Extraneous Solutions
Extraneous solutions are those solutions encountered when solving an equation or system of equations that at first glance appear reasonable, but do not actually satisfy the original conditions of the problem. Extraneous solutions can occur with any type of equation--including differential equations--but most often occur when a problem involves applying the square, quartic, or any other even-numbered power to both sides of an equation. Extraneous solutions also sometimes occur when the problem involves a radical term (that is, a term with a non-integer exponent) or trigonometric function, or when the restrictions on the domains of terms in the equation(s) are not well-considered. Extraneous solutions are also quite common when a mathematical model is applied to a real-world event, scientific experiment, or engineering problem, as models generally apply only to a well-defined domain, such as the time between the start of a race and the time that the last competitor crosses the finish line. Extraneous solutions are particularly common with a quadratic equation.
How an extraneous solution is created in a just few simple steps is easily demonstrated. Consider, for example, the equation x = 1. Squaring both sides of the equation, which is a perfectly legitimate operation, results in the equation x2 = 1. However, the second equation (x2 = 1) has two solutions--x = 1 and x = -1! Clearly, x = -1 is not a valid solution of the first equation (that is, x = 1 ( -1). The solution x = -1 is extraneous; although the solution x = -1 was derived as the result of applying a perfectly legitimate operation to an equation, the outcome of the operation led to two results--one that was correct and one that was incorrect. Failing to isolate and then eliminate extraneous solutions is a common failing in mathematical problem-solving, which can lead to false assumptions and operating parameters. Such missteps are particularly troublesome in computer programming, as a computer program may generate and utilize many mathematical solutions without a human in the loop to review the reasonableness (that is, correctness) of the solution. Establishing tight parameters in which a computer-generated solution will be accepted and then used in a subsequent part of the computer program is an important element in locating and correcting errors.
One of the most common mathematical models that results in an extraneous solution is the path of an object through free space (usually considered a vacuum or frictionless air, which is physically unattainable). Using the Cartesian coordinate system and parametric equations--as well as assuming the object is moving through a frictionless medium--the path of an object can be modeled by the equations x -x0 = (v0 * cos0) * t and y - y0 = (v0 * sin0) * t -1/2 gt2, where x0 is the initial displacement in the x-direction, y0 is the initial displacement in the y-direction, v0 is the initial displacement of the object, t is time, g is the gravitational constant, and 0 is the initial angle of projection for the object. (Note that x0, y0, v0, g, and 0 are all constants, so they do not vary with time.)
In the example, suppose that x0 = 0 (that is, there is no displacement x-direction at t = 0, such as the starting line for a shotput competition), that y0 = 5 feet (the shoulder height of a shotput athlete), that z0 = 30 degrees, that the initial velocity v0 is 40 feet per second, and g, the gravitational constant, is approximately equal to 16 feet per second squared. For the y-variable, which represents the height of the shotput as it travels through the air, the equation becomes y - 5 = (40 * sin 30o) * t - 1/2 * 16 * t2, or y = -8t2 + 20t + 5. If one solves this equation for y = 0 (that is, the height of the ground), one finds t is approximately equal to -0.23 seconds and 2.73 seconds. The first solution, however, is not possible, as it does not account for the fact that the shotput started its parabolic path of travel at the athlete's shoulder height, not the ground. The second solution is the amount of time elapsed before the shotput reaches the ground after it is tossed. In the first case, the solution represents a solution in which the model of the object's path is not appropriate and is therefore an extraneous solution.
This is the complete article, containing 733 words
(approx. 2 pages at 300 words per page).
