Đặt: \(a=x-1\Rightarrow x-2=a-1\)
Ta có:
\(a^4+\left(a-1\right)^4=1\)
\(\Leftrightarrow a^4+a^4-4a^3+6a^2-4a+1=1\)
\(\Leftrightarrow2a^4-4a^3+6a^2-4a=0\)
\(\Leftrightarrow a\left(2a^3-4a^2+6a-4\right)=0\)
\(\Leftrightarrow a\left(2a^3-2a^2-2a^2+2a+4a-4\right)=0\)
\(\Leftrightarrow a\left[2a^2\left(a-1\right)-2a\left(a-1\right)+4\left(a-1\right)\right]=0\)
\(\Leftrightarrow a\left(a-1\right)\left(2a^2-2a+4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=0\\a=1\\2a^2-2a+4=2\left(a-\dfrac{1}{2}\right)^2+\dfrac{7}{2}=0\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\x-1=1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=2\end{matrix}\right.\)
\(\left(x-1\right)^4+\left(x-2\right)^4=1\)
Đặt: \(a=x-1\Rightarrow x-2=a-1\)
Ta có:
\(a^4+\left(a-1\right)^4=1\)
\(\Leftrightarrow a^4-a^4+4a^3-6a^2+4a-1=1\)
\(\Leftrightarrow4a^3-6a^2+4a-2=0\)
\(\Leftrightarrow4a^3-4a^2-2a^2+2a+2a-2=0\)
\(\Leftrightarrow4a^2\left(a-1\right)-2a\left(a-1\right)+2\left(a-1\right)=0\)
\(\Leftrightarrow\left(a-1\right)\left(4a^2-2a+2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a-1=0\\4a^2-2a+2=4\left(a-\dfrac{1}{4}\right)^2+\dfrac{7}{4}=0\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow a=1\Rightarrow x=2\)
Đặt x - 2 = t
=> t4 + (t - 1)4 = 1
<=> t4 + t4 - 4t3 + 6t2 - 4t + 1 - 1 = 0
<=> 2t4 - 4t3 + 6t2 - 4t = 0
<=> t(2t4 - 4t3 + 6t - 4) = 0
<=> t( 2t3 - 2t2 + 4t - 2t2 + 2t - 4 ) = 0
<=> t[t(2t2 - 2t + 4) - 1(2t2 - 2t + 4)] = 0
<=> t(t - 1)(2t2 - 2t + 4) = 0
=> t = 0
=> t - 1 = 0
=> 2t2 - 2t + 4 = 0
=> t = 0
=> t = 1
=> Không có nghiệm
=> x - 2 = 0
=> x = 2
=> x - 2 = 1
=> x = 3
