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.

Is there a simpler formula solved for x?
Why are there 3 ,x = in the formula in the attached image
What is I which is present 2 times in front of the Roots in the formula in the attached image
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

For some specific values of $y$ the formula may simplify, but in general the roots of a cubic equation are ugly. Do you know Cardano's formula? — Gary, Mar 01 '22 at 12:08
Does this answer your question? What are the ways to solve cubic equations? — Dietrich Burde, Mar 01 '22 at 12:08
The formula in your title and the formula in the body of the question are different. Which did you intend? — N. F. Taussig, Mar 01 '22 at 12:11
$i$ is the imaginary unit, it has the property that $i^2 =-1$. Do you know about complex numbers? — Gary, Mar 01 '22 at 12:16
To understand your problem you will need to study complex numbers and how to solve cubic equations. There is no other way. — Gary, Mar 01 '22 at 12:26
You say "I need to solve...". Is it an isolated question (homework) or part of a larger problem ? In this case, which one ? One thing is sure, in such questions, if you haven't studied complex numbers, you will not go far. — Jean Marie, Mar 01 '22 at 12:27
Not homework but an isolated question. I could use the solved for y formula in a loop to figure out what I need, however, these loops have limitations i wouldn't have if i had a formula solved for x — Lars_und_so, Mar 01 '22 at 12:29

score 3 · Answer 1 · answered Mar 01 '22 at 14:09

3

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 ?

answered Mar 01 '22 at 14:09

Claude Leibovici

289,558

I tried running the second solution in JavaScript ```const y = 1150; const x = -(9/2) + Math.sqrt(65/3) * Math.cosh((1/3) * Math.cosh((y + 30) * (36/325) * Math.sqrt(3 / 65)) **-1 );
console.log(x);
```
`x = 0.15474668125631386`
```
– Lars_und_so Mar 01 '22 at 14:58
@Lars_und_so You have made an error m you have used $\cosh$ twice but you should have used the reciprocal function $\cosh^{-1}$ the second time (I don't know its name in javascript) which is not at all the same as taking the power $-1$ ! – Jean Marie Mar 01 '22 at 23:24
@JeanMarie Calling it the reciprocal function is confusing. I would call it the inverse function. What I would call the reciprocal function is the hyperbolic secant. – Gary Mar 01 '22 at 23:55
1

@Lars_und_so. For $y=1150$, $x=5.019603301445864643357247$. Enter this number in the ISC. Click Integer Relation Algorithms and look at the first line (which is your equation). – Claude Leibovici Mar 02 '22 at 06:32
@Gary You are right; the reason of my confusion is that in french "bijection réciproque" means $f^{-1}$ . I will take care next time. – Jean Marie Mar 02 '22 at 07:09
@Claude Leibovici Hi, Claude: Here is the way to access ISC at present : http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html – Jean Marie Mar 02 '22 at 07:20

score 0 · Answer 2 · answered Mar 01 '22 at 13:56

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 $$x=-\frac{27}{4}+\frac{\sqrt{17}}{4}$$ $$x=-\frac{27}{4}-\frac{\sqrt{17}}{4}$$

Wouldn't it be more direct to say that the problem is that the given function is not bijective, therefore needs restriction to certain intervals to warrant this bijectivity ? — Jean Marie, Mar 01 '22 at 23:33

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

2 Answers2