ĐKXĐ: \(x\ge2\)

pt \(\Leftrightarrow\left(2x-6\right)+\left(3\sqrt{x-2}-\sqrt{x+6}\right)=0\)

\(\Leftrightarrow2\left(x-3\right)+\frac{9\left(x-2\right)-\left(x+6\right)}{3\sqrt{x-2}+\sqrt{x+6}}=0\)

\(\Leftrightarrow2\left(x-3\right)+\frac{8\left(x-3\right)}{3\sqrt{x-2}+\sqrt{x+6}}=0\)

\(\Leftrightarrow\left(x-3\right)\left(2+\frac{8}{3\sqrt{x-2}+\sqrt{x+6}}\right)=0\) (1)

Với \(x\ge2\Rightarrow2+\frac{8}{3\sqrt{x-2}+\sqrt{x+6}}>0\)

(1) <=> x-3=0 <=> x=3 (tm ĐKXĐ)

Vậy x=3

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

\(\Leftrightarrow\left(\sqrt{x^2-3x+2}-\sqrt{x-2}\right)-\left(\sqrt{x^2+2x-3}+\sqrt{x+3}\right)=0\)

\(\Leftrightarrow\dfrac{\left(x^2-3x+2\right)-\left(x-2\right)}{\sqrt{x^2-3x+2}+\sqrt{x-2}}-\dfrac{\left(x^2+2x-3\right)-\left(x+3\right)}{\sqrt{x^2+2x-3}-\sqrt{x+3}}=0\)

\(\Leftrightarrow\dfrac{\left(x-2\right)^2}{\sqrt{\left(x-2\right)\left(x-1\right)}+\sqrt{x-2}}-\dfrac{\left(x-2\right)\left(x+3\right)}{\sqrt{\left(x+3\right)\left(x-1\right)}-\sqrt{x+3}}=0\)

\(\Leftrightarrow\left(x-2\right)\left[\dfrac{x-2}{\sqrt{x-2}\left(\sqrt{x-1}+1\right)}-\dfrac{x+3}{\sqrt{x+3}\left(\sqrt{x-1}-1\right)}\right]=0\)

\(\Leftrightarrow\left(x-2\right)\left[\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}-\dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}\right]=0\)

Pt \(\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}-\dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}=0\) vô n_o

(vì \(\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}< \dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}\forall x\ge2\Rightarrow VT< 0\))

=> x - 2 = 0

<=> x = 2 (nhận)

\(\sqrt{4x+1}-\sqrt{3x-2}=\dfrac{x+3}{5}\)

\(\Leftrightarrow\dfrac{\left(4x+1\right)-\left(3x-2\right)}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{x+3}{5}=0\)

\(\Leftrightarrow\dfrac{x+3}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{x+3}{5}=0\)

\(\Leftrightarrow\left(\dfrac{1}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{1}{5}\right)\left(x+3\right)=0\)

TH1:

x + 3 = 0

<=> x = - 3 (loại)

TH2:

\(\dfrac{1}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{1}{5}=0\)

\(\Leftrightarrow\sqrt{4x+1}+\sqrt{3x-2}=5\)

\(\Leftrightarrow\left(\sqrt{4x+1}-3\right)+\left(\sqrt{3x-2}-2\right)=0\)

\(\Leftrightarrow\dfrac{4x+1-9}{\sqrt{4x+1}+3}+\dfrac{3x-2-4}{\sqrt{3x-2}+2}=0\)

\(\Leftrightarrow\dfrac{4\left(x-2\right)}{\sqrt{4x+1}+3}+\dfrac{3\left(x-2\right)}{\sqrt{3x-2}+2}=0\)

\(\Leftrightarrow\left(\dfrac{4}{\sqrt{4x+1}+3}+\dfrac{3}{\sqrt{3x-2}+2}\right)\left(x-2\right)=0\)

Pt \(\dfrac{4}{\sqrt{4x+1}+3}+\dfrac{3}{\sqrt{3x-2}+2}>0\forall x\ge\dfrac{2}{3}\) => vô n_o

=> x - 2 = 0

<=> x = 2 (nhận)

~ ~ ~

Vậy x = 2

\(\sqrt{2x^2+8x+6}+\sqrt{x^2-1}=2x+2\)

\(\Leftrightarrow\sqrt{2\left(x^2+4x+3\right)}-\left[\left(2x+2\right)-\sqrt{x^2-1}\right]=0\)

\(\Leftrightarrow\sqrt{2\left(x+3\right)\left(x+1\right)}-\dfrac{\left(4x^2+8x+4\right)-\left(x^2-1\right)}{\sqrt{x^2-1}+2x+2}=0\)

\(\Leftrightarrow\sqrt{2\left(x+3\right)\left(x+1\right)}-\dfrac{\left(x+1\right)\left(3x+5\right)}{\sqrt{\left(x-1\right)\left(x+1\right)}+2\left(x+1\right)}=0\)

\(\Leftrightarrow\sqrt{x+1}\left[2\sqrt{x+3}-\dfrac{\sqrt{x+1}\left(3x+5\right)}{\sqrt{x+1}\left(\sqrt{x-1}+2\sqrt{x+1}\right)}\right]=0\)

\(\Leftrightarrow\sqrt{x+1}\left[2\sqrt{x+3}-\dfrac{3x+5}{\sqrt{x-1}+2\sqrt{x+1}}\right]=0\)

