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?
Best Answer
Hint:
$$p^3+q^3=4$$ $$p^3q^3=(\frac{2}{3})^3 \Rightarrow q^3=\frac{8}{27p^3}$$
The equation $$p^3+\frac{8}{27p^3}=4$$ is quadratic in $p^3$.