Extraneous solutions to rational equations

The  exercise appears under the Algebra II Math Mission. This exercise solves equations and experiments with understanding extraneous solutions.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the solution(s) to the rational equation: This problem has a rational equation that possibly has some extraneous solutions. The student is expected to solve the equation and determine if any of the solutions were actually extraneous.Estre1-too.png
 * 2) Determine if there is an extraneous solution: This problem has a rational equation and a possible extraneous solution. The student is asked to determine if the given value is a solution, an extraneous solution or nothing.Estre2.png

Strategies
Knowledge of basic rational equations and a motivation to "check" answers are encouraged to ensure success on this exercise.
 * 1) The equations conveniently have the same denominator. Ignore the denominator, solve the resulting quadratic for the initial solutions.
 * 2) Extraneous solutions are initial solutions to an equation that are restricted due to some domain restriction.
 * 3) Not being a solution is not the same as being an extraneous solution.
 * 4) Extraneous solutions occur because operations are performed on an equation that do not (necessarily) maintain equality.

Real-life Applications

 * 1) Rational equations are used to solve certain work rate and distance word problems.
 * 2) Extraneous solutions occur frequently with restricted domains. They are subtle but important in some applications.
 * 3) Extraneous solutions not only happen with rational equations, but also with radical equations.