solve this formula for $x$: $y = \frac{5x(2x^2 + 27x + 89)}{6}$

Question

I need to solve this formula for x:

$$y = \frac{5x(2x^2+27x+89)}{6}$$

When inserting this into a Formula calculator it gave me the formula in the attached image, however, I have no clue how to use this formula at all.

1. Is there a simpler formula solved for x?
2. Why are there 3 ,x = in the formula in the attached image
3. What is I which is present 2 times in front of the Roots in the formula in the attached image
4. How do I apply this, or any other solved for x, formula in JavaScript

Attached image

I don't know if any of you all know Maple, but here's the Maple Link https://maple.cloud/#doc=5772900943790080

Answer 1

Now, I will tried to answer all of your question:

Question 1 : Is there a simpler formula solved for x?

It really depends on how you want to solve this kind on the problems. You can solve it using general Cubic formula from this great video from Mathologer. In some cases, you can try to input some small numbers and hopefully you will get a root (cubic polynomial will have at least one real root). Then, use long division to solve other two roots.

Question 2 : Why are there 3 ,x = in the formula in the attached image?

It is because cubic polynomial will have three roots where it should have a least one real roots.

Question 3 : What is I which is present 2 times in front of the Roots in the formula in the attached image

I is stands for complex numbers which means that the cubic formula has one real roots and two complex roots but from my calculation, your equation should have all three real roots. (refer to my calculation below)

Question 4 : How do I apply this, or any other solved for x, formula in JavaScript

Here I do refer to solution in StackOverFlow for JavaScript. I do not tested it yet but from the comments, you can try.

Bonus

The equation given is : $$y=\frac{5x\left(2x^2+27x+89\right)}{6}$$ Let $y=0$ and we get $$0=x\left(2x^2+27x+89\right)$$ Notice that $x=0$ is one of the roots. Thus, from $$0=2x^2+27x+89$$ Use quadratic formula and you will get $$x=-\frac{27}{4}+\frac{\sqrt{17}}{4}$$ $$x=-\frac{27}{4}-\frac{\sqrt{17}}{4}$$

Answer 2

I strongly suggest to avoid Cardano formula and Idid follow the steps given here using the hyperbolic method when there is only one real root. The equation being $$10x^3+135x^2+445x-6y=0$$

for $y \geq 0$, the result is simply $$x=-\frac{9}{2}+\sqrt{\frac{65}{3}} \cosh \left(\frac{1}{3} \cosh ^{-1}\left((y+30)\frac{36}{325} \sqrt{\frac{3}{65}} \right)\right)$$ What do you prefer ?