  By the construction of suitable graphs and the determination of their
chromatic polynomials, we resolve two open questions concerning real chromatic
roots. First we exhibit graphs for which the Beraha number $B_{10} = (5 +
\sqrt{5})/2$ is a chromatic root. As it was previously known that no other
non-integer Beraha number is a chromatic root, this completes the determination
of precisely which Beraha numbers can be chromatic roots. Next we construct an
infinite family of $3$-connected graphs such that for any $k \geqslant 1$,
there is a member of the family with $q=2$ as a chromatic root of multiplicity
at least $k$. The former resolves a question of Salas and Sokal [J. Statist.
Pys. 104 (2001) pp. 609--699] and the latter a question of Dong and Koh [J.
Graph Theory 70 (2012) pp. 262--283].
