a.

ĐKXĐ: \(x^2+2x-1\ge0\)

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

Đặt \(\sqrt{x^2+2x-1}=t\ge0\)

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

\(\Delta'=\left(x-1\right)^2+4x=\left(x+1\right)^2\)

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

\(\Rightarrow\left[{}\begin{matrix}\sqrt{x^2+2x-1}=2\\\sqrt{x^2+2x-1}=-2x\left(x\le0\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+2x-5=0\\3x^2-2x+1=0\left(vn\right)\end{matrix}\right.\)

\(\Rightarrow x=-1\pm\sqrt{6}\)

b.

ĐKXĐ: \(x\ge\dfrac{1}{5}\)

\(2x^2+x-3+2x-\sqrt{5x-1}+2-\sqrt[3]{9-x}=0\)

\(\Leftrightarrow\left(x-1\right)\left(2x+3\right)+\dfrac{\left(x-1\right)\left(4x-1\right)}{2x+\sqrt[]{5x-1}}+\dfrac{x-1}{4+2\sqrt[3]{9-x}+\sqrt[3]{\left(9-x\right)^2}}=0\)

\(\Leftrightarrow\left(x-1\right)\left(2x+3+\dfrac{4x-1}{2x+\sqrt[]{5x-1}}+\dfrac{1}{4+2\sqrt[3]{9-x}+\sqrt[3]{\left(9-x\right)^2}}\right)=0\)

\(\Leftrightarrow x=1\) (ngoặc đằng sau luôn dương)

1.

ĐKXĐ: \(x\ge-\dfrac{1}{3}\)

\(\Leftrightarrow3x^2-3x+\left(x+1-\sqrt{3x+1}\right)+\left(x+2-\sqrt{5x+4}\right)=0\)

\(\Leftrightarrow3\left(x^2-x\right)+\dfrac{x^2-x}{x+1+\sqrt{3x+1}}+\dfrac{x^2-x}{x+2+\sqrt{5x+4}}=0\)

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

\(\Leftrightarrow x^2-x=0\)

\(\Leftrightarrow...\)

2.

Đặt \(\left\{{}\begin{matrix}2x=a\\\sqrt[3]{2-8x^3}=b\end{matrix}\right.\)

Ta được hệ:

\(\left\{{}\begin{matrix}\left(2a-1\right)b=a\\a^3+b^3=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=2ab\\\left(a+b\right)^3-3ab\left(a+b\right)=2\end{matrix}\right.\)

\(\Rightarrow8\left(ab\right)^3-6\left(ab\right)^2=2\)

\(\Leftrightarrow\left(ab-1\right)\left[4\left(ab\right)^2+ab+1\right]=0\)

\(\Leftrightarrow ab=1\Rightarrow a+b=2\)

\(\Rightarrow\left\{{}\begin{matrix}a+b=2\\ab=1\end{matrix}\right.\) \(\Leftrightarrow a=b=1\)

\(\Rightarrow2x=1\Rightarrow x=\dfrac{1}{2}\)

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

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

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

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

\(\Leftrightarrow a=b\)

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

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

\(\Leftrightarrow...\)