x-1 + x-3 =1 <=> 2x -4=1 tu giai not

1   ĐKXD \(x\ge1\)

.\(2x^2+5x-1=7\sqrt{\left(x-1\right)\left(x^2+x+1\right)}\)

Đặt \(\sqrt{x-1}=a;\sqrt{x^2+x+1}=b\left(a,b\ge0\right)\)

=> \(2b^2+3a^2=2x^2+5x-1\)

=> \(2b^2+3a^2-7ab=0\)

<=> \(\orbr{\begin{cases}a=2b\\a=\frac{1}{3}b\end{cases}}\)

+ \(a=2b\)

=> \(2\sqrt{x^2+x+1}=\sqrt{x-1}\)

=> \(4x^2+3x+5=0\)vô nghiệm

+ \(a=\frac{1}{3}b\)

=> \(\sqrt{x^2+x+1}=3\sqrt{x-1}\)

=> \(x^2-8x+10=0\)

<=> \(\orbr{\begin{cases}x=4+\sqrt{6}\left(tmĐK\right)\\x=4-\sqrt{6}\left(kotmĐK\right)\end{cases}}\)

Vậy \(x=4+\sqrt{6}\)

ĐKXĐ:\(2x^2-1\ge0;x^2-3x-2\ge0;2x^2+2x+3\ge0;x^2-x+2\ge0\)

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

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

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

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

Vì \(\frac{1}{\sqrt{2x^2+2x+3}+\sqrt{2x^2-1}}+\frac{1}{\sqrt{x^2-x+2}+\sqrt{x^2-3x-2}}>0\)

nên pt(1) <=> \(2x+4=0\Leftrightarrow x=-2\)(tmđk)

Vậy x=-2

Em kiểm tra lại đề bài câu trên nhé

2, 

PT

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

<=> \(\frac{2x+4}{\sqrt{2x^2+2x+3}+\sqrt{2x^2-1}}+\frac{2x+4}{\sqrt{x^2-x+2}+\sqrt{x^2-3x-2}}=0\)

<=> \(\orbr{\begin{cases}x=-2\left(tmĐK\right)\\\frac{1}{\sqrt{2x^2+2x+3}+\sqrt{2x^2-1}}+\frac{1}{\sqrt{x^2-x+2}+\sqrt{x^2-3x-2}}=0\left(2\right)\end{cases}}\)

PT (2) vô nghiệm do VT>0 với x tm ĐKXĐ
Vậy x=-2

\(ĐK:x\ge\dfrac{1}{2}\\ PT\Leftrightarrow2x-2\sqrt{2x^2+5x-3}=1+x\sqrt{2x-1}-2x\sqrt{x+3}\\ \Leftrightarrow\left(2x-2\right)-\left(2\sqrt{2x^2+5x-3}-4\right)=\left(x\sqrt{2x-1}-x\right)-\left(2x\sqrt{x+3}-4x\right)-3x+3\\ \Leftrightarrow2\left(x-1\right)-\dfrac{2\left(2x^2+5x-7\right)}{\sqrt{2x^2+5x-3}+4}=\dfrac{x\left(2x-2\right)}{\sqrt{2x-1}+1}-\dfrac{2x\left(x-1\right)}{\sqrt{x+3}+4x}-3\left(x-1\right)\\ \Leftrightarrow2\left(x-1\right)-\dfrac{2\left(x-1\right)\left(2x+7\right)}{\sqrt{2x^2+5x-3}+4}-\dfrac{2x\left(x-1\right)}{\sqrt{2x-1}+1}+\dfrac{2x\left(x-1\right)}{\sqrt{x+3}+4x}+3\left(x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left[2-\dfrac{2\left(2x+7\right)}{\sqrt{2x^2+5x-3}+4}-\dfrac{2x}{\sqrt{2x-1}+2}+\dfrac{2x}{\sqrt{x+3}+4x}+3\right]=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\left(tm\right)\\2-\dfrac{2\left(2x+7\right)}{\sqrt{2x^2+5x-3}+4}-\dfrac{2x}{\sqrt{2x-1}+2}+\dfrac{2x}{\sqrt{x+3}+4x}+3=0\left(1\right)\end{matrix}\right.\)

Với \(x\ge\dfrac{1}{2}\Leftrightarrow-\dfrac{2\left(2x+7\right)}{\sqrt{2x^2+5x-3}+4}>-\dfrac{2\cdot8}{4}=-4\)

\(-\dfrac{2x}{\sqrt{2x-1}+2}>-\dfrac{1}{2};\dfrac{2x}{\sqrt{x+3}+4x}>0\)

Do đó \(\left(1\right)>2-4-\dfrac{1}{2}+3=\dfrac{1}{2}>0\) nên (1) vô nghiệm

Vậy PT có nghiệm duy nhất \(x=1\)

Đặt \(\left\{{}\begin{matrix}\sqrt[3]{x^2+3x+1}=a\\\sqrt[3]{5x+1}=b\end{matrix}\right.\)

\(\Rightarrow a+a^3-b^3=b\)

\(\Leftrightarrow a-b+\left(a-b\right)\left(a^2+ab+b^2\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(a^2+ab+b^2+1\right)=0\)

\(\Leftrightarrow a=b\)

\(\Leftrightarrow\sqrt[3]{x^2+3x+1}=\sqrt[3]{5x+1}\)

\(\Leftrightarrow x^2+3x+1=5x+1\)

\(\Leftrightarrow...\)

\(pt\Leftrightarrow3\left(x-1\right)+2\left(x^2+x+1\right)=7\sqrt{\left(x-1\right)\left(x^2+x+1\right)}\)

Ta thấy x=1 không là nghiệm chia 2 vế cho x-1 ta đc:

\(3+\frac{2\left(x^2+x+1\right)}{x-1}=7\sqrt{\frac{x^2+x+1}{x-1}}\)

Đặt \(t=\sqrt{\frac{x^2+x+1}{x-1}}\left(t\ge0\right)\)

\(\Leftrightarrow3+2t^2=7t\)

\(\Leftrightarrow\orbr{\begin{cases}t=3\\t=\frac{1}{2}\end{cases}\left(tm\right)}\)

\(\Leftrightarrow\frac{x^2+x+1}{x-1}=9\)

\(\Leftrightarrow x^2+x+1=9x-9\)

\(\Leftrightarrow x^2-8x+10=0\)

\(\Leftrightarrow x=4\pm\sqrt{6}\)

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

\(\Leftrightarrow x^2+\frac{3}{4}x+\frac{5}{4}=0\)

\(\Leftrightarrow\left(x+\frac{3}{8}\right)^2+\frac{71}{64}>0\)(vô nghiệm)

Vậy pt trên có nghiệm thỏa mãn là \(x=4\pm\sqrt{6}\)