-- -- -- 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)