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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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up vote 2 down vote accepted

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.

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Could you specify? Newton's method will do? I mean I'm not a mathematician, I just recently learned what numerical analysis is in my university, so I'm not good very good in this kind of things. – mrpyo Nov 4 '11 at 0:06
@mrpyo: It will, if you manage to figure out good initial estimates for the roots. – J. M. Nov 4 '11 at 0:28
Chapter 9 of Numerical Recipes (obsolete versions are free) has a discussion of available methods with C code. So will any numerical analysis text. – Ross Millikan Nov 4 '11 at 1:03

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