  We define a discrete-time, coined quantum walk on weighted graphs that is
inspired by Szegedy's quantum walk. Using this, we prove that many
lackadaisical quantum walks, where each vertex has $l$ integer self-loops, can
be generalized to a quantum walk where each vertex has a single self-loop of
real-valued weight $l$. We apply this real-valued lackadaisical quantum walk to
two problems. First, we analyze it on the line or one-dimensional lattice,
showing that it is exactly equivalent to a continuous deformation of the
three-state Grover walk with faster ballistic dispersion. Second, we generalize
Grover's algorithm, or search on the complete graph, to have a weighted
self-loop at each vertex, yielding an improved success probability when $l < 3
+ 2\sqrt{2} \approx 5.828$.
