--
--
-- Equations
--
--

solve1 f expr x := inverse expr f x

solve eqs := solve' eqs []

solve' eqs rets :=
  match eqs as list (mathExpr, mathExpr, symbolExpr) with
    | [] -> rets
    | ($f, $expr, $x) :: $rs ->
      solve'
        rs
        (rets ++ [(x, solve1 (substitute rets f) (substitute rets expr) x)])

--
-- Quadratic Equations
--
quadraticFormula := qF

qF f x :=
  match coefficients f x as list mathExpr with
    | $a_0 :: $a_1 :: $a_2 :: [] -> qF' a_2 a_1 a_0

qF' a b c :=
  ( ((- b) + sqrt (b ^ 2 - 4 * a * c)) / 2 * a
  , ((- b) - sqrt (b ^ 2 - 4 * a * c)) / 2 * a )

--
-- Cubic Equations
--
cubicFormula := cF

cF f x :=
  match coefficients f x as list mathExpr with
    | $a_0 :: $a_1 :: $a_2 :: $a_3 :: [] -> cF' a_3 a_2 a_1 a_0

cF' a b c d :=
  match (a, b, c, d) as (mathExpr, mathExpr, mathExpr, mathExpr) with
    | (#1, #0, $p, $q) ->
      let (s1, s2) := 2#(rt 3 %1, rt 3 %2) (qF' 1 (27 * q) ((-27) * p ^ 3))
       in ( (s1 + s2) / 3               -- r1
          , (w ^ 2 * s1 + w * s2) / 3   -- r2
          , (w * s1 + w ^ 2 * s2) / 3)  -- r3
    | (#1, _, _, _) ->
      3#(%1 - b / 3, %2 - b / 3, %3 - b / 3)
        (withSymbols [x, y]
          cF (substitute [(x, y - b / 3)] (x ^ 3 + b * x ^ 2 + c * x + d)) y)
    | (_, _, _, _) -> cF' 1 (b / a) (c / a) (d / a)