I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I have found is not proven to be a part of the cubic formula in accordance ot the Factor Theorem. I attempted to plug in my found root hoping for a returned value of either 0 or any number relatively close to 0 to no avail.
Function: $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$
Depressed Function: $0.08y^3 + 104.1y = 110.5625$
Simplified Tri-Quadratic: $t^6 + 110.5625t^3 - 41781.923 $
Invalid Root Found: 17.050929844523631089522642277055993836395567319866
If anyone could, please let me know what I may have potentially done wrong.
EDIT: Tried to re-do my depressed function. Factor theorem still verifies that its wrong.
$$0.08x^3 - 3.84x^2 + 42.66x - 137.7625$$
$$x = y - \frac b{3a}$$ $$x = y - \frac{-3.84}{0.24}$$ $$x = y + 16$$
$$0.08(y+16)^3 - 3.84(y+16)^2 + 42.66(y+16) - 137.7625 = 0$$
$$0.08y^3 - 18.78y = 110.5625$$
$$3st = -18.78$$ $$st = -6.26$$ $$s = \frac{-6.26}t$$
$$s^3 - t^3 = 110.5625$$
$$\left(\frac{-6.26}t\right)^3 - t^3 = 110.5625$$
$$-245.314376 = t^3(110.5625 + t^3) t^6 + 110.5625t^3 + 245.314376$$
$$\frac{-b + \sqrt{b^2 - 4ac}} {2a}$$
$$t^3 = -2.265193700398649889770836478972638144674233105284$$ $$t = -1.3133136204762414303468003214424883499397557722014$$
$$s^3 = 110.5625 + (-2.265193700398649889770836478972638144674233105284)$$ $$s = (110.5625 + (-2.265193700398649889770836478972638144674233105284)) ^ {\frac 13}$$
$$x = y + 16 = (110.5625 + (-2.265193700398649889770836478972638144674233105284)) ^{\frac13} - (-1.3133136204762414303468003214424883499397557722014 ) + 16$$
$$x = 22.079882627603390814664491720652545369380018281066$$
But $f(x) = -6379.1161837130485049747130791414393334925879583964$