a/ Đặt \(\sqrt{x^2-3x+5}=t>0\)

\(\Leftrightarrow t^2-5-t>1\Leftrightarrow t^2-t-6>0\)

\(\Rightarrow\left[{}\begin{matrix}t>3\\t< -2\left(l\right)\end{matrix}\right.\) \(\Rightarrow\sqrt{x^2-3x+5}>3\)

\(\Leftrightarrow x^2-3x+5>9\Leftrightarrow x^2-3x-4>0\Rightarrow\left[{}\begin{matrix}x>4\\x< -1\end{matrix}\right.\)

b/ ĐKXĐ: \(x\ge1\)

Đặt \(\sqrt[4]{x-\sqrt{x^2-1}}=t>0\Rightarrow\sqrt[4]{x+\sqrt{x^2-1}}=\frac{1}{t}\)

\(\Leftrightarrow t+\frac{4}{t^2}-3< 0\)

\(\Leftrightarrow t^3-3t^2+4< 0\)

\(\Leftrightarrow\left(t+1\right)\left(t-2\right)^2< 0\)

Do \(t>0\Rightarrow t+1>0\Rightarrow VT\ge0\Rightarrow\) BPT vô nghiệm

ĐKXĐ: \(-\dfrac{4}{3}\le x\le5\)

\(\left(\sqrt{3x+4}-4\right)+\left(1-\sqrt{5-x}\right)+\left(3x^2-8x-16\right)=0\)

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

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

\(\Leftrightarrow x-4=0\)

\(\Leftrightarrow x=4\)

\(\sqrt{3x+4}-\sqrt{5-x}+3x^2-8x-19=0\) (\(5\ge x\ge\dfrac{-4}{3}\))

Vì 2 vế không âm, theo BĐT Cô-si ta được:

\(\dfrac{3x+4+1}{2}\ge\sqrt{3x+4}\)

\(\dfrac{5-x+1}{2}\ge\sqrt{5-x}\) \(\Rightarrow\) \(\dfrac{x-6}{2}\le-\sqrt{5-x}\)

Dấu "=" xảy ra khi và chỉ khi \(\left[{}\begin{matrix}3x+4=1\\5-x=1\end{matrix}\right.\)

\(\Leftrightarrow\) \(\left[{}\begin{matrix}x=-1\left(KTM\right)\\x=4\left(TM\right)\end{matrix}\right.\)

Thay vào pt trên thấy pt luôn đúng nên x = 4 TMĐK

Vậy ...

Chúc bn học tốt! (Có gì sai mong bạn bỏ qua)

1. Đợi chút t tìm cách ngắn gọn.

2. ĐK: \(\left\{{}\begin{matrix}2x^2+8x+6\ge0\\x^2-1\ge0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x\le-3\\x\ge1\\x=-1\end{matrix}\right.\) (*)

BPT\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\3x^2+8x+5+2\sqrt{\left(2x^2+8x+6\right)\left(x^2-1\right)}\le\left(2x+2\right)^2\left(1\right)\end{matrix}\right.\)

Giải (1) \(\Leftrightarrow x^2-1-2\sqrt{\left(2x^2+8x+6\right)\left(x^2-1\right)}\ge0\)

\(\Leftrightarrow\sqrt{x^2-1}\left(\sqrt{x^2-1}-2\sqrt{2x^2+8x+6}\right)\ge0\)

TH1: \(\sqrt{x^2-1}=0\Leftrightarrow x=\pm1\) (tm)

TH2: \(x^2-1\ne0\)

\(\Leftrightarrow\sqrt{x^2-1}-2\sqrt{2x^2+8x+6}\ge0\)

\(\Leftrightarrow\sqrt{x^2-1}\ge2\sqrt{2x^2+8x+6}\)

\(\Leftrightarrow x^2-1\ge8x^2+32x+24\)

\(\Leftrightarrow7x^2+32x+25\le0\)

\(\Leftrightarrow-\frac{25}{7}\le x\le-1\) kết hợp đk (*) và đk để giải bpt

=>\(x=-1\)

Vậy \(x=\pm1\)

3. ĐK: \(x\ge\frac{4}{5}\)

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

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

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

\(\Leftrightarrow x-1>0\) \(\Leftrightarrow x>1\)

Vậy \(x>1\)

c:

ĐKXĐ: 6-5x>=0

=>5x<=6

=>x<=1,2

\(2\sqrt[3]{3x-2}-3\cdot\sqrt{6-5x}+16=0\)

=>\(2\cdot\sqrt[3]{3x-2}+4+12-3\cdot\sqrt{6-5x}=0\)

=>\(2\cdot\left(\sqrt[3]{3x-2}+2\right)+3\left(4-\sqrt{6-5x}\right)=0\)

=>\(2\cdot\frac{3x-2+8}{\sqrt[3]{\left(3x-2\right)^2}-2\cdot\sqrt[3]{3x-2}+4}+3\cdot\frac{16-6+5x}{4+\sqrt{6-5x}}=0\)

