2020 AIME II Problem 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.