i)\(\left\{{}\begin{matrix}ab=2\\a^3+b^3=9\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}ab=2\\\left(a+b\right)^3-3ab\left(a+b\right)=9\end{matrix}\right.\)
\(\Rightarrow\left(a+b\right)^3-6\left(a+b\right)-9=0\)
\(\Leftrightarrow\left(a+b\right)^3-3\left(a+b\right)^2+3\left(a+b\right)^2-9\left(a+b\right)+3\left(a+b\right)-9=0\)
\(\Leftrightarrow\left(a+b-3\right)\left[\left(a+b\right)^2+3\left(a+b\right)+3\right]=0\)
\(\Leftrightarrow a+b=3\)( \(\left(a+b\right)^2+3\left(a+b\right)+3>0;\forall a,b\)
ii) \(\left\{{}\begin{matrix}a+b+ab=23\\a^2+b^2=34\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}ab=23-\left(a+b\right)\\\left(a+b\right)^2-2ab=34\end{matrix}\right.\)
\(\Rightarrow\left(a+b\right)^2-2\left[23-\left(a+b\right)\right]=34\)
\(\Leftrightarrow\left(a+b\right)^2+2\left(a+b\right)-80=0\)
\(\Leftrightarrow\left(a+b-8\right)\left(a+b+10\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a+b=8\\a+b=-10\end{matrix}\right.\)