=>\(2\cdot\frac{3x+6}{\sqrt[3]{\left(3x-2\right)^2}-2\cdot\sqrt[3]{3x-2}+4}+3\cdot\frac{5x+10}{4+\sqrt{6-5x}}=0\)

=>\(\left(2\cdot\frac{3}{\sqrt[3]{\left(3x-2\right)^2}-2\cdot\sqrt[3]{3x-2}+4}+3\cdot\frac{5}{4+\sqrt{6-5x}}\right)\left(x+2\right)=0\)

=>x+2=0

=>x=-2(nhận)

d: ĐKXĐ: x>=1

\(\sqrt[3]{x+6}-2\cdot\sqrt{x-1}=4-x^2\)

=>\(\sqrt[3]{x+6}-2-2\cdot\sqrt{x-1}+2=4-x^2\)

=>\(\frac{x+6-8}{\sqrt[3]{\left(x+6\right)^2}+2\cdot\sqrt[3]{x+6}+4}+2\left(1-\sqrt{x-1}\right)=\left(2-x\right)\left(2+x\right)\)

=>\(\frac{x-2}{\sqrt[3]{\left(x+6\right)^2}+2\cdot\sqrt[3]{x+6}+4}+2\cdot\frac{1-x+1}{1+\sqrt{x-1}}=\left(2-x\right)\left(2+x\right)\)

=>\(\frac{x-2}{\sqrt[3]{\left(x+6\right)^2}+2\cdot\sqrt[3]{x+6}+4}-2\cdot\frac{x-2}{1+\sqrt{x-1}}-\left(2-x\right)\left(2+x\right)=0\)

=>\(\frac{x-2}{\sqrt[3]{\left(x+6\right)^2}+2\cdot\sqrt[3]{x+6}+4}-2\cdot\frac{x-2}{1+\sqrt{x-1}}+\left(x-2\right)\left(2+x\right)=0\)

=>\(\left(x-2\right)\left(\frac{1}{\sqrt[3]{\left(x+6\right)^2}+2\cdot\sqrt[3]{x+6}+4}-\frac{2}{1+\sqrt{x-1}}+\left(2+x\right)\right)=0\)

=>x-2=0

=>x=2(nhận)

2: ĐKXĐ: x>=0

\(\sqrt{3x}-2\sqrt{12x}+\dfrac{1}{3}\cdot\sqrt{27x}=-4\)

=>\(\sqrt{3x}-2\cdot2\sqrt{3x}+\dfrac{1}{3}\cdot3\sqrt{3x}=-4\)

=>\(\sqrt{3x}-4\sqrt{3x}+\sqrt{3x}=-4\)

=>\(-2\sqrt{3x}=-4\)

=>\(\sqrt{3x}=2\)

=>3x=4

=>\(x=\dfrac{4}{3}\left(nhận\right)\)

3: 

ĐKXĐ: x>=0

\(3\sqrt{2x}+5\sqrt{8x}-20-\sqrt{18}=0\)

=>\(3\sqrt{2x}+5\cdot2\sqrt{2x}-20-3\sqrt{2}=0\)

=>\(13\sqrt{2x}=20+3\sqrt{2}\)

=>\(\sqrt{2x}=\dfrac{20+3\sqrt{2}}{13}\)

=>\(2x=\dfrac{418+120\sqrt{2}}{169}\)

=>\(x=\dfrac{209+60\sqrt{2}}{169}\left(nhận\right)\)

4: ĐKXĐ: x>=-1

\(\sqrt{16x+16}-\sqrt{9x+9}=1\)

=>\(4\sqrt{x+1}-3\sqrt{x+1}=1\)

=>\(\sqrt{x+1}=1\)

=>x+1=1

=>x=0(nhận)

5: ĐKXĐ: x<=1/3

\(\sqrt{4\left(1-3x\right)}+\sqrt{9\left(1-3x\right)}=10\)

=>\(2\sqrt{1-3x}+3\sqrt{1-3x}=10\)

=>\(5\sqrt{1-3x}=10\)

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

=>1-3x=4

=>3x=1-4=-3

=>x=-3/3=-1(nhận)

6: ĐKXĐ: x>=3

\(\dfrac{2}{3}\sqrt{x-3}+\dfrac{1}{6}\sqrt{x-3}-\sqrt{x-3}=-\dfrac{2}{3}\)

=>\(\sqrt{x-3}\cdot\left(\dfrac{2}{3}+\dfrac{1}{6}-1\right)=-\dfrac{2}{3}\)

=>\(\sqrt{x-3}\cdot\dfrac{-1}{6}=-\dfrac{2}{3}\)

=>\(\sqrt{x-3}=\dfrac{2}{3}:\dfrac{1}{6}=\dfrac{2}{3}\cdot6=\dfrac{12}{3}=4\)

=>x-3=16

=>x=19(nhận)