# How to solve following equation?

How to solve for $t$: $\|X + Vt + At^2\| = r$

$\|x\|$ is vector norm.

I would prefer explanation of how to get to exact solutions, I know definitions, it's more of an algebraic difficulty for me.

Equation in other form: $\sum (a_it^2 + v_it + x_i)^2 = r^2$

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If you expand the squares on the left, you will get a quartic in $t$. These can be solved algebraically, but it is a mess. Maybe there is something special about your constants so it simplifies, or maybe you should just solve it numerically.