We can obtain the result with some geometry:
The equation has not two distinct real roots if and only if its discriminant $b^2-20a\le 0$, i.e. if and only if, in the $(a,b)$-plane, the point $(a,b)$ is inside the parabola $\mathcal P$ with equation $b^2=20a$.
Consider the family of straight lines with equations $\;5a+b=k\enspace(k\in\mathbf R)$. The minimum sought for is the value of $k$ for which the corresponding straight line has a double intersection with $\mathcal P$.
This let's determine the intersection points: they satisfy the equations $b^2=20a\;$ and $\;5a=k-b$, whence
$$b^2=4(k-b)\iff b^2+4b=4k\iff(b+2)^2=4(k+1).$$
We have a double intersection if and only if $b=-2$, which corresponds to $\color{red}{k=-1}$.