what t gives y(t)=0?

0 = y(t) y(t_a) + y'(t_a) (t-t_a)
so t t_a - y(t_a)/y'(t_a)

Newton-Raphson algorithm: t_n+1 = t_n - y(t_n)/y'(t_n)

Multi-dimensional version:

J*(X-X_o) = -F(X_o)