The distance x of a particle moving along a straight line from a fixed point on line at time t after start is given by t = ax² + bx + c (a, b, c are positive constants). If v be velocity of particle and u(≠0) be initial velocity of particle n which one is correct?

We have, t = ax² + bx + c  ???.(1)
Differentiating both sides of (1) with respect to x we get,
dt/dx = d(ax² + bx + c)/dx = 2ax + b
Thus, v = velocity of the particle at time t
= dx/dt = 1/(dt/dx) = 1/(2ax + b) = (2ax + b)^-1  ???.(2)
Initially, when t = 0 and v = u, let x = x₀; hence, from (1) we get,
ax₀² + bx₀ + c = 0
Or ax₀² + bx₀ = -c   ???.(3)
And from (2) we get, u = 1/(2ax₀ + b)
Thus, 1/v² ? 1/u² = (2ax + b)² ? (2ax₀ + b)²
= 4a²x² + 4abx ? 4a²x₀² ? 4abx₀
= 4a²x² + 4abx ? 4a(ax₀² ? bx₀)
= 4a²x² + 4abx ? 4a(-c)   [using (3)]
= 4a(ax² + bx + c)
Or 1/v² ? 1/u² = 4at