Error in my algebra for differential equations (A level maths)

I did everything right up until 2h^1/2 = -kt+c. I wrote my incorrect working on the whiteboard. I am struggling to understand why the mark scheme converts values to A and B. Also why do you need to do this? Why is my way incorrect.

Your goal is to determine what k and C are subject to the boundary conditions. So you need two equations at least.

Your error comes from assuming C is 2.25 - it is not. You might want to write again "When t=... and h=..." for the initial condition with 2.25m.

With the confusion with A and B...
A is just -k/2, and B is just -C/2. They are arbitrary constants, we can write them however we like (in some sense).

Thank you! I redid the question and I got it right. Do you recommend using arbitrary constants?

Uh... As long as you keep track of your constants, arbitrary constants and variables, all is good.
For me, it really depends on how ugly the arbitrary constants get. In this case, k/2 and C/2 are nice enough that I wouldn't bother.