a.

ĐKXĐ: \(x\ne-1\)

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

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

Đặt \(\sqrt{x^2+x+2}=t>0\)

\(\Rightarrow t^2-2\left(x+1\right)t+4x=0\)

\(\Leftrightarrow t\left(t-2x\right)-2\left(t-2x\right)=0\)

\(\Leftrightarrow\left(t-2\right)\left(t-2x\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=2\\t=2x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+x+2}=2\\\sqrt{x^2+x+2}=2x\left(x\ge0\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+x+2=4\\x^2+x+2=4x^2\left(x\ge0\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-2\\\end{matrix}\right.\)

b.

ĐKXĐ: \(x\ge-1\)

\(x^2-5x+14-4\sqrt{x+1}=0\)

\(\Leftrightarrow\left(x^2-6x+9\right)+\left(x+1-4\sqrt{x+1}+4\right)=0\)

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

\(\Leftrightarrow\left\{{}\begin{matrix}x-3=0\\\sqrt{x+1}-2=0\end{matrix}\right.\)

\(\Leftrightarrow x=3\)

Lời giải:
ĐKXĐ: $x\geq -1$

PT $\Leftrightarrow (x^2-6x+9)+[(x+1)-4\sqrt{x+1}+4]=0$

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

Vì $(x-3)^2; (\sqrt{x+1}-2)^2\geq 0$ với mọi $x\geq -1$
Do đó để tổng của chúng $=0$ thì:

$(x-3)^2=(\sqrt{x+1}-2)^2=0$
$\Leftrightarrow x=3$ (tm)

ĐKXĐ: \(x\ge-1\)

\(\Leftrightarrow\left(x^2-6x+9\right)+\left(x+1-4\sqrt{x+1}+4\right)=0\)

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

\(\Leftrightarrow\left\{{}\begin{matrix}x-3=0\\\sqrt{x+1}-2=0\end{matrix}\right.\) \(\Leftrightarrow x=3\)

\(a,PT\Leftrightarrow x\sqrt{3}=x+2\\ \Leftrightarrow3x^2=x^2+4x+4\\ \Leftrightarrow2x^2-4x-4=0\Leftrightarrow x^2-2x-2=0\\ \Delta=4+8=12\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{2-2\sqrt{3}}{2}=1-\sqrt{3}\\x=\dfrac{2+2\sqrt{3}}{2}=1+\sqrt{3}\end{matrix}\right.\)

\(b,ĐK:x\ge\dfrac{2}{3}\\ PT\Leftrightarrow3x-2=7-4\sqrt{3}\\ \Leftrightarrow3x=9-4\sqrt{3}\\ \Leftrightarrow x=\dfrac{9-4\sqrt{3}}{3}\left(tm\right)\)

\(c,ĐK:x\ge-1\\ PT\Leftrightarrow\left(x+1-4\sqrt{x+1}+4\right)+\left(x^2-6x+9\right)=0\\ \Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(x-3\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}\sqrt{x+1}=2\\x-3=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+1=4\\x=3\end{matrix}\right.\Leftrightarrow x=3\left(tm\right)\)

Ta có x2−5x+14≐(x−3)2+x+5≥x+5≥x+1+4≥4√x+1x2−5x+14≐(x−3)2+x+5≥x+5≥x+1+4≥4x+1
⇒VT≥VP⇒VT≥VP
Để VT=VP thì x=3.(dấu "=" xảy ra) 

ĐK: x + 1 ≥ 0 <=> x ≥ - 1

<=> 16( x + 1) = x⁴ + 25x² + 196 - 10x³ + 28x² - 140x

<=> 16x + 16 = x⁴ - 10x³ + 53x² - 140x +196

<=> x⁴ - 10x³ + 53x² - 156x + 180 = 0

<=> ( x - 3)²(x² - 4x + 20 ) = 0

<=> x = 3

\(\left(ĐKXĐ:x\ge-1\right)\)

\(4\sqrt{x-1}-x-5=x^2-6x+9\)

\(\Leftrightarrow\frac{16\left(x+1\right)-\left(x+5\right)^2}{4\sqrt{x+1}+x+5}=\left(x-3\right)^2\)

\(\Leftrightarrow\frac{16x+16-x^2-10x-25}{4\sqrt{x+1}+x+5}-\left(x-3\right)^2\)\(=0\)

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

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

\(\Leftrightarrow\left(x-3\right)^2=0\)

\(\Leftrightarrow x=3\)

ĐKXĐ: \(x\in R\)

\(\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2\)

=>\(\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}+x^2+2x-4=0\)

\(\Leftrightarrow\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}+x^2+2x+1-5=0\)

=>\(\sqrt{3x^2+6x+7}-2+\sqrt{5x^2+10x+14}-3+\left(x+1\right)^2=0\)

=>\(\dfrac{3x^2+6x+7-4}{\sqrt{3x^2+6x+7}+2}+\dfrac{5x^2+10x+14-9}{\sqrt{5x^2+10x+14}+3}+\left(x+1\right)^2=0\)

=>

\(\dfrac{3x^2+6x+3}{\sqrt{3x^2+6x+7}+2}+\dfrac{5x^2+10x+5}{\sqrt{5x^2+10x+14}+3}+\left(x+1\right)^2=0\)

=>\(\dfrac{3\left(x^2+2x+1\right)}{\sqrt{3x^2+6x+7}+2}+\dfrac{5\left(x^2+2x+1\right)}{\sqrt{5x^2+10x+14}+3}+\left(x+1\right)^2=0\)

\(\Leftrightarrow\dfrac{3\left(x+1\right)^2}{\sqrt{3x^2+6x+7}+2}+\dfrac{5\left(x+1\right)^2}{\sqrt{5x^2+10x+14}+3}+\left(x+1\right)^2=0\)

