarrow P=\frac{4\sqrt{\frac{1}{4}}}{2-\sqrt{\frac{1}{4}}}=\frac{4\cdot\frac{1}{2}}{2-\frac{1}{2}}=\frac{2}{\frac{3}{2}}=\frac{4}{3}\)

4) Để \(P=\frac{\sqrt{x}+3}{2\sqrt{x}-1}\)

\(\Leftrightarrow\frac{4\sqrt{x}}{2-\sqrt{x}}=\frac{\sqrt{x}+3}{2\sqrt{x}-1}\)

\(\Leftrightarrow8x-4\sqrt{x}=-x-\sqrt{x}+6\)

\(\Leftrightarrow9x-3\sqrt{x}-6=0\)

\(\Leftrightarrow3x-\sqrt{x}-2=0\)

\(\Leftrightarrow\sqrt{x}=3x-2\)

\(\Leftrightarrow x=9x^2-12x+4\)

\(\Leftrightarrow9x^2-13x+4=0\)

\(\Leftrightarrow\left(9x-4\right)\left(x-1\right)=0\)

\(\Leftrightarrow\hept{\begin{cases}9x-4=0\\x-1=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{4}{9}\\x=1\end{cases}}\)

Thử lại ta được kết quá : \(x=\frac{4}{9}\left(ktm\right)\); \(x=1\left(tm\right)\)

Vậy để \(P=\frac{\sqrt{x}+3}{2\sqrt{x}-1}\Leftrightarrow x=1\)

5) Để biểu thức nhận giá trị nguyên

\(\Leftrightarrow\frac{4\sqrt{x}}{2-\sqrt{x}}\inℤ\)

\(\Leftrightarrow4\sqrt{x}⋮2-\sqrt{x}\)

\(\Leftrightarrow-4\left(2-\sqrt{x}\right)+8⋮2-\sqrt{x}\)

\(\Leftrightarrow8⋮2-\sqrt{x}\)

\(\Leftrightarrow2-\sqrt{x}\inƯ\left(8\right)=\left\{\pm1;\pm2;\pm4;\pm8\right\}\)

\(\Leftrightarrow\sqrt{x}\in\left\{1;3;0;4;-2;6;-6;10\right\}\)

Ta loại các giá trị < 0

\(\Leftrightarrow\sqrt{x}\in\left\{1;3;0;4;6;10\right\}\)

\(\Leftrightarrow x\in\left\{1;9;0;16;36;100\right\}\)

Vậy để \(P\inℤ\Leftrightarrow x\in\left\{1;9;0;16;36;100\right\}\)

\(\)

\(đkxđ\Leftrightarrow\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

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

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

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

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

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

\(P\in Z\Leftrightarrow1+\frac{3}{\sqrt{x}-1}\in Z\Rightarrow\frac{3}{\sqrt{x}-1}\in Z\)

\(\Rightarrow\sqrt{x}-1\inƯ_3\)

Mà \(Ư_3=\left\{\pm1;\pm3\right\}\)

\(Th1:\sqrt{x}-1=1\Rightarrow\sqrt{x}=2\Rightarrow x=4\)

\(Th2:\sqrt{x}-1=-1\Rightarrow\sqrt{x}=0\Rightarrow x=0\)

\(Th3:\sqrt{x}-1=3\Rightarrow\sqrt{x}=4\Rightarrow x=16\)

\(Th4:\sqrt{x}-1=-3\Rightarrow\sqrt{x}=-2\Rightarrow x\in\varnothing\)

\(\Rightarrow x\in\left\{0;4;16\right\}\)

\(M=\frac{x+12}{\sqrt{x}-1}.\left(1\div\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)

\(=\frac{x+12}{\sqrt{x}-1}.\frac{\sqrt{x}-1}{\sqrt{x}+2}=\frac{x+12}{\sqrt{x}+2}\)

\(=\frac{x-4+16}{\sqrt{x}+2}=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}+2}+\frac{16}{\sqrt{x}+2}\)

\(=\sqrt{x}-2+\frac{16}{\sqrt{x}+2}=\sqrt{x}+2+\frac{16}{\sqrt{x}+2}-4\)

Áp dụng Bất đẳng thức Cô - Si cho hai số nguyên dương \(\sqrt{x}+2;\frac{16}{\sqrt{x}+2}\)ta có :

\(\sqrt{x}+2+\frac{16}{\sqrt{x}+2}\ge2\sqrt{\left(\sqrt{x}+2\right).\frac{16}{\sqrt{x}+2}}\)

\(\Rightarrow\sqrt{x}+2+\frac{16}{\sqrt{x}+2}\ge2.\sqrt{16}=2.4=8\)

\(\Rightarrow\sqrt{x}+2+\frac{16}{\sqrt{x}+2}-4\ge4\)

\(\Rightarrow M_{min}=4\Leftrightarrow\sqrt{x}+2=\frac{16}{\sqrt{x}+2}\)

\(\Rightarrow\left(\sqrt{x}+2\right)^2=16\)

\(\Rightarrow\sqrt{x}+2=4\Rightarrow\sqrt{x}=2\Rightarrow x=4\)

\(KL:M_{min}=4\Leftrightarrow x=4\)

\(a,đkxđ\Leftrightarrow\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

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

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

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

\(c,A_{max}\Leftrightarrow1-x\)lớn nhất \(\Rightarrow x\)nhỏ nhất

Mà \(x\ge0\)\(\Rightarrow x\)nhỏ nhất \(\Leftrightarrow x=0\)

\(\Rightarrow A_{max}=1\Leftrightarrow x=0\)

ĐKXĐ:   \(x\ge0;\)\(x\ne1\)

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

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

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

\(=\frac{\sqrt{x}+1}{\sqrt{x}}:\frac{1}{\sqrt{x}-1}\)

\(=\frac{x-1}{\sqrt{x}}\)

a) bổ sung ĐKXĐ nhé:   \(x>0;\)\(x\ne1\)

b)  \(P< 0\)

=>  \(\frac{x-1}{\sqrt{x}}< 0\) 

=>  \(x-1< 0\)   (do \(\sqrt{x}>0\))

=>  \(x< 1\)

=>  \(0< x< 1\)

\(P=\frac{x-1}{\sqrt{x}}\)

\(=\frac{4-2\sqrt{3}-1}{\sqrt{4-2\sqrt{3}}}\)

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

\(=\frac{\sqrt{3}-3}{2}\)