If a + b = 2 and a^3 + b^3 = 14, then find all possible values of a^2 + b^2.
+26367 If \(a + b = 2\) and \(a^3 + b^3 = 14\), then find all possible values of \(a^2 + b^2\).
\(\begin{array}{|rcll|} \hline (a+b)^3 &=& a^3+3a^2b+3ab^2+b^3 \\ (a+b)^3 &=& a^3+b^3 + 3ab(a+b) \quad & | \quad a+b = 2,\ a^3+b^3=14 \\ 2^3 &=& 14+ 3ab\times 2 \\ 8 &=& 14+ 6ab \quad & | \quad :2 \\ 4 &=& 7+ 3ab \\ 3ab &=& -3 \\ \mathbf{ab} &=& \mathbf{-1} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline (a+b)^2 &=& a^2+2ab+b^2 \\ (a+b)^2 &=& a^2+b^2+2ab \quad & | \quad \mathbf{ab=-1} \\ (a+b)^2 &=& a^2+b^2+2(-1) \quad & | \quad a+b = 2 \\ 2^2 &=& a^2+b^2 -2 \\ 4 &=& a^2+b^2 -2 \\ a^2+b^2 &=& 4+2 \\ \mathbf{ a^2+b^2} &=&\mathbf{ 6} \\ \hline \end{array}\)
