I want to solve $t'$ in:
$$ut + \dfrac{at^2}{2} = ut' + \dfrac{a't'^2}{2}$$
Where I will know the values for constants $u$, $t$, $a$ and $a'$.
I believe I can reduce the above to a quadratic equation:
$$2ut + at^2 = 2ut' + a't'^2$$
$$a't'^2 + 2ut' - 2ut + at^2 = 0$$
Is that correct? Then once I have the constants I can simply substitute them for $a$ $b$ and $c$ in the quadratic formula to solve for $t'$?
(Background: my question comes from my physics question here: https://physics.stackexchange.com/questions/87490/how-long-would-it-take-a-projectile-accelerating-twice-as-fast-as-another-to-cat, but I now want to solve for any, positive, acceleration of the second projectile).
Yes, this is a quadratic equation and can be solved with the quadratic formula.
$$a = a', b = 2u, c = 2ut + at^2$$