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

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

\(\Leftrightarrow x^3=5x\left(x-2\right)\left(x+2\right)\)

\(\Leftrightarrow x\left(x^2-5x^2+20\right)=0\)

\(\Leftrightarrow4x\left(5-x^2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\sqrt{5}\\x=-\sqrt{5}\end{matrix}\right.\)

.

Đặt \(\sqrt{x^2-5x+5}=t>0\)

\(\Rightarrow log_2\left(t+1\right)+log_3\left(t^2+2\right)-2=0\)

Nhận thấy \(t=1\) là 1 nghiệm của pt

Xét hàm \(f\left(t\right)=log_2\left(t+1\right)+log_3\left(t^2+2\right)-2\)

\(f'\left(t\right)=\dfrac{1}{\left(t+1\right)ln2}+\dfrac{2t}{\left(t^2+2\right)ln3}>0\Rightarrow f\left(t\right)\) đồng biến

\(\Rightarrow f\left(t\right)\) có tối đa 1 nghiệm

\(\Rightarrow t=1\) là nghiệm duy nhất của pt

\(\Rightarrow\sqrt{x^2-5x+5}=1\Rightarrow\left[{}\begin{matrix}x=1\\x=4\end{matrix}\right.\)

\(ĐK:x\in R\)

Đặt \(\sqrt{x^2+3}=t\left(t\ge0\right)\)

\(PT\Leftrightarrow2t^2-\left(7x+1\right)t+3x^2+3x=0\\ \Delta=\left(7x+1\right)^2-4\cdot2\left(3x^2+3x\right)=25x^2-10x+1=\left(5x-1\right)^2\ge0\\ \Leftrightarrow\left[{}\begin{matrix}t=\dfrac{7x+1-5x+1}{4}\\t=\dfrac{7x+1+5x-1}{4}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}t=\dfrac{2x+2}{4}=\dfrac{x+1}{2}\\t=\dfrac{12x}{4}=3x\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+3}=\dfrac{x+1}{2}\\\sqrt{x^2+3}=3x\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x^2+3=\dfrac{x^2+2x+1}{4}\\x^2+3=9x^2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}3x^2-2x+11=0\\x^2=\dfrac{3}{8}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\Delta=4-132< 0\\\left[{}\begin{matrix}x=\dfrac{\sqrt{6}}{4}\\x=-\dfrac{\sqrt{6}}{4}\end{matrix}\right.\end{matrix}\right.\)

Vậy \(S=\left\{-\dfrac{\sqrt{6}}{4};\dfrac{\sqrt{6}}{4}\right\}\)

\(Đk:x\ge\dfrac{3}{2}\Rightarrow x>0\)

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

\(\Leftrightarrow2x^3-8x^2+10x-2-2\sqrt{2x-3}=0\)

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

\(\Leftrightarrow2x\left(x-2\right)^2+\left(\sqrt{2x-3}-1\right)^2=0\)

Ta có: \(\left\{{}\begin{matrix}2x\left(x-2\right)^2\ge0\left(x>0\right)\\\left(\sqrt{2x-3}-1\right)^2\ge0\end{matrix}\right.\)

\(\Rightarrow2x\left(x-2\right)^2+\left(\sqrt{2x-3}-1\right)^2\ge0\)

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

Thử lại ta có x=2 là nghiệm duy nhất của phương trình đã cho.

x^3-4x^2+5x-1-căn 2x-3=0

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

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

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

=>x-2=0

=>x=2

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