TITLE: Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.
QUESTION [0 upvotes]: 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 $f(x) = -6379.1161837130485049747130791414393334925879583964$

REPLY [0 votes]: The problem exists in the step, after you write down this equation:
$$0.08y^3−18.78y=110.5625$$
If you check the website the $3st$ and $s^3-t^3$ step comes after you equation has been put into the following form:
$$y^3+Ay=B$$
If your equation is put into this form, it becomes:
$$y^3-234.75y=1382.03125$$
$$3st=-234.75$$
$$st=-78.25$$
$$s^3-t^3=1382.03125$$
$$t^6+1382.03125t^3+479129.640625=0$$
All 6 solutions to this equation are imaginary.
This means that the equation $0.08y^3−18.78y=110.5625$ has atleast 2 imaginary roots. 
This means the whole system fails and use some other method to solve the original $0.08x^3−3.84x^2+42.66x−137.7625=0$.
Use this site for checking whatever I said.
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