Lời giải:
\(a^3+8b^3=1-6b\)
\(\Leftrightarrow a^3+8b^3+6ab-1=0\)
\(\Leftrightarrow (a+2b)^3-3.a.2b(a+2b)+6ab-1=0\)
\(\Leftrightarrow [(a+2b)^3-1]-6ab(a+2b-1)=0\)
\(\Leftrightarrow (a+2b-1)[(a+2b)^2+(a+2b)+1]-6ab(a+2b-1)=0\)
\(\Leftrightarrow (a+2b-1)(a^2+4b^2+1+a+2b-2ab)=0\)
\(\Rightarrow \left[\begin{matrix} a+2b-1=0(1)\\ a^2+4b^2+1+a+2b-2ab=0(2)\end{matrix}\right.\)
Với \((1)\Rightarrow a+2b=1\)
Với \((2)\Rightarrow 2a^2+8b^2+2+2a+4b-4ab=0\)
\(\Leftrightarrow (a^2+4b^2-4ab)+(a^2+2a+1)+(4b^2+4b+1)=0\)
\(\Leftrightarrow (a-2b)^2+(a+1)^2+(2b+1)^2=0\)
\(\Rightarrow (a-2b)^2=(a+1)^2=(2b+1)^2=0\)
\(\Rightarrow \left\{\begin{matrix} a=-1\\ 2b=-1\end{matrix}\right.\Rightarrow a+2b=-2\)
Vậy $a+2b\in \left\{1;-2\right\}$
Lời giải:
\(a^3+8b^3=1-6b\)
\(\Leftrightarrow a^3+8b^3+6ab-1=0\)
\(\Leftrightarrow (a+2b)^3-3.a.2b(a+2b)+6ab-1=0\)
\(\Leftrightarrow [(a+2b)^3-1]-6ab(a+2b-1)=0\)
\(\Leftrightarrow (a+2b-1)[(a+2b)^2+(a+2b)+1]-6ab(a+2b-1)=0\)
\(\Leftrightarrow (a+2b-1)(a^2+4b^2+1+a+2b-2ab)=0\)
\(\Rightarrow \left[\begin{matrix} a+2b-1=0(1)\\ a^2+4b^2+1+a+2b-2ab=0(2)\end{matrix}\right.\)
Với \((1)\Rightarrow a+2b=1\)
Với \((2)\Rightarrow 2a^2+8b^2+2+2a+4b-4ab=0\)
\(\Leftrightarrow (a^2+4b^2-4ab)+(a^2+2a+1)+(4b^2+4b+1)=0\)
\(\Leftrightarrow (a-2b)^2+(a+1)^2+(2b+1)^2=0\)
\(\Rightarrow (a-2b)^2=(a+1)^2=(2b+1)^2=0\)
\(\Rightarrow \left\{\begin{matrix} a=-1\\ 2b=-1\end{matrix}\right.\Rightarrow a+2b=-2\)
Vậy $a+2b\in \left\{1;-2\right\}