Question

a particle moves along the x-axis in such a way that its acceleration at any time t is given by a(t)= 6t - 18. At time t = 0, the velocity v(t) = 24. and at time t = 1, the position x(t) = 20
i have to find the expression for the velocity and what values t is at rest

Answer 1

since a(t) = 6t -18
v(t) = 3t^2 - 18 + c
but we are told v(0) = 24
so 24 = 0 + 0 + c, ----> c = 24

if v(t) = 3t^2 - 18t + 24
x(t) = t^3 - 9t^2 + 24t + k
but we are told x(1) = 20
so 20 = 1 - 9 + 24 + k , ---> k = 4

x(t) = t^3 - 9t^2 + 24t + 4

Your last sentence is confusing.
Did you mean " ... and what values of t the particle is at rest" ?

anyway, since you now have all 3 equations, I am pretty sure you can proceed from there