Let $p^3+q^3=4$ and $pq=\frac{2}{3}$ . Find $p+q$.
A graphing calculator can find values of $p$ and $q$ numerically. As one can see from the graph below, the two solutions are approximately $(0.4, 1.6)$ and $(1.6, 0.4)$:
However, I am interested in a symbolic solution. Is there a method to solve this problem quickly without having to use a graphing calculator?