# Approximate zeros

Define a procedure `cubic` that can be used together with the `newtons-method` procedure in expressions of the form

``````(newtons-method (cubic a b c) 1)
``````

to approximate zeros of the cubic `x³ + ax² + bx + c` .

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``````(define tolerance 0.00001)

(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2)) tolerance))
(define (try guess)
(let ((next (f guess)))
(if (close-enough? guess next)
next
(try next))))
(try first-guess))

(define (deriv g)
(lambda (x)
(/ (- (g (+ x tolerance)) (g x))
tolerance)))

(define (newton-transform g)
(lambda (x)
(- x (/ (g x) ((deriv g) x)))))

(define (newtons-method g guess)
(fixed-point (newton-transform g) guess))

(check-equal? (round (newtons-method (cubic 0 0 0) 1)) 0.0)
(check-equal? (round (newtons-method (cubic 0 0 (- 27)) 10)) 3.0)
(check-equal? (round (newtons-method (cubic 0 0 (- 81)) 10)) 4.0)
(check-equal? (round (newtons-method (cubic 2 8 (- 32)) 10)) 2.0)``````