The quadratic equation $x^2-5x+t =-x^2+3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.

Answer 1

Rearranging the equation, we have \[2x^2-8x + t = 0.\]By Vieta's formulas, the sum of the possible roots of this equation is $$\frac 82 = 4$$Since the quadratic factors into $(2x-a)(x-b)$ for some positive integers $a,b,$ we have $ab = 2t$, and $a + b = 4$, which is possible only for $\{a,b\} = \{1,8\}$ or $\{2,4\}$. In either case, $t=ab/2=4$, so the average of all possible $t$ is $\boxed{4}$. (Though, to be thorough, we could verify that both of these cases actually do produce integer roots of the original quadratic.)