The quadratic equation $x^2-5x+t =3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.
\(t\in \{0,7,12,15,16\}\\ {\color{blue}The\ average}\ of\ all\ distinct\ possible values\ of\ t\ \color{blue}is\ \frac{50}{5}=10.\)
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