Question 11
Let $P(x) = x^2 - 3x - 7,$ and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1.$ David computes each of the three sums $P + Q,$ $P + R,$ and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2,$ then $R(0) = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$