Lời giải:

ĐKXĐ: $x\geq 1$
PT $\Leftrightarrow \sqrt{(x-1)+2\sqrt{x-1}+1}-\sqrt{(x-1)-2\sqrt{x-1}+1}=2$
$\Leftrightarrow \sqrt{(\sqrt{x-1}+1)^2}-\sqrt{(\sqrt{x-1}-1)^2}=2$

$\Leftrightarrow |\sqrt{x-1}+1|-|\sqrt{x-1}-1|=2$
Nếu $2\geq x\geq 1$ thì:

$\sqrt{x-1}+1+(1-\sqrt{x-1})=2$
$\Leftrightarrow 2=2$ (luôn đúng)

Nếu $x>2$ thì: $\sqrt{x-1}+1+(\sqrt{x-1}-1)=2$
$\Leftrightarrow 2\sqrt{x-1}=2$
$\Leftrightarrow x-1=1$

$\Leftrihgtarrow x=2$ (loại)

Vậy $2\geq x\geq 1$

$

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

\(b,x=6-2\sqrt{5}=\left(\sqrt{5}-1\right)^2\\ \Rightarrow P=\dfrac{\sqrt{5}-1}{\sqrt{5}-1+1}=\dfrac{\sqrt{5}-1}{\sqrt{5}}=\dfrac{5-\sqrt{5}}{5}\\ c,\dfrac{P}{\sqrt{x}}=\dfrac{\sqrt{x}}{\sqrt{x}-1}\cdot\dfrac{1}{\sqrt{x}}=\dfrac{1}{\sqrt{x}-1}\le\dfrac{1}{0-1}=-1\)

Vậy \(\left(\dfrac{P}{\sqrt{x}}\right)_{max}=-1\Leftrightarrow x=0\)

Mình sửa lại đề tí:

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

ĐKXĐ: \(x\ge1\)

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

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

\(\Leftrightarrow\left|\sqrt{x-1}+1\right|-\left|\sqrt{x-1}-1\right|=2\)

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

TH1: \(x\ge2\Rightarrow\sqrt{x-1}-1\ge0\) pt trở thành:

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

TH2: \(1\le x< 2\)

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

\(\Leftrightarrow2\sqrt{x-1}=2\Rightarrow x=2\) (ktm)

Vậy nghiệm của pt là \(x\ge2\)

\(a,P=\left[\dfrac{\left(1-\sqrt{x}\right)\left(x+\sqrt{x}+1\right)}{1-\sqrt{x}}+\sqrt{x}\right]\left[\dfrac{\left(1+\sqrt{x}\right)\left(x-\sqrt{x}+1\right)}{1+\sqrt{x}}-\sqrt{x}\right]\\ P=\left(x+2\sqrt{x}+1\right)\left(x-2\sqrt{x}+1\right)\\ P=\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2\\ P=\left(x-1\right)^2\\ b,x=\sqrt{3+2\sqrt{2}}=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\\ \Leftrightarrow P=\left(\sqrt{2}+1-1\right)^2=\left(\sqrt{2}\right)^2=2\)

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

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

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

ĐKXĐ: x>=0; x<>1

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

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

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

b: Khi x=4-2căn 3=(căn 3-1)^2 thì \(B=\dfrac{\sqrt{3}-1}{\sqrt{3}-1+1}=\dfrac{\sqrt{3}-1}{\sqrt{3}}=\dfrac{3-\sqrt{3}}{3}\)

c: B=2/3

=>căn x/căn x+1=2/3

=>căn x=2

=>x=4

d: \(B-1=\dfrac{\sqrt{x}-\sqrt{x}-1}{\sqrt{x}+1}=-\dfrac{1}{\sqrt{x}+1}< 0\)

=>B<1

e: B>1

=>-1/căn x+1>0

=>căn x+1<0(vô lý)

=>KO có x thỏa mãn

f: B nguyên khi căn x chia hết cho căn x+1

=>căn x+1-1 chia hết cho căn x+1

=>căn x+1=1 hoặc căn x+1=-1(loại)

=>căn x=0

=>x=0

\(1,\\ a,ĐK:\left\{{}\begin{matrix}x\ge0\\x+5\ge0\end{matrix}\right.\Leftrightarrow x\ge0\\ b,Sửa:B=\left(\sqrt{3}-1\right)^2+\dfrac{24-2\sqrt{3}}{\sqrt{2}-1}\\ B=4-2\sqrt{3}+\dfrac{2\sqrt{3}\left(\sqrt{2}-1\right)}{\sqrt{2}-1}\\ B=4-2\sqrt{3}+2\sqrt{3}=4\\ 3,\\ =\left[1-\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{1+\sqrt{x}}\right]\cdot\dfrac{\sqrt{x}-3+2-2\sqrt{x}}{\left(1-\sqrt{x}\right)\left(\sqrt{x}-3\right)}-2\\ =\left(1-\sqrt{x}\right)\cdot\dfrac{-\sqrt{x}-1}{\left(1-\sqrt{x}\right)\left(\sqrt{x}-3\right)}-2\\ =\dfrac{-\sqrt{x}-1}{\sqrt{x}-3}-2=\dfrac{-\sqrt{x}-1-2\sqrt{x}+6}{\sqrt{x}-3}=\dfrac{-3\sqrt{x}+5}{\sqrt{x}-3}\)

\(A=P:Q=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}:\dfrac{\sqrt{x}+4}{\sqrt{x}+2}\)

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

Điều kiện : \(x\ge4\Rightarrow\sqrt{x}+4\ge4\Rightarrow-\dfrac{5}{\sqrt{x}+4}\le-\dfrac{5}{4}\Rightarrow\dfrac{5}{\sqrt{x}+4}\ge\dfrac{5}{4}\)

Dấu ''='' xảy ra \(\Leftrightarrow x=0\)

Vậy \(min_A=\dfrac{5}{4}\Leftrightarrow x=0\)