$v (dv / dx) = g$
$v \, dv = g \, dx$
$\displaystyle \int v \, dv = g \int dx$
$\dfrac{v^2}{2} = gx + \dfrac{c}{2}$
$v^2 = 2gx + c$
when x = xo, v = vo
${v_o}^2 = 2gx_o + c$
$c = {v_o}^2 - 2gx_o$
thus,
$v^2 = 2gx + ({v_o}^2 - 2gx_o)$
$v^2 - {v_o}^2 = 2gx - 2gx_o$
$v^2 - {v_o}^2 = 2g(x - x_o)$ answer