Why do these slightly different methods for finding roots yield different results?

The essential difference is in what function is being repeated twice. In the correct one average-damp is being applied twice, with the net effect of more damping ((repeated average-damp 2) f) reduces, mathematically, to (lambda (x) (+ (* 0.75 x) (* 0.25 (f x)))) (apologies if my syntax is off, my lisp is very, very rusty). This makes the algorithm less susceptible to the wild fluctuations of the transformation.

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