I have this information: $$u = 2i + 3j$$ $$r0(\text{initial position}) = 40i + 20j$$ $$r = 52i + 128j$$ $$a = -0.06i -0.04j$$
I need to find the $t(\text{time})$ at this point. I can use the equation $r = ut + \frac{1}{2}at^2 + r0 $.
Substituting in the vectors I have I get:
$$ 52i + 128j = 2ti + 3tj + \frac{1}{2}\left(-0.06t^2\right)i + \frac{1}{2}\left(-0.04t^2\right)j + 40i + 20j $$
Solving down, I get:
$$ 12i + 108j = \left(2t - 0.03t^2\right)i + \left(3t - 0.02t^2\right)j $$
From here, I tried just solving as a quadratic with just the $i$ components, and get $t = 60$(correct answer) and $t = 6.66666$. If I solve using the $j$ component, I get $t = 90$ and $t = 60$. I don't understand how to find out which is the correct answer to use in a situation without the answers to hand.