Tuning k Until the Roots Collide

For a real parameter $k$, let

$A = \begin{bmatrix} 5 & k \\ 3 & -1 \end{bmatrix}$

There is exactly one value of $k$ for which $A$ has a repeated eigenvalue — that is, its characteristic equation has a single root of multiplicity two instead of two distinct ones.

Find that $k$, and the repeated eigenvalue it produces.