Quadratic Solver for Casio FX-3650P

[Albert Chan]

Another way to get discriminant, from Kahans cubic equation paper (page 11)
BTW, page 12 example 2 typo, should be: 1,160,928,203² - 3,234,424,451 * 416,690,636 = -89,060,331,627

PHP Code:

function b2_less_ac(a, b, c)  -- assume a, c > 0
    if a < c then a, c = c, a end
    if verbal then io.write(math.abs(b), '\xfd - ', a, ' * ', c, '\n') end
    local n = 0
    if c>0 then n = math.floor(b/c + 0.5) end
    if n == 0 then return b*b - a*c end
    local h = math.floor(n/2)
    a = a - (n-h)*b - h*b
    b = b - (n-h)*c - h*c
    return b2_less_ac(c, b, a - n*b)
end 


To show that reduction code work, let reduced a = a', reduced b = b':
b'² = (b - nc)² = b² - nc * (2b - nc) = b² - nc * (b+b')
a'c = (a - nb - nb') * c = ac - nc * (b+b')
-> b'² - a'c = b² - ac

lua> A, B, C = 8441600, -22222222, 14624809 -- previous post example
lua> verbal = true
lua> b2_less_ac(4*A, B, C)
22222222² - 33766400 * 14624809
7027396² - 14624809 * 3376748
273900² - 3376748 * 22217
7296² - 22217 * 2396
108² - 2396 * 5
2² - 64 * 5
-316

Doing by hand, we can do slightly better:

(-22222222)² - (4)(8441600)(14624809)
= 22222222² - (5849923600)(84416) -- n = round(22222222/84416) = 263, b' = b-nc = 20814
= 20814² - (84416)(5849923600 - 263*22222222 - 263*20814)
= 20814² - (84416)(5132)
= -316