Answer
Equation (d) $v^2=v_0^2+2ax$ is dimensionally correct.
Work Step by Step
(a) $x=v_0t+at^3$
LHS - $x$ has SI unit $m$.
RHS -
$v_0t$ has SI unit $m/s\times s$ = $m$.
$at^3$ has SI unit $m/s^2\times s^3$ = $ms$.
LHS $\ne$ RHS.
(b) $v^2=v_0^2+2at$
LHS - $v^2$ has SI unit $(m/s)^2$ = $m^2/s^2$.
RHS -
$v_0^2$ has SI unit $(m/s)^2$ = $m^2/s^2$.
$2at$ has SI unit $m/s^2\times s$ = $m/s$.
LHS $\ne$ RHS.
(c) $x=at+vt^2$
LHS - $x$ has SI unit $m$.
RHS -
$at$ has SI unit $m/s^2\times s$ = $m/s$.
$vt^2$ has SI unit $m/s\times s^2$ = $ms$.
LHS $\ne$ RHS.
(d) $v^2=v_0^2+2ax$
LHS - $v^2$ has SI unit $(m/s)^2$ = $m^2/s^2$.
RHS -
$v_0^2$ has SI unit $(m/s)^2$ = $m^2/s^2$.
$2ax$ has SI unit $m/s^2\times m$ = $m^2/s^2$.
LHS = RHS.
