# Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

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 $f(x) = -6379.1161837130485049747130791414393334925879583964$

• How did you get $104.1$ for the coefficient of $y$? I got $-18.78$. – David Oct 10 '14 at 5:29
• How did you get the coefficients of your cubic? i.e. 0.08, 3.84, 42.66 137.7625? – andre Oct 10 '14 at 8:32
• @David: By manually expanding it and simplifying it myself, I had gotten an alternative depressed function where the coefficient for $y$ is -18.78. However, that still ceased to provide me an invalid root. The present depressed function was found through using the extended form found on the SOSMath page. Does the alternative depressed function work for you? – Dranithix Oct 10 '14 at 8:50
• @andre: These coefficients were collected as I was trying to find the intersection between a quadratic and a cubic function. I had to equate them and subtract one from another which led me to a cubic function. – Dranithix Oct 10 '14 at 8:52
• @Kenta My depressed cubic worked. It produced an $x$ value which I substituted back into the original, giving zero to $9$ decimal places. – David Oct 10 '14 at 9:34

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.