I am not sure why you would attempt "to solve it using energy methods" when you are given forces. If there is no driving force them "force= mass times acceleration" gives $m\frac{dv}{dt}= -m(\alpha v+ \beta v^2)$. The "m"s cancel and we can "separate" the variables v and t as $\frac{dv}{\alpha v+ \beta v^2}= dt$. Integrate both sides.

$\int \frac{dv}{\alpha v+ \beta v^2}= \int \frac{dv}{v(\beta v+ \alpha)}$.

Use "partial fractions".

("decreased twice" is a very strange phrase. Going from $v_0$ to $\frac{v_0}{2}$ would be "decreased by half".)