Assume an object moves from $0$ m to $a$ m with an unknown constant acceleration $c$ m s$^{-2}$. The moment it gets to $a$ m we know that its velocity is $b$ m s$^{-1}$. So basically:
$$ \begin{align*} s(t_1)&=a\\ v(t_1)&=b\\ a(t)&=c \text{ for all }t \end{align*} $$
Now what will $t_1$ be? I have been working on this problem for quite a long time and I couldn't figure it out. I have tried some integrals but they didn't seem helpful.
Usual notation:
$\left \{ \begin{align*} v&=u+at \\ s&=ut+\frac{1}{2} at^{2} \end{align*} \right. \implies s=vt-\frac{1}{2}at^{2}$
Switch to notation in this case: \begin{align*} a &= bt_{1}-\frac{c}{2} t_{1}^{2} \\ 0 &= a - bt_{1}+\frac{c}{2} t_{1}^{2} \\ t_{1} &= \frac{b\pm \sqrt{b^{2}-2ac}}{c} \end{align*}
Check carefully for the validity of $t_{1}$