Quadratic Solver for Casio FX-3650P

[Albert Chan]

Discovered a trivia: discriminant is the same if quadratic is "shifted"

AX² + BX + C, let Y = X - k

Using synthetic division, we get: AY² + B'Y + C', where B' = 2Ak + B, C' = Ak² + Bk + C

B'² - 4 A C'
= (2Ak + B)² - 4A*(Ak² + Bk + C)
= (4A²k² + 4ABk + B²) - (4A²k² + 4ABk + 4AC)
= B² - 4 A C

Using previous post example: A, B, C = 8441600, -22222222, 14624809

-B/(2A) ≈ 1.316232823
Let Y = X - 1.3, quadratic => 8441600Y² - 274062 Y + 2224.4

4AC' has 10 significant digits, thus can evaluate exactly.
Slight adjustment made B'² part also exact ...

B'² - 4 A C'
= -4 A C' + (B'-2)(B'+2) + 4
= (-4)(8441600)(2224.4) + (274060)(274064) + 4
= -316

Edit: there is a simpler prove that discriminant unchanged when shifted.
Shifting is just another perspective to say where is considered zero.
The gap between the roots remains the same.
Since gap = √(D / A²), and A unchanged when shifted, D also unchanged.

Update: we could avoid B' adjustment by doing synthetic division again
let Z = Y - 0.016, quadratic => 8441600Z² - 3930.8 Z + 0.4576 = 0
Solve for Z, then X = Z + 1.316