<s>[INST] Let $p^3+q^3=4$ and $pq=2/3$ . Find $p+q$. 
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?
 [/INST] we have $p^3 + q^3 = 4$ and $pq = 2/3$ so $p^3 q^3 = 8/27.$  therefore 
$$0= (x - p^3)(x-q^3) = x^2 - (p^3 + q^3)x + p^3q^3 = x^2 - 4x + 8/27$$ solving this quadratic equation we find 
$x = \dfrac{4 \pm \sqrt{16 - 4*8/27}}{2} =\dfrac{6\sqrt 3 \pm 10}{ 3\sqrt 3}$
so $$p+q = \left( \dfrac{6\sqrt 3 + 10}{ 3\sqrt 3}\right)^{1/3} + \left( \dfrac{6\sqrt 3 - 10}{ 3\sqrt 3}\right)^{1/3} $$
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