;;;;;
;;;;;
;;;;; Equations
;;;;;
;;;;;

(define $solve1
  (lambda [$f $expr $x]
    (inverse expr f x)))

(define $solve
  (lambda [$eqs]
    (solve' eqs {})))

(define $solve'
  (lambda [$eqs $rets]
    (match eqs (list [math-expr math-expr symbol-expr])
      {[<nil> rets]
       [<cons [$f $expr $x] $rs>
        (solve' rs {@rets [x (solve1 (substitute rets f) (substitute rets expr) x)]})]})))

;;;
;;; Quadratic Equations
;;;
(define $quadratic-formula q-f)

(define $q-f
  (lambda [$f $x]
    (match (coefficients f x) (list math-expr)
      {[<cons $a_0 <cons $a_1 <cons $a_2 <nil>>>>
        (q-f' a_2 a_1 a_0)]})))

(define $q-f'
  (lambda [$a $b $c]
    [(/ (+ (* -1 b) (sqrt (- (** b 2) (* 4 a c)))) (* 2 a))
     (/ (- (* -1 b) (sqrt (- (** b 2) (* 4 a c)))) (* 2 a))]))

;;;
;;; Cubic Equations
;;;
(define $cubic-formula c-f)

(define $c-f
  (lambda [$f $x]
    (match (coefficients f x) (list math-expr)
      {[<cons $a_0 <cons $a_1 <cons $a_2 <cons $a_3 <nil>>>>>
        (c-f' a_3 a_2 a_1 a_0)]})))

(define $c-f'
  (lambda [$a $b $c $d]
    (match [a b c d] [math-expr math-expr math-expr math-expr]
      {[[,1 ,0 $p $q]
        (let {[[$u3 $v3] (q-f' 1 q (/ (* -1 p^3) 27))]}
          [(+ (rt 3 u3) (rt 3 v3))
           (+ (* w (rt 3 u3)) (* w^2 (rt 3 v3)))
           (+ (* w^2 (rt 3 u3)) (* w (rt 3 v3)))]
          )]
       [[,1 $a2 $a1 $a0]
        (let {[[$y1 $y2 $y3] (c-f' 1 0 (- a1 (/ a2^2 3)) (+ a0 (* (/ -1 3) a1 a2) (* (/ 2 27) a2^3)))]}
          [(- y1 (/ a2 3)) (- y2 (/ a2 3)) (- y3 (/ a2 3))]
          )]
       [[_ _ _ _] (c-f' 1 (/ b a) (/ c a) (/ d a))]})))