Quadratic, Cubic, and Quartic Equations
Overview
An equation often describes a function, a rule that relates numbers in one set to numbers in another. Rather than listing all the numbers related by a function, letters, also termed variables, are often used to stand in for the numbers.
Fundamental Mathematical Concepts and Terms
The function y = 2x says that for every number x in some set there is some other number, y, in some other set that is twice as large as x. Some functions consist of a sum of powers of x, like y = x3 + 3x2 + 2x + 1.
Here the number just above each x tells us how many times to multiply x times itself: that is, x3 = x × x × x, and so forth. Functions of this form are named by the highest power of x they contain, which is the rank or order of the equation. For example, the highest power of x in y = 2x is 1 (because x = x1), so this is a first-order equation. The highest power of x in y = x3 + 3x2 + 2x + 1 is 3, so this is a third-order equation.
The first four orders have special names, namely linear, quadratic, cubic, and quartic. Quadratic and higher-order equations appear constantly in science, engineering, and business mathematics. They are used literally millions of times a day in these fields, designing electronics, analyzing data, implementing codes, predicting profits, and performing other tasks.
Examples of equations of the first four orders are given in Table 1. In the examples, the letters A through E are used to stand for any constants (fixed numbers), with the exception that A cannot equal 0. These constants are called the coefficients of the equation.
A "solution" to an equation is an x, y pair for which the equation holds true. For example, a solution to the linear equation y = 2x is x = 5, y = 10, because 10 = 2 × 5. In this equation—in fact, in all linear equations—there is one x for each y. Finding solutions to equations is one of the most common tasks in the mathematics of science, engineering, and business. Often we know what y is, or what we want it to be—the cost of an item to be manufactured, say—and we want to know what x (or x's) will produce that y. The variable x often stands for something that we can chose or control, such as the length of an assembly line or the amount of a chemical added to a reaction.
For equations where y is equal to a sum of powers of x, including linear, quadratic, cubic, and quartic equations, the x's for which the equation is true are called its roots. Often the y value is subtracted from both sides of the equation to produce a nice, neat 0 on the left-hand side of the equation, but this is a minor detail. What is important is that the number of roots is equal to the order of the equation. A linear (first-order) equation has one root, a quadratic (second-order) equation has two roots, and so on.
We can find the roots of any linear, quadratic, cubic, or quartic equation by writing down certain equations containing the coefficients of the original equation. This cannot be done for equations of order higher than 4, as mathematicians have known since the 1820s. The first four orders are therefore special. The equation that gives the roots of a quadratic equation, y = Ax2 + Bx + C, is one of the most commonly used formulas in all math and science, and has been known since mathematicians of Babylon discovered it some 4,000 years ago:
This formula is known as "the quadratic equation." In the equation 0 = 2x2 + 3x − 1, we have A = 2, B = 3, and C = −1 and the quadratic equation gives us the two roots:
These roots are the two values of x for which 0 = 2x2 + 3x −1 is true. If you plug either of them in for x and do the arithmetic on a calculator, you'll see that 0 really is the answer. (The small numbers hanging off x1 and x2 are just labels to set them apart.)
Real-life Applications
AREA AND VOLUME
The most basic uses of quadratic and cubic equations are for determining area and volume. In fact, it was the need to calculate land areas that motivated the Babylonians to discover the quadratic equation to begin with. You already know that the area of a square with edges x units long is x × x or x2. If we call the area of a square S, then we have the quadratic equation S = x2 (which can also be written 0 = x2 − S). The formulas for the area of a circle, a triangle, or even of the surface areas of solids like spheres and cubes, all contain x2; all are quadratic equations. Surface area is important in real estate, medicine, physics, and engineering. It affects how
| Type of equation |
General form |
Example |
| Linear |
y = Ax + B |
y = x + 10 |
| Quadratic |
y = Ax2 |
y = 2x2 + 3−1 |
| Cubic |
y = Ax3 + Bx2 + Cx + D |
y = 12 x3 + x + 5 |
| Quartic |
y = Ax4 + Bx3 + Cx2 + Dx + E |
y = x4 − 12x3 + x2 + 100 |
fast an object cools off (greater area equals quicker cooling), which is why machines that need to get rid of extra heat sometimes have little metal fins stuck on them to increase their surface area. It affects how quickly a droplet evaporates (greater area equals quicker evaporation). It affects how quickly a chemical reaction proceeds (greater area equals quicker reaction).
