(12C Platinum) Cubic Equation

[Albert Chan]

Another way to get good guess is "look" at shape of curve

Example 1: x^3 - 4x^2 + 6x - 24 = 0
x((x-2)² + 2) = 24 ~ 2.88³

Any guess > 2 work. Say, guess = 3
X = 3 -> 4.6667 -> 4.1220 -> 4.0051 -> 4.0000

Example 2: -2x^3 + 3x^2 + 4x - 5 = 0
x^3 - 1.5x^2 - 2x + 2.5 = 0
x((x-0.75)² - 2.5625) = -2.5 ~ -1.36³

If RHS is 0, we have roots -0.8508, 0, 2.3508, with LHS interval signs = - + - +
Any guess < -0.8508 work. Say, guess = -1.5
X = -1.5 -> -1.3649 -> -1.3509 -> -1.3508

Since LHS interval (0, 2.3508) also is negative, eqn might have 3 real roots.
Indeed it does, 3 roots = -1.3508, 1.0000, 1.8508