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I'm having trouble understanding the method/reasoning behind finding the root of an equation though iterative convergence.

x^{2}- 4x + 1 = 0

x^{2}= + 4x - 1

x = 4 - 1/x

I can understand that once we input a 'root' the equation will equal be equal on both sides. (Due to the remainder theorem) However I can't grasp why it converges to a root in all cases...

I also can't seem to understand how if we set x as 3, then we get x or x_{0}as 3.667...

x = 4 - 1/(3) = 3.667

I obviously understand the equation it just doesn't quite fit how x is equal to 2 different values neither of which are a root?

How would it account for a equation that would have a 'higher' Y value for: x_{0}. Which 'goes past' the root? (I've drawn a blue line over the graph).

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# Numerical method: iteratation to converge on root

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