\(x^4-16x^2+32x-16=0\)
\(\Leftrightarrow x^4-2x^3+2x^3-4x^2-12x^2+24x+8x-16=0\)
\(\Leftrightarrow x^3\left(x-2\right)+2x^2\left(x-2\right)-12x\left(x-2\right)+8\left(x-2\right)\)
\(\Leftrightarrow\left(x-2\right)\left(x^3+2x^2-12x+8\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^3-2x^2+4x^2-8x^2-4x+8\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left[x^2\left(x-2\right)+4x\left(x-2\right)-4\left(x-2\right)\right]=0\)
\(\Leftrightarrow\left(x-2\right)^2\left(x^2+4x-4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-2+2\sqrt{2}\\x=-2-2\sqrt{2}\end{matrix}\right.\)
Vậy.............
\(x^4-16x^2+32x-16=0\)
\(\Leftrightarrow x^4-16\left(x^2-2x+1\right)=0\)
\(\Leftrightarrow x^4-16\left(x-1\right)^2=0\)
\(\Leftrightarrow x^4-\left(4\left(x-1\right)\right)^2=0\)
\(\Leftrightarrow\left(x^2-4\left(x-1\right)\right).\left(x^2+4\left(x-1\right)\right)=0\)
\(\Leftrightarrow\left(x^2-4x+4\right).\left(x^2+4x-4\right)=0\)
\(\Leftrightarrow\left(x-2\right)^2.\left(x^2+4x-4\right)=0\)
\(\Leftrightarrow\)\(\left(x-2\right)^2=0\) hoặc \(x^2+4x-4=0\)
1) \(\left(x-2\right)^2=0\Leftrightarrow x-2=0\Leftrightarrow x=2\)
\(2\)) \(x^2+4x-4=0\Leftrightarrow x^2+4x+4-8=0\)
\(\Leftrightarrow\left(x+2\right)^2=8\)
\(\Leftrightarrow x+2=\sqrt{8}\) hoặc \(x+2=-\sqrt{8}\)
\(\Leftrightarrow x=\sqrt{8}-2\) \(x=-\sqrt{8}-2\)
Vậy tập nghiệm của phương trình là \(S=\left\{2;\sqrt{8}-2;-\sqrt{8}-2\right\}\)
\(x^4-16x^2+32x-16=0\)
\(\Leftrightarrow\left(x^4-16\right)-\left(16x^2-32x\right)=0\)
\(\Leftrightarrow\left(x^2-4\right)\left(x^2+4\right)-16x\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x+2\right)\left(x^2+4\right)-16x\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^3+2x^2-12x+8\right)=0\)
\(\Leftrightarrow\left(x-2\right)\left(x^3-2x^2+4x^2-8x^2-4x+8\right)=0\)
\(\Leftrightarrow\left(x-2\right)^2\left(x^2+4x-4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2=0\\x^2+4x-4=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-2+2\sqrt{2}\\x=-2-2\sqrt{2}\end{matrix}\right.\)
Vậy x có tập \(n_o\) \(S=\left\{2;-2+2\sqrt{2};-2-2\sqrt{2}\right\}\)
