Cubic equations are equations that have a variable that is cubed (that is, raised to the third power) in them. There are several common ones, two being:

• ax³ + bx² + cx + d = 0
• y³ + ay = b

The method of solution of the first equation requires some work to arrange it into the form of the second equation. Cubic equations have three roots (due to quadratics having 2 and quartics having 4). The roots are often a combination of real and complex numbers, however the roots can be all real and all complex. Here is the method to change the first equation into the second equation:

• Take ax³ + bx² + cx + d = 0.
• Note that if a variable equals zero, then the computations are much easier.
• We need to use an extra variable to make the substitution; call it y.
• x = y - b/3a
• Apply this to the equation.
• This gives: a(y - b/3a)³ + b(y - b/3a)² + c(y - b/3a) + d = 0.
• Multiplying out and simplifying, we find:
• ay³ + (c - b²/3a)y + (d + 2b³/27a² - bc/3a) = 0.
• This is the depressed cubic.
• Note that c - b²/3a means c take away (b²/3a).

After some simplifying, we get a cubic of the form y³ + ay = b. This is solvable by another method:

• Take y³ + ay = b.
• We need to use two variables, s and t, so that:
• 3st = a and s³ - t³ = b.
• Replacing a, b and y gives (s-t)³+3st (s-t)=s³-t³.
• y = s - t is a solution of the depressed cubic. So how do we find s and t?
• We find that (a/3t)³ - t³ = b.
• Simplify this into the tri-quadratic equation t^6 + bt³ - a³/27 = 0.
• Note that t^6 means t to the power of 6.
• If we use a variable u to equal t³, we can simplify into a quadratic.
• This quadratic is u² + bu - a³/27 = 0.
• We can use the quadratic formula to find u. t is then the cube root of u.
• We can now find s by dividing a by 3t.
• y is now findable.

x can be found easily once you know y. Once you know one root of the equation, you can use this knowledge to easily find the other two roots. Most cubics can be solved by this method. You may find you are working out roots of negative numbers. As a result to solve most cubics you need a working knowledge of complex numbers.

The Cubic Formula

A cubic formula does exist, however it is very cumbersome and is needed extremely rarely. It has a possible common use in computer programs that need to solve cubics for situations such as 3D graphics. Here is the monster formula:

You are not recommended to memorize this formula. Even though a cubic equation may have 3 real roots, in the formula the roots of negative numbers are needed (they cancel out in the rest of the formula). This formula hardly has any practical use; it is merely interesting. Home