If the quadratic $x^2+4mx+m$ has exactly one real root, find the positive value of $m$.
Consider the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Since the quadratic has exactly one real root, then its discriminant must be 0. Thus, this gives us  \begin{align*} 0&=b^2-4ac
\\\Rightarrow\qquad0&=(4m)^2-4m
\\\Rightarrow\qquad0&=16m^2-4m
\\\Rightarrow\qquad0&=4m(4m-1).
\end{align*}This gives us the two possible values of $m$: $0$ and $\frac{1}{4}$. Since the question only asks for the positive value, our final answer is $\boxed{\frac14}$.