\(\left(x-6\right)^4+\left(x-8\right)^4=16\)
\(\Leftrightarrow\left(x-6\right)^4+\left(x-8\right)^4-16=0\)
\(\Leftrightarrow\left(x-8\right)\left[\left(x-6\right)^3+\left(x-6\right)^2.2+\left(x-6\right).2^2+2^3\right]+\left(x-8\right)^4=0\)
\(\Leftrightarrow\left(x-8\right)\left[\left(x-6\right)^3+\left(x-6\right)^2.2+\left(x-6\right).2^2+2^3+\left(x-8\right)^3\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[x^3-3x^2.6+3x.6^2-6^3+\left(x^2-12x+36\right).2+4x-24+8+x^3-3x^2.8+3x.8^2-8^3\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[x^3-18x^2+108x-216+2x^2-24x+72+4x-24+8+x^3-24x^2+192x-512\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[2x^3-40x^2+280x-672\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[x^3-20x^2+140x-336\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[x^3-6x^2-14x^2+84x+56x-336\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left[x^2\left(x-6\right)-14x\left(x-6\right)+56\left(x-6\right)\right]=0\)
\(\Leftrightarrow\left(x-8\right)\left(x-6\right)\left(x^2-14x+56\right)=0\)
\(\Leftrightarrow x-8=0\) hay \(x-6=0\) hay \(x^2-14x+56=0\)
\(\Leftrightarrow x=8\) hay \(x=6\) hay \(\left(x-7\right)^2+7=0\) (vô nghiệm).
\(\Leftrightarrow S=\left\{8;6\right\}\).
-Vậy tổng các nghiệm của phương trình là 14.