TH1

x + 1 = 0

<=> x = - 1 (loại)

TH2

\(2\sqrt{x+3}-\dfrac{3x+5}{\sqrt{x-1}+2\sqrt{x+1}}=0\)

mà \(2\sqrt{x+3}=\dfrac{4x+12}{2\sqrt{x+3}}>\dfrac{3x+5}{\sqrt{x-1}+2\sqrt{x+1}}\forall x\ge1\)

=> VT > 0

=> vô n_o

~ ~ ~

Vậy pt vô n_o

a/ ĐKXĐ: \(x\ge\frac{1}{2}\)

\(\Leftrightarrow x+1-\sqrt{2x+2}+\sqrt{2x-1}-1=0\)

\(\Leftrightarrow\frac{x^2+2x+1-2x-2}{x+1+\sqrt{2x+2}}+\frac{2x-1-1}{\sqrt{2x-1}+1}=0\)

\(\Leftrightarrow\left(x-1\right)\left(\frac{x+1}{x+1+\sqrt{2x+2}}+\frac{2}{\sqrt{2x-1}+1}\right)=0\)

\(\Rightarrow x=1\)

2/ ĐKXĐ:\(\left[{}\begin{matrix}x=0\\x\ge2\\x\le-3\end{matrix}\right.\)

- Nhận thấy \(x=0\) là 1 nghiệm

- Với \(x\ge2\):

\(\Leftrightarrow\sqrt{x-1}+\sqrt{x-2}=2\sqrt{x+3}=\sqrt{4x+12}\)

Ta có \(VT\le\sqrt{2\left(x-1+x-2\right)}=\sqrt{4x-6}< \sqrt{4x+12}\)

\(\Rightarrow VT< VP\Rightarrow\) pt vô nghiệm

- Với \(x\le-3\)

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

\(\Leftrightarrow3-2x+2\sqrt{x^2-3x+2}=-4x-12\)

\(\Leftrightarrow2\sqrt{x^2-3x+2}=-2x-15\) (\(x\le-\frac{15}{2}\))

\(\Leftrightarrow4x^2-12x+8=4x^2+60x+225\)

\(\Rightarrow x=-\frac{217}{72}\left(l\right)\)

Vậy pt có nghiệm duy nhất \(x=0\)

Bài 3: ĐKXĐ: \(-3\le x\le6\)

Đặt \(\sqrt{3+x}+\sqrt{6-x}=t\) \(\Rightarrow3\le t\le3\sqrt{2}\)

\(t^2=9+2\sqrt{\left(3+x\right)\left(6-x\right)}\Rightarrow-\sqrt{\left(3+x\right)\left(6-x\right)}=\frac{9-t^2}{2}\)

Phương trình trở thành:

\(t+\frac{9-t^2}{2}=m\Leftrightarrow-t^2+2t+9=2m\) (2)

a/ Với \(m=3\Rightarrow t^2-2t-3=0\Rightarrow\left[{}\begin{matrix}t=-1\left(l\right)\\t=3\end{matrix}\right.\)

\(\Rightarrow\sqrt{3+x}+\sqrt{6-x}=3\)

\(\Leftrightarrow2\sqrt{\left(3+x\right)\left(6-x\right)}=0\Rightarrow\left[{}\begin{matrix}x=-3\\x=6\end{matrix}\right.\)

b/ Xét hàm \(f\left(t\right)=-t^2+2t+9\) trên \(\left[3;3\sqrt{2}\right]\)

\(-\frac{b}{2a}=1< 3\Rightarrow\) hàm số nghịch biến trên \(\left[3;3\sqrt{2}\right]\)

\(f\left(3\right)=6\) ; \(f\left(3\sqrt{2}\right)=6\sqrt{2}-9\)

\(\Rightarrow6\sqrt{2}-9\le2m\le6\Rightarrow\frac{6\sqrt{2}-9}{2}\le m\le3\)

Bài 4 làm tương tự bài 3

a) 

ĐK x >= 0  (1)

pt <=> \(\sqrt{x+1}=\frac{1}{\sqrt{x}}-\sqrt{x}\)

ĐK \(\frac{1}{\sqrt{x}}-\sqrt{x}\ge0\) => \(\frac{1-x}{\sqrt{x}}\ge0\) => \(x\le1\) (2)

pt <=> \(x+1=\frac{1}{x}+x-2\Leftrightarrow\frac{1}{x}=3\Rightarrow x=\frac{1}{3}\) ( TM (1) và (2) ) 

Vậy x = 1/3 là n* của pt 

b) ĐKXĐ: t lười lắm, c tự tìm nhe :D

đặt a=x+3

b=x-3

khi đó ptr trở thành:

\(\frac{a+2\sqrt{ab}}{2b+\sqrt{ab}}\)=\(\sqrt{2}\)

<=>\(\frac{\sqrt{a}.\left(\sqrt{a}+2\sqrt{b}\right)}{\sqrt{b}\left(\sqrt{a}+2\sqrt{b}\right)}\)=\(\sqrt{2}\)

<=>\(\frac{\sqrt{a}}{\sqrt{b}}\)=\(\sqrt{2}\)

<=>a/b=2

<=>a=2b

<=>x+3=2(x-3)

<=>x+3=2x-6

<=>x=9(chắc chắn là thỏa mãn ĐKXĐ nhưng mà sao thay vào ko đc nhỉ.phát hiện lỗi sai sửa giùm t nhe! :D)