Cubic equations come up just as naturally. Recall that the volume of a cube with an edges x units long is x3. If we call the volume of the cube V, then we have the quadratic equation 0 = x3 − V. And, just as with surface area, this cubic relationship pops up not only in the formula for the volume of a cube, but in the formula for the volume of a sphere or cylinder or any other three-dimensional object.
The fact that area is described by a quadratic equation and volume by a cubic equation affects many things in nature. Any object's surface area is proportional to x2—where x stands for how wide the object is—but its volume is proportional to x3. And as you make the object bigger, that is, increase x, x3 will always grow faster than x2. This is why insects can't (lucky for us) grow to the size of dogs or whales: they breathe using surface area (x2) but their need for oxygen goes by body volume (x3). This is why elephants have fat legs: the strength of a leg-bone goes by cross-sectional area (x2), but the weight the bone has to bear goes by the volume of the elephant (x3).
ACCELERATION
Quadratic equations are needed to predict the paths of accelerating objects. Acceleration is any change in speed. When the driver of a car steps on the gas or hits the brakes, the car accelerates (goes faster or slower). When you drop a ball or throw it up in the air it accelerates. And almost any time a machine with moving parts is designed, from a CD player to a car engine to a jet plane, the people designing the product must deal with accelerations.
CAR TIRES
Car tires are made of rubber-like plastics derived from petroleum and interwoven with metal wires, and must work well despite thousands of miles of use, violent blows from bumps, fast turns, and other stresses. Your life depends on them every day, and their design is a complex art. Computer calculations are used to predict how a new tire design will behave, as this is much cheaper than casting actual tires in a trial-and-error way. One of the most important factors in modeling a tire using calculations is describing the mechanical properties of the "rubber" used in the tire: how it responds to stretching, squeezing, and twisting. In a class of new synthetic tire materials called "carbon black filled rubber compounds," it has been found that a cubic equation best describes the stress-strain relationship—that is, how much the material gives in response to a certain amount of force. This cubic equation is used in writing a computer program that will accurately predict how a tire made with these compounds will behave.
JUST IN TIME MANUFACTURING
Traditional economics treated supply and demand as the two factors deciding profitability in manufacturing. However, in the 1990s some Japanese manufacturers introduced a philosophy called "just in time" (JIT) manufacturing. In this approach, a manufacturer—say of cars, computers, or cameras—tries to produce as many items as possible just in time to deliver them to a buyer. Manufacturing a product and then having it sit in a warehouse, waiting to be sold, reduces profit. But a manufacturer must balance certain variables: they must announce a price and stick to it, they must guess at how much delay or "lead time" they will need to deliver a product, and they must guess at how much demand for the product there will be. The goal, as always, is to earn maximum profit. It turns out that the solution of a cubic equation is central to solving the equation for maximizing profit.
HOSPITAL SIZE
Since the 1980s, hospitals have found it increasingly difficult to make a profit—or even to stay out of debt. Mathematical cost-profit analysis has therefore been brought into play to help hospitals make more profit. One basic decision that a hospital must make is how many beds to have. Having too few or too many beds makes it harder for a hospital to be profitable. Traditionally, profitability has been described as a quadratic function of bed size (the number of beds in the hospital, not how big each bed is); more recent work has shown that a cubic equation works even better. (Other factors are involved, such as where the hospital is located and how affluent the surrounding population is. But if these assumptions are held steady, profitability is a cubic function of bed size.) Using a cubic equation, researchers have found that there isn't just one bed size that is most profitable, but two; or, rather, a point this is typical of a cubic equation, which can have two maximum points rather than one (as a quadratic equation does). From 0 to 238 beds, profit increases. After 238 beds it decreases until 560, after which it goes up indefinitely (but other factors prevent us from building infinitely large hospitals). A hospital is therefore most profitable, in the United States under current conditions, if it is either medium-sized (about 238 beds) or as big as it can be (560 beds or larger).
GUIDING WEAPONS
In steering weapons such as missiles and planes, it is necessary to tell the computer that guides the weapon where it is. Each position is coded as a set of numbers, the "coordinates" of the weapon or vehicle. These can be given in traditional terms as latitude and longitude (numbers derived from a network of imaginary lines laid down on the Earth's surface by map-makers) plus altitude (height above the surface), or in terms of an "Earthcentered coordinate system." Since one type of coordinates is better for some purposes and the other is better for other purposes, it is sometimes necessary to translate between them—to take position information given in one form and turn it into the other form. Going from latitude-longitude coordinates to Earth-centered coordinates is mathematically easy, but going the other way requires the solution of a quartic equation.
