key term - Extraneous Solution

Definition

An extraneous solution is a solution to an equation that satisfies the equation algebraically but does not satisfy the original problem statement or the domain restrictions of the equation. These solutions are not considered valid answers to the original problem and must be discarded.

5 Must Know Facts For Your Next Test

1. Extraneous solutions can arise when solving rational equations due to the division by zero restriction.
2. When solving radical equations, extraneous solutions can occur if the original equation is squared or raised to a power to isolate the variable.
3. In a system of nonlinear equations, extraneous solutions can be introduced if one or more of the equations are manipulated in a way that changes the solution set.
4. Identifying and discarding extraneous solutions is a crucial step in solving rational, radical, and systems of nonlinear equations.
5. Extraneous solutions do not satisfy the original problem statement or the domain restrictions of the equation and must be excluded from the final solution set.

Review Questions

• Explain how extraneous solutions can arise when solving rational equations and why it is important to identify them.
  • Extraneous solutions can arise when solving rational equations due to the division by zero restriction. When solving a rational equation, we may need to cross-multiply or perform other algebraic manipulations that can introduce solutions that satisfy the equation algebraically but do not satisfy the original problem statement or the domain restrictions. It is crucial to identify and discard these extraneous solutions, as they are not valid answers to the original problem.
• Describe the process of identifying and discarding extraneous solutions when solving radical equations.
  • When solving radical equations, extraneous solutions can occur if the original equation is squared or raised to a power to isolate the variable. This can introduce solutions that satisfy the transformed equation but do not satisfy the original radical equation. To identify and discard extraneous solutions, students must carefully check each solution by substituting it back into the original radical equation and verifying that it satisfies the equation and the domain restrictions.
• Analyze the role of extraneous solutions in the context of solving systems of nonlinear equations and explain the importance of recognizing them.
  • In a system of nonlinear equations, extraneous solutions can be introduced if one or more of the equations are manipulated in a way that changes the solution set. For example, squaring or raising an equation to a power can introduce extraneous solutions that satisfy the transformed equation but not the original system. Recognizing and discarding these extraneous solutions is crucial, as they do not represent valid solutions to the original problem and can lead to incorrect conclusions if not properly identified and removed from the final solution set.

Related terms

Rational Equation:

An equation that can be written in the form $\frac{P(x)}{Q(x)} = 0$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

Radical Equation:

An equation that contains one or more radical expressions, such as $\sqrt{x} = 5$ or $\sqrt{x} + 2 = 7$.

System of Nonlinear Equations: A set of two or more equations that are not linear, meaning they contain variables with exponents other than 1, or variables multiplied together.