Why Equivalent?

Newton's method is an iterative technique for approximating the zeros of a smooth (differentiable) function f(x). In the picture below,

the right triangle with vertices

A = (x₂,0), B = (x₁, f(x₁)), and C = (x₁,0)

satisfies

Solving for x₂ now yields

In case f(x) = ax² + bx + c, this yields

Newton's iterator is always "flat" at a simple zero of f(x). This is because

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