# Cubic Equation

A **cubic equation** is an equation of the form:

.

A cubic equation has 3 roots, either all real OR one real, two complex.

## Contents

## Solving Cubic Equations

If you're too lazy to follow, look at subsection "TLDR" for each section.

### Converting to a Depressed Equation

You start with the equation .

Divide both sides by a: .

Now we change the coefficient of to (e.g. change it to a depressed cubic). We do this by substituting , giving:

.

is and is , so now we have .

#### TLDR?

The equation is where and .

### Solving the Depressed Equation

Now here comes the smart part. Substitute .

The equation becomes . Simplification:

We want that last term to equal , so we can set . (We can't use , because then , which is not necessarily true.) Solving this equation gives us . If , then . We now have a system of equations:

.

We can solve this via the quadratic formula. After and are obtained, we have and .

#### TLDR?

where u and v are roots of the system .

## The Cubic Formula

The cubic formula can be obtained by using the above method. These are the steps:

The depressed cubic is of the form .

and are the roots of the system of equations . We can solve this by substitution:

(We are still using p and q because they might get a little messy if we use p and q in terms of a, b, c, and d.)

(comes from )

(See? I told you it would be messy.) I'm not going to simplify all that squaring and cubing right now: maybe soon! Also, if you select for the first equation and for the second (or vice versa--they lead to the same number), you will always get a real number.

One last piece of advice: * Don't try to memorize this.* Memorize the process (shortcut: just look at TLDR for each section). Here is another way to do it.

### If you're just asking for the formula for a monic cubic...

Here is the formula for :

### If you're asking for the formula for a depressed monic cubic...

Here is the formula for :