=>\(\left(x+1\right)^2\left(\dfrac{3}{\sqrt{3x^2+6x+7}+2}+\dfrac{5}{\sqrt{5x^2+10x+14}+3}+1\right)=0\)

=>\(\left(x+1\right)^2=0\)

=>x+1=0

=>x=-1(nhận)

a)  \(\sqrt {2{x^2} - 14}  = x - 1\quad \left( 1 \right)\)

ĐK: \(x - 1 \ge 0\,\, \Leftrightarrow \,\,x \ge 1.\)

\( \Rightarrow \) TXĐ: \(D = \left[ {1; + \infty } \right)\)

\(\begin{array}{l}\left( 1 \right)\,\, \Leftrightarrow \,\,{\left( {\sqrt {2{x^2} - 14} } \right)^2} = {\left( {x - 1} \right)^2}\\ \Leftrightarrow \,\,2{x^2} - 14 = {x^2} - 2x + 1\\ \Leftrightarrow \,\,{x^2} + 2x - 15 = 0\\ \Leftrightarrow \,\,\left[ {\begin{array}{*{20}{c}}{x = 3}\\{x =  - 5}\end{array}} \right.\end{array}\)

Nhận thấy \(x = 3\) thỏa mãn điều kiện

Vậy nghiệm của phương trình \(\left( 1 \right)\)  là: \(x = 3\)

b)  \(\sqrt { - {x^2} - 5x + 2}  = \sqrt {{x^2} - 2x - 3} \quad \left( 2 \right)\)

ĐK: \(\left\{ {\begin{array}{*{20}{c}}{ - {x^2} - 5x + 2 \ge 0}\\{{x^2} - 2x - 3 \ge 0}\end{array}} \right.\,\, \Leftrightarrow \,\,\frac{{ - 5 - \sqrt {33} }}{2} \le x \le  - 1.\)

\( \Rightarrow \) TXĐ: \(D = \left[ {\frac{{ - 5 - \sqrt {33} }}{2}; - 1} \right].\)

\(\begin{array}{l}\left( 2 \right)\,\, \Leftrightarrow \,\,{\left( {\sqrt { - {x^2} - 5x + 2} } \right)^2} = {\left( {\sqrt {{x^2} - 2x - 3} } \right)^2}\\ \Leftrightarrow \,\, - {x^2} - 5x + 2 = {x^2} - 2x - 3\\ \Leftrightarrow \,\,2{x^2} + 3x - 5 = 0\\ \Leftrightarrow \,\,\left[ {\begin{array}{*{20}{c}}{x = 1}\\{x =  - \frac{5}{2}}\end{array}} \right.\end{array}\)

Nhận thấy \(x =  - \frac{5}{2}\) thỏa mãn điều kiện

Vậy nghiệm của phương trình \(\left( 2 \right)\) là: \(x =  - \frac{5}{2}\)

Câu a:

ĐKXĐ: \(x\geq 1\)

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

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

\(\Rightarrow x-1=8x-3+2\sqrt{(3x-2)(5x-1)}\) (bình phương 2 vế)

\(\Leftrightarrow 7x-2+2\sqrt{(3x-2)(5x-1)}=0\)

(Vô lý với mọi \(x\geq 1\) )

Do đó PT vô nghiệm.

Câu b)

PT \(\Leftrightarrow \sqrt{3(x^2+2x+1)+4}+\sqrt{5(x^2+2x+1)+9}=5-(x^2+2x+1)\)

\(\Leftrightarrow \sqrt{3(x+1)^2+4}+\sqrt{5(x+1)^2+9}=5-(x+1)^2\)

Vì \((x+1)^2\geq 0, \forall x\) nên:

\(\sqrt{3(x+1)^2+4}\geq \sqrt{4}=2\)

\(\sqrt{5(x+1)^2+9}\geq \sqrt{9}=3\)

\(\Rightarrow \sqrt{3(x+1)^2+4}+\sqrt{5(x+1)^2+9}\geq 5(1)\)

Mặt khác ta cũng có: \(5-(x+1)^2\leq 5-0=5(2)\)

Từ \((1);(2)\Rightarrow \sqrt{3(x+1)^2+4}+\sqrt{5(x+1)^2+9}\geq 5\geq 5-(x+1)^2\)

Dấu "=" xảy ra khi $(x+1)^2=0$ hay $x=-1$ (thỏa mãn)

Vậy pt có nghiệm $x=-1$

\(a,PT\Leftrightarrow\left|x+3\right|=3x-6\\ \Leftrightarrow\left[{}\begin{matrix}x+3=3x-6\left(x\ge-3\right)\\x+3=6-3x\left(x< -3\right)\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{9}{2}\left(tm\right)\\x=\dfrac{3}{4}\left(ktm\right)\end{matrix}\right.\\ \Leftrightarrow x=\dfrac{9}{2}\\ b,PT\Leftrightarrow\left|x-1\right|=\left|2x-1\right|\\ \Leftrightarrow\left[{}\begin{matrix}x-1=2x-1\\1-x=2x-1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{2}{3}\end{matrix}\right.\)

\(c,ĐK:x\le\dfrac{2}{5}\\ PT\Leftrightarrow4-5x=25x^2-20x+4\\ \Leftrightarrow25x^2-15x=0\\ \Leftrightarrow5x\left(5x-3\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\left(tm\right)\\x=\dfrac{3}{5}\left(ktm\right)\end{matrix}\right.\Leftrightarrow x=0\\ d,ĐK:x\le\dfrac{2}{5}\\ PT\Leftrightarrow4-5x=2-5x\\ \Leftrightarrow x\in\varnothing\)