Cubic equations were known since ancient times, even from the Babylonians. However they did not know how to solve all cubic equations. There are many mathematicians that attempted to solve this “impossible equation”. Scipione del Ferro in the 16th century, made progress on the cubic by figuring out how to solve a 3rd degree equation that lacks a 2nd degree. He passes the solution onto his student, Fiore, right on his deathbed. In 1535 Niccolò Tartaglia figures out how to solve x3+px2=q and later Cardano begs Tartalia for the methods. Cardano finally publishes the methods of solving the cubic and quartic equations.
The easiest way to solve a cubic equation is to use either grouping or factoring. Here is an example:
Solve x3 + 12x2 − 9x − 108=0 by grouping.
(x3 + 12x2) + (−9x − 108) =0 In this step, group 2 pairs of terms.
x2 (x + 12) +(−9) (x −12)=0 Factor out the common term in each group. x2 and (−9)
(x+12) (x2 −9) = 0 Factor out the common term again (x+12).
(x+3) (x−3) (x+12)=0 Factor difference of perfect square.
The roots to this equation are −3, 3, −12.
To find the cubic equation from a graph, find the points where the function passes through the x axis. Then you multiply together (X-point1) (X-point2) (X-point2) to get the equation. However
this does not work for all cubic equations.
Now here is the cubic formula for any cubic equation! The objective of solving cubic equations is to find a real root. The other two roots, which can be either real or complex, can be found by other familiar means. There are two main...