giải phương trình, tìm đkxd đầy đủ, giải thường, k giải bằng máy tính
\(\Leftrightarrow\left\{{}\begin{matrix}x+1\ge0\Leftrightarrow x\ge-1\\\left(x+1\right)^2=2\left(x+1\right)+2\sqrt{2\left(x+1\right)+2\sqrt{4\left(x+1\right)}}\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow2\sqrt{2\left(x+1\right)+2\sqrt{4\left(x+1\right)}}=x^2-1\ge0\Leftrightarrow\left[{}\begin{matrix}x\ge1\\x\le-1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\left(thỏa\right)\\x\ge1\end{matrix}\right.\)
\(với:x\ge1\Rightarrow\left(1\right)\Leftrightarrow\left(x+1\right)^2\left(x-1\right)^2=8\left(x+1\right)+8\sqrt{4\left(x+1\right)}\)
\(đặt:x+1=t\ge2\)
\(pt\Leftrightarrow t^2\left(t-2\right)^2=8t+16\sqrt{t}\)
\(\Leftrightarrow t\left(t-2\right)^2=8+\dfrac{16}{\sqrt{t}}\Leftrightarrow\left(t-2\right)^2\left(t-4+4\right)-8-\dfrac{16}{\sqrt{t}}=0\)
\(\)\(\Leftrightarrow\left(t-4\right)\left(t-2\right)^2+4\left(t-2\right)^2-16+8-\dfrac{16}{\sqrt{t}}=0\)
\(\Leftrightarrow\left(t-4\right)\left(t-2\right)^2+4\left(\left(t-2\right)^2-2^2\right)+8\left(\dfrac{\sqrt{t}-2}{\sqrt{t}}\right)=0\)
\(\Leftrightarrow\left(t-4\right)\left(t-2\right)^2+4t\left(t-4\right)+8\left(\dfrac{\dfrac{t-4}{\sqrt{t}+2}}{\sqrt{t}}\right)=0\)
\(\Leftrightarrow\left(t-4\right)\left(t-2\right)^2+4t\left(t-4\right)+8\left(\dfrac{t-4}{\sqrt{t}\left(\sqrt{t}+2\right)}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}t-4=0\Leftrightarrow t=4\Rightarrow x=3\left(tm\right)\\\left(t-2\right)^2+4t+\dfrac{8}{\sqrt{t}\left(\sqrt{t}+2\right)}=0\left(2\right)\end{matrix}\right.\)
\(\left(2\right)>0\forall t\ge2\Rightarrow\left(2\right)vô\) \(nghiệm\Rightarrow S=\left\{-1;3\right\}\)
\(\)
