Problem of solving the cubic function

In summary, solving a cubic function can be done through various methods, but the most efficient way is to use software such as Maple, MATLAB, or Mathematica to plot the function and then locate one of the roots. This root can then be used as a factor to divide the cubic and obtain a quadratic equation, which is easier to solve. Another approach is to use Newton's method or the substitution method of x = z +\frac{\gamma}{z}. However, it is important to verify the solution afterwards due to potential limitations of the substitution method.
  • #1
Martin Zhao
8
0
Guys, I may need your help. There is a question saying that how to solve the cubic function in general form, which means that y=ax^3+bx^2+cx+d. How do you guys solve for x? To be honest, I have no idea of this question. Probably, it uses the same way as the quartic function. Thanks!
 
Mathematics news on Phys.org
  • #3
There are numerous ways of solving cubic functions, but the most efficient way would be to plot it using some type of software--most easily maple, MATLAB, or Mathematica. Once you do that, locate one of the roots. Use the root as a factor, and divide the cubic by that factor to obtain a quadratic. Quadratics are easy to solve, thus you can easily find the remaining two roots.
 
  • #4
AMenendez said:
There are numerous ways of solving cubic functions, but the most efficient way would be to plot it using some type of software--most easily maple, MATLAB, or Mathematica. Once you do that, locate one of the roots. Use the root as a factor, and divide the cubic by that factor to obtain a quadratic. Quadratics are easy to solve, thus you can easily find the remaining two roots.

Well, if you're going to be content with numerical answers, then there are many good techniques to approximate the roots to a very high degree: http://en.wikipedia.org/wiki/Newton's_method
 
  • #5
For the cubic equation [itex]ax^3+bx^2+cx+d=0[/itex] (in your case the constant term is [itex]d-y[/itex], not [itex]d[/itex]), try substituting [itex]x = z +\frac{\gamma}{z}[/itex], and solve for [itex]z[/itex] by choosing the constant [itex]\gamma[/itex] correctly. If fairly certain that for a good choice of [itex]\gamma[/itex] (it will become apparent what [itex]\gamma[/itex] must be) you will end up with a quadratic function in [itex]z^2[/itex].

This way you may arrive at the formula yourself, it's a neat exercise. You probably need to be careful verifying your solution afterwards, as [itex]z +\frac{\gamma}{z}[/itex] is not defined everywhere, and does not attain all values. To make calculations easier, you can assume [itex]a = 1[/itex] first, and make the necessary modification afterwards.
 
Last edited:

1. What is the cubic function?

The cubic function is a type of polynomial function that has the general form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and x is the independent variable. It is called "cubic" because the highest degree of the variable x is 3.

2. What is the problem of solving the cubic function?

The problem of solving the cubic function refers to finding the values of the independent variable x that make the function equal to a specific output value. This can be challenging because not all cubic functions have real solutions, and even when they do, the solutions may not be easily obtained.

3. What are the methods for solving the cubic function?

There are several methods for solving the cubic function, including the algebraic method, the graphical method, and the numerical method. The algebraic method involves using algebraic techniques such as factoring, completing the square, and the quadratic formula. The graphical method involves plotting the function on a graph and finding the x-intercepts. The numerical method involves using algorithms to approximate the solutions.

4. How do you know if a cubic function has real solutions?

A cubic function has real solutions if the discriminant, which is b^2-4ac, is greater than or equal to 0. If the discriminant is less than 0, then the function has complex solutions. If the discriminant is equal to 0, then the function has one real solution.

5. Can the cubic function be solved with just one formula?

No, there is no single formula that can solve all cubic functions. The algebraic method may use different techniques depending on the form of the function, while the graphical and numerical methods require different approaches as well. However, there is a general formula called the cubic formula, but it is complex and not commonly used in practice.

Similar threads

Replies
15
Views
1K
  • General Math
Replies
16
Views
3K
Replies
9
Views
1K
  • General Math
Replies
13
Views
1K
Replies
6
Views
1K
Replies
3
Views
773
  • General Math
Replies
10
Views
2K
  • Precalculus Mathematics Homework Help
Replies
12
Views
387
Replies
17
Views
2K
Replies
5
Views
2K
Back
Top