Suppose that $P(x)$ is a polynomial with rational coefficients that has a root at $x = 8 + 3\sqrt{3}$. Find another root of $P(x)$.

Quadratic formula: Roots of ax^2 + bx + c = 0 are x = (-b pm sqrt(b^2 - 4ac))/2a

 

If one root is 8 + 3*sqrt(3), then another possible root is 3 + 8*sqrt(3).