  The metallicity dependence of the wide-binary fraction in stellar populations
plays a critical role in resolving the open question of wide binary formation.
In this paper, we investigate the metallicity ([Fe/H]) and age dependence of
the wide-binary fraction (binary separations between $10^3$ and $10^4$ AU) for
field F and G dwarfs within 500 pc by combining their metallicity and radial
velocity measurements from LAMOST DR5 with the astrometric information from
Gaia DR2. We show that the wide-binary fraction strongly depends on the
metallicity: as metallicity increases, the wide-binary fraction first
increases, peaks at [Fe/H]$\simeq 0$, and then decreases at the high
metallicity end. The wide-binary fraction at [Fe/H]$=0$ is about two times
larger than that at [Fe/H]$=-1$ and [Fe/H]$=+0.5$. This metallicity dependence
is dominated by the thin-disk stars. Using stellar kinematics as a proxy of
stellar age, we show that younger stars have a higher wide-binary fraction at
fixed metallicity close to solar. We propose that multiple formation channels
are responsible for the metallicity and age dependence. In particular, the
positive metallicity correlation at [Fe/H]$<0$ and the age dependence may be
due to the denser formation environments and higher-mass clusters at earlier
times. The negative metallicity correlation at [Fe/H]$>0$ can be inherited from
the similar metallicity dependence of close binaries, and radial migration may
play a role in enhancing the wide-binary fraction around the solar metallicity.
