Extraneous solution

Definition

An extraneous solution is a solution derived from an equation that is not valid within the original equation. Extraneous solutions often arise when both sides of an equation are manipulated.

5 Must Know Facts For Your Next Test

1. Extraneous solutions commonly occur when solving exponential and logarithmic equations, especially after squaring both sides or applying logarithms.
2. Identifying extraneous solutions involves substituting them back into the original equation to verify their validity.
3. Extraneous solutions can be introduced by operations that are not reversible, such as squaring both sides of an equation.
4. When working with logarithmic functions, ensuring the argument of the logarithm remains positive is crucial to avoid extraneous solutions.
5. In exams, students should always check for extraneous solutions as part of their final answer verification process.

Review Questions

• What is an extraneous solution and how does it arise?
• How can you check if a solution is extraneous in an exponential or logarithmic equation?
• Why do operations like squaring both sides of an equation often lead to extraneous solutions?

Related terms

Logarithmic Equation:

An equation involving a logarithm with a variable inside its argument.

Exponential Equation:

An equation in which variables appear as exponents.

Inverse Function:

\text{A function that reverses another function, such that } f(f^{-1}(x)) = x \text{ and } f^{-1}(f(x)) = x.