Quartic Equations

Quartic Equations

Quartic equations are polynomial equations with one unknown variable (usually denoted by x), which is never raised to a power greater than 4. Symbolically, they can always be written as follows: ax⁴ + bx³ + cx² + dx + e = 0, where a, b, c, d, and e are numbers. The problem of solving polynomial equations in general, and quartic equations in particular, has been a central theme in algebra from antiquity to the present.

For much of this period, the interest in such problems was purely theoretical. Nowadays, polynomial equations have practical significance as well. To position the end of a robot arm correctly to screw in a bolt, for instance, the robot (or its programmer) sets up a system of polynomial equations. The solution to this system tells the robot what angle at each joint will cause the end of the arm to end up in the desired location. In essence, the robot uses mathematics to replace the senses of sight and touch.

Quadratic equations were solved by the ancient Babylonians, but mathematicians did not master cubic equations until 1545, when the Italian mathematician Girolamo Cardano published the first method for solving them. One might have expected quartic equations to be even harder, but in fact they were conquered at the same time. In remarkably offhand fashion, Cardano attributed the method to his student Luigi Ferrari, "who invented it at my request."

Though the computations can be backbreaking to do by hand, the solution method can be broken down into a simple sequence of steps that are easily programmed into a computer. Even so, the solution to a quartic equation typically requires many lines of type, and involves square roots inside of cube roots inside of square roots. Sadly, Ferrari's ingenious method is almost useless for practical purposes. Nevertheless, its discovery demonstrated the power of algebra, and prepared the way for a shocking discovery over 250 years later: the unsolvability of quintic equations.