a/ ĐKXĐ: \(\left|x\right|\ge1\)
- Với \(x\le-1\Rightarrow\left\{{}\begin{matrix}\sqrt{x^2+6}>0\\x-2\sqrt{x^2-1}< 0\end{matrix}\right.\) \(\Rightarrow\) pt vô nghiệm
- Với \(x>1\) ta luôn có \(\sqrt{x^2+6}>x\) (dễ dàng chứng minh bằng cách bình phương 2 vế)
Mà \(x>x-2\sqrt{x^2-1}\Rightarrow\sqrt{x^2+6}>x-2\sqrt{x^2-1}\)
Phương trình vô nghiệm
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b/ ĐKXĐ: \(x\ge1\)
\(\sqrt[3]{2-x}+\sqrt{x-1}=1\)
Đặt \(\left\{{}\begin{matrix}\sqrt[3]{2-x}=a\\\sqrt{x-1}=b\end{matrix}\right.\) ta có hệ:
\(\left\{{}\begin{matrix}a+b=1\\a^3+b^2=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}b=1-a\\a^3+b^2=1\end{matrix}\right.\) \(\Rightarrow a^3+\left(1-a\right)^2=1\)
\(\Leftrightarrow a^3+a^2-2a=0\) \(\Leftrightarrow a\left(a-1\right)\left(a+2\right)=0\Rightarrow\left[{}\begin{matrix}a=0\\a=1\\a=-2\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\sqrt[3]{2-x}=0\\\sqrt[3]{2-x}=1\\\sqrt[3]{2-x}=-2\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=2\\x=1\\x=10\end{matrix}\right.\)
c/
Đặt \(\left\{{}\begin{matrix}\sqrt[3]{x+1}=a\\\sqrt[3]{x-1}=b\end{matrix}\right.\) ta có hệ:
\(\left\{{}\begin{matrix}a^3-b^3=2\\a^2+b^2+ab=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}\left(a-b\right)\left(a^2+ab+b^2\right)=2\\a^2+b^2+ab=1\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=2\Rightarrow a=b+2\\a^2+b^2+ab=1\end{matrix}\right.\) \(\Rightarrow\left(b+2\right)^2+b^2+\left(b+2\right)b-1=0\)
\(\Leftrightarrow3b^2+6b+3=0\Rightarrow3\left(b+1\right)^2=0\Rightarrow b=-1\)
\(\Rightarrow\sqrt[3]{x-1}=-1\Rightarrow x=0\)