# Thread: Simplification of root finding problem

1. ## Simplification of root finding problem

Hi,
I was wondering whether there is any simplification available to find roots for something like
$
\sum_{j=0}^n A_j(t) = 0.
$

where,
$
A_n(t) = \frac{p_{n2} t^2 + p_{n1} t + p_{n0}}{q_{n2} t^2 + q_{n1} t + q_{n0}}
$

Thanks

2. Originally Posted by krindik
Hi,
I was wondering whether there is any simplification available to find roots for something like
$
\sum_{j=0}^n A_j(t) = 0.
$

where,
$
A_n(t) = \frac{p_{n2} t^2 + p_{n1} t + p_{n0}}{q_{n2} t^2 + q_{n1} t + q_{n0}}
$
Almost certainly not. Equations of the form $\sum_{j=0}^n A_j(t) = 0$ would include (as a very special case) anything of the form $at^2 + bt + c + \frac dt + \frac e{t^2} = 0$. So any general method for solving such equations would include a method for solving the general quartic equation. To say the least, that would make it quite an elaborate procedure.

3. Thanks a lot.
Infact A_n(t) have the exact same form below.

$
A_n(t) = \frac{(x-a_n) t^2 + (y-b_n) t + b_n^2}{ (y-a_n)^2 t^2 + 2(x-a_n) t + 2b_n}
$

The parameters p, q consists of only x, y and a_n, b_n differ for each A_n.
So isnt there any simplification i can do so that roots in terms of x and y can be found?