\(\left(x-1\right)\left(x-3\right)\left(x-5\right)\left(x-7\right)=20\\ \Leftrightarrow\left[\left(x-1\right)\left(x-7\right)\right]\left[\left(x-3\right)\left(x-5\right)\right]-20=0\\ \Leftrightarrow\left(x^2-8x+7\right)\left(x^2-8x+15\right)-20=0\\ \Leftrightarrow\left(x^2-8x+7\right)\left[\left(x^2-8x+7\right)+8\right]-20=0\\ \Leftrightarrow\left(x^2-8x+7\right)^2+8\left(x^2-8x+7\right)-20=0\\ \Leftrightarrow\left(x^2-8x+7\right)^2-2\left(x^2-8x+7\right)+10\left(x^2-8x+7\right)-20=0\\ \Leftrightarrow\left(x^2-8x+7\right)\left(x^2-8x+7-2\right)+10\left(x^2-8x+7-2\right)=0\)
\(\Leftrightarrow\left(x^2-8x+7+10\right)\left(x^2-8x+7-2\right)=0\\ \Leftrightarrow\left(x^2-8x+17\right)\left(x^2-8x+5\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x^2-8x+16+1=0\\x^2-8x+16-11=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}\left(x-4\right)^2+1=0\left(vô.lí\right)\\\left(x-4\right)^2-11=0\end{matrix}\right.\\ \Leftrightarrow\left(x-4-\sqrt{11}\right)\left(x-4+\sqrt{11}\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=4+\sqrt{11}\\x=4-\sqrt{11}\end{matrix}\right.\)
\(\left(x-1\right)\left(x-3\right)\left(x-5\right)\left(x-7\right)-20\)
\(=\left[\left(x-1\right)\left(x-7\right)\right]\left[\left(x-3\right)\left(x-5\right)\right]-20\)
\(=\left(x^2-8x+7\right)\left(x^2-8x+15\right)-20\)
Đặt \(x^2-8x+7=t\),ta có :
\(t\left(t+8\right)-20\)
\(=t^2+8t-20\)
\(=\left(t^2+8t+16\right)-16-20\)
\(=\left(t+4\right)^2-36\)
\(=\left(t+4\right)^2-6^2\)
\(=\left(t+4-6\right)\left(t+4+6\right)\)
\(=\left(t-2\right)\left(t+10\right)\)
\(=\left(x^2-8x+7-2\right)\left(x^2-8x+7+10\right)\)
\(=\left(x^2-8x+5\right)\left(x^2-8x+17\right)\)
