\(x\sqrt{3x-2}+\sqrt{3-2x}=\sqrt{x^3+x^2+x+1}\)
\(\Leftrightarrow x\sqrt{3x-2}-1+\sqrt{3-2x}-1=\sqrt{x^3+x^2+x+1}-2\)
\(\Leftrightarrow\dfrac{x\left(3x-2\right)-1}{x\sqrt{3x-2}+1}+\dfrac{3-2x-1}{\sqrt{3-2x}+1}=\dfrac{x^3+x^2+x+1-4}{\sqrt{x^3+x^2+x+1}+2}\)
\(\Leftrightarrow\dfrac{3x^2-2x-1}{x\sqrt{3x-2}+1}+\dfrac{2-2x}{\sqrt{3-2x}+1}-\dfrac{x^3+x^2+x-3}{\sqrt{x^3+x^2+x+1}+2}=0\)
\(\Leftrightarrow\dfrac{\left(x-1\right)\left(3x+1\right)}{x\sqrt{3x-2}+1}+\dfrac{-2\left(x-1\right)}{\sqrt{3-2x}+1}-\dfrac{\left(x-1\right)\left(x^2+2x+3\right)}{\sqrt{x^3+x^2+x+1}+2}=0\)
\(\Leftrightarrow\left(x-1\right)\left(\dfrac{3x+1}{x\sqrt{3x-2}+1}-\dfrac{2}{\sqrt{3-2x}+1}-\dfrac{x^2+2x+3}{\sqrt{x^3+x^2+x+1}+2}\right)=0\)
Dễ thấy: \(\dfrac{3x+1}{x\sqrt{3x-2}+1}-\dfrac{2}{\sqrt{3-2x}+1}-\dfrac{x^2+2x+3}{\sqrt{x^3+x^2+x+1}+2}=0\) vô nghiệm
\(\Rightarrow x-1=0\Rightarrow x=1\)
\(x\sqrt{3x-2}+\sqrt{3-2x}=\sqrt{x^3+x^2+x+1}\)
\(\Leftrightarrow x\sqrt{3x-2}-1+\sqrt{3-2x}-1=\sqrt{x^3+x^2+x+1}-2\)
\(\Leftrightarrow\dfrac{x^2\left(3x-2\right)-1}{x\sqrt{3x-2}+1}+\dfrac{3-2x-1}{\sqrt{3-2x}+1}=\dfrac{x^3+x^2+x+1-4}{\sqrt{x^3+x^2+x+1}+2}\)
\(\Leftrightarrow\dfrac{3x^3-2x^2-1}{x\sqrt{3x-2}+1}+\dfrac{2-2x}{\sqrt{3-2x}+1}=\dfrac{x^3+x^2+x-3}{\sqrt{x^3+x^2+x+1}+2}\)
\(\Leftrightarrow\dfrac{\left(x-1\right)\left(3x^2+x-1\right)}{x\sqrt{3x-2}+1}+\dfrac{-2\left(x-1\right)}{\sqrt{3-2x}+1}-\dfrac{\left(x-1\right)\left(x^2+2x+3\right)}{\sqrt{x^3+x^2+x+1}+2}=0\)
\(\Leftrightarrow\left(x-1\right)\left(\dfrac{3x^2+x-1}{x\sqrt{3x-2}+1}-\dfrac{2}{\sqrt{3-2x}+1}-\dfrac{x^2+2x+3}{\sqrt{x^3+x^2+x+1}+2}\right)=0\)
Dễ thấy: \(\dfrac{3x^2+x-1}{x\sqrt{3x-2}+1}-\dfrac{2}{\sqrt{3-2x}+1}-\dfrac{x^2+2x+3}{\sqrt{x^3+x^2+x+1}+2}=0\) vô nghiệm
\(\Rightarrow x-1=0\Rightarrow x=1\)
