Quadratic Equations and Common Roots

Product of roots of one equation is 35. Since roots are positive integers, the pair is either $(5, 7)$ or $(1, 35)$. Case 1: Roots are $5, 7$ and $C$. $5 + 7 + C = 41 \implies C = 29$. The roots are $5, 7, 29$. Case 2: Roots are $1, 35$ and $C$. $1 + 35 + C = 41 \implies C = 5$. The roots are $1, 35, 5$. In both cases, 5 is always one of the three roots.