An algebra problem by Theodore Jonathan

According to Vi_et we have: $x_1+x_2=\frac{2015-c}{2}$ $x_1x_2=84$ Try several pair of two roots to get: $x_1=7, x_2=12$ $c=\boxed{1977}$

Dividing this quadratic equation through by $2$ , one obtains:

$x^2 + [(c-2015)/2 ]x + 84 = 0$

and we'd like to factor this into $(x - A)(x - B) = 0$ . Knowing that $84 = 2^{2} 3^{1} 7^{1}$ for its prime factorization, we can write the following:

$(x-2)(x-42) = 0;$

$(x-3)(x-28) = 0;$

$(x-7)(x-12) = 0.$

such that we have the maximum number of prime roots available. The third equation yields the largest integral value $\boxed{c = 1977}$ (using coefficient matching).