Use Newton's method with initial approximation x1 = −1 to find x2, the second approximation to the solution of the following equation.

Question

x³ + x + 4 = 0

Answer 1

Newton's formula in this situation is

x_n+1 = x_n - [x_n³ + x_n + 4]/[3x_n² +1]

Substituting x₁ = -1 yields x₂ = -3/2

and that is a pretty good estimate for the root already!