I have an object that can accelerate at a known rate and decelerate at a different known rate that needs to cover a known distance as quickly as possible.
It must start and end stationary so it would have to accelerate at its acceleration rate for a certain (unknown) distance and then immediately start decelerating at its deceleration rate for the rest of the journey.
How can I work out the time the whole journey would take?
Let $t_1$ be the time your object accelerates. Write down a function $f(t_1)$ that gives the time it takes to decelerate to a standstill afterwards. Write down a function $h(t_1)$ that gives the distance covered by this. Set $h(t_1)$ equal to your desired distance, and solve for $t_1$. Plug the solution you find into $t_1+f(t_1)$ and evaluate.