MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Find X given Y in a cubic function. [closed]

Hello... community or saviours?

I have a cubic function in the form:

y = ax^3 + bx^2 + cx + d

...(where a, b, c and d are all known constants e.g -0.3, 3.5, 3.83 and 0 respectively) that produces a curve and allows me to calculate a point on that curve given it's x co-ordinate.

What I'd really appreciate help with is: How could I rewrite this equation to calculate the value of x if I knew the value of y?

I understand that cubic curves of this nature can have multiple (upto 3) values for x for any given y though I do not know how to calculate them, and my specific interest is only in the 'rightmost' x (i.e that with highest positive value).

Through research to date I gather my question relates some what to finding the roots of the cubic equation (where the given Y is specifically 0, so the values of x are where the curve crosses the horizontal axis) but I can't expand my admittedly hazy comprehension to calculating for different y values.

The only additional hint I've been able to glean is that because in my circumstances d is always 0 that I may not be dealing with a 'true' cubic function at all and instead might want to "factor out an 'x' and a quadratic".

Give me anything but mathematics to Google and I'm usually able to work something out for myself but in the sea of coefficients and polynomials I'm afraid I'm lost.

Thank you in adavnce for any pointers, John.

-
Look for Cardano: en.wikipedia.org/wiki/… – András Bátkai Nov 24 at 19:52
Hint: give yourself an extra degree of freedom by writing x=s+t. Now expand everything out and make some reasonable guesses about which terms are equal to each others. – Steven Landsburg Nov 24 at 20:50
I understand the Cardano method (among others) can give you the roots, as is the case in the link you provided where there are three formulas providing the values for x1, x2 and x3. But as stated in my original question, roots are for the constant y value 0, so I'm not quite sure how to modify these to accept a given y value. In addition, I'm unfamiliar with the different expression I see used when referring to the Cardano method, i.e. the original cubic formula is constructed with different coefficients: y = ax^3 + bx^2 + cx + d vs x^3 + ax^2 + bx + c = 0 – John Nov 24 at 20:55
Thanks for the help Steve, unfortunately I'm trying to come up with a formula or process that allows me to convert a given cubic (expressed by it's 3 coefficients) into a form in which a y value can be input and one of the values for x can be derived. If I 'eyeball' a specific example how do I use that in the construction of such a solution? – John Nov 24 at 21:08
Ironically I'm breaking one of the rules (MathOverflow is not for questions about MathOverflow) but could someone who closed this question give me a quick pointer as to why it was closed so I do not repeat my mistake in the future. As I understand it from the FAQ: My question is not homework, is not a general discussion, and has a very specific answer which I have not been able to resolve fully through independent study. – John Nov 24 at 21:14