A car accelerates from rest at a constant rate α for some time...

${v}_{0} = \alpha{t}_{1}$ and $0 = {v}_{0} - \beta{t}_{2} ⇒ {v}_{0} = \beta{t}_{2}$

${t}_{1} + {t}_{2} = t$

${v}_{0}\left(\dfrac{1}{\alpha} + \dfrac{1}{\beta}\right) = t$

$⇒ {v}_{0} = \dfrac{\alpha\betat}{\alpha + \beta}$

Distance = area of $v - t$ graph

$= \dfrac{1}{2} \times t \times {v}_{0} = \dfrac{1}{2} \times t \times \dfrac{\alpha\betat}{\alpha + \beta} = \dfrac{\alpha\beta{t}^{2}}{2 ( \alpha + \beta )}$