31.
ĐKXĐ: \(\left[{}\begin{matrix}x\ge5\\x\le-1\end{matrix}\right.\)
\(\Leftrightarrow2\left(x^2-4x-5\right)+\sqrt{x^2-4x-5}-3=0\)
Đặt \(\sqrt{x^2-4x-5}=t\ge0\)
\(\Rightarrow2t^2+t-3=0\Rightarrow\left[{}\begin{matrix}t=1\\t=-\dfrac{3}{2}\left(loại\right)\end{matrix}\right.\)
\(\Rightarrow\sqrt{x^2-4x-5}=1\)
\(\Leftrightarrow x^2-4x-5=1\)
\(\Leftrightarrow x^2-4x-6=0\)
\(\Leftrightarrow x=2\pm\sqrt{10}\)
32.
ĐKXĐ: \(x\ge2\)
\(3\left(2+\sqrt{x-2}\right)=2x+\sqrt{x+6}\)
\(\Leftrightarrow6+3\sqrt{x-2}=2x+\sqrt{x+6}\)
\(\Leftrightarrow3\sqrt{x-2}-\sqrt{x+6}=2\left(x-3\right)\)
\(\Leftrightarrow\dfrac{8\left(x-3\right)}{3\sqrt{x-2}+\sqrt{x+6}}=2\left(x-3\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=0\Rightarrow x=3\\\dfrac{4}{3\sqrt{x-2}+\sqrt{x+6}}=1\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow3\sqrt{x-2}+\sqrt{x+6}=4\)
\(\Leftrightarrow10x-12+6\sqrt{\left(x-2\right)\left(x+6\right)}=16\)
\(\Leftrightarrow3\sqrt{x^2+4x-12}=14-5x\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le\dfrac{14}{5}\\9\left(x^2+4x-12\right)=\left(14-5x\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le\dfrac{14}{5}\\x^2-11x+19=0\end{matrix}\right.\)
\(\Rightarrow x=\dfrac{11-3\sqrt{5}}{2}\)
