Associated Topics || Dr. Math Home || Search Dr. Math

### Cubic Equations in One Formula

```
Date: 11/27/2001 at 06:37:34
From: Robert Sullivan
Subject: Cubic equations in one formula

To Dr. Math,

I know the formula for solving quadratic equations of the type
ax^2 + bx + c = 0, but is there a formula for cubic equations? I've
seen supposed cubic formulas, but they take several steps. Is there
one giant formula that I can use?

Robert Sullivan
```

```
Date: 11/27/2001 at 08:40:58
From: Doctor Paul
Subject: Re: Cubic equations in one formula

Hi Robert,

I think you'll find what you're looking for in the section of
the Dr. Math FAQ called "Cubic and quartic equations":

http://mathforum.org/dr.math/faq/faq.cubic.equations.html

any other questions.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 11/27/2001 at 12:50:54
From: Robert Sullivan
Subject: Re: Cubic equations in one formula

Thanks for the tip. The only problem is, I looked at the FAQ and it
says simplify into an equation. The equation has two variables e and f
in it, but the FAQ does not tell you how to find these. What I really
wanted was a big formula to solve the equation in one simple step,
like you would use for a program.

Robert Sullivan
```

```
Date: 11/27/2001 at 13:27:59
From: Doctor Paul
Subject: Re: Cubic equations in one formula

The FAQ does tell you how to find e and f. Start with the general
form:

a*x^3 + b*x^2 + c*x + d = 0

and make the leading coefficient a one by dividing both sides by a

This gives:

x^3 + (b/a)*x^2 + (c/a)*x + d/a = 0

so

e = b/a
f = c/a
g = d/a

Then proceed as instructed in the FAQ by making the appropriate
substitution to get rid of the x^2 term (if necessary).

This is as close as you're going to get to one big formula. An
algorithm that does this in "one simple step" does not exist. The
method outlined in the FAQ is how to solve a general cubic equation.

some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 11/27/2001 at 13:50:53
From: Doctor Rob
Subject: Re: Cubic equations in one formula

Thanks for writing to Ask Dr. Math, Robert.

The Quadratic Formula is actually two equations:

x1 = (-b+sqrt[b^2-4*a*c])/(2*a),
x2 = (-b-sqrt[b^2-4*a*c])/(2*a).

For the Cubic Formula, you need three equations. They are seldom
written out because of how complicated they are. They involve cube
roots ("cbrt" below) of expressions involving square roots ("sqrt"
below), and it is necessary to work with complex numbers (hence the
"i" below). That said, here is one way of writing the three solutions
to a*x^3 + b*x^2 + c*x + d = 0:

x1 = -b/(3*a) +
cbrt(-2*b^3+9*a*b*c-27*a^2*d+
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a) +
cbrt(-2*b^3+9*a*b*c-27*a^2*d-
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a),
x2 = -b/(3*a) +
(-1+i*sqrt[3])/2*cbrt(-2*b^3+9*a*b*c-27*a^2*d+
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a) +
(-1-i*sqrt[3])/2*cbrt(-2*b^3+9*a*b*c-27*a^2*d-
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a),
x3 = -b/(3*a) +
(-1-i*sqrt[3])/2*cbrt(-2*b^3+9*a*b*c-27*a^2*d+
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a) +
(-1+i*sqrt[3])/2*cbrt(-2*b^3+9*a*b*c-27*a^2*d-
sqrt[4*(-b^2+3*a*c)^3+(-2*b^3+9*a*b*c-27*a^2*d)^2])/
(3*cbrt[2]*a).

If the quantity inside the sqrt[] above is negative, you will be faced
with finding the cube roots of two complex numbers (which are complex
conjugates of each other). Of course there are three of these. Pick
any one of the three for the first number, and pick the complex
conjugate of that choice for the second number, and use those values
in all three equations above. In this case, all three roots x1, x2,
and x3 are real numbers. (Surprise!)

If the quantity inside the sqrt[] above is nonnegative, you should use
the real cube root of the real numbers in each case. In this case,
only x1 is a real number and x2 and x3 are a complex conjugate pair of
complex numbers.

Notice: the expressions -b^2+3*a*c and -2*b^3+9*a*b*c-27*a^2*d appear
more than once in the above equations.

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search