Extraneous solutions

Definition

Extraneous solutions are solutions derived from the algebraic manipulation of an equation that do not satisfy the original equation. They often arise when both sides of an equation are squared or when absolute value functions are involved.

5 Must Know Facts For Your Next Test

1. Extraneous solutions must be checked against the original equation to determine their validity.
2. They commonly occur in equations involving square roots, logarithms, and rational expressions.
3. Squaring both sides of an equation can introduce extraneous solutions because it removes information about the sign of the quantities involved.
4. To identify and exclude extraneous solutions, substitute them back into the original equation to see if they hold true.
5. Understanding how extraneous solutions arise can help prevent mistakes during problem-solving.

Review Questions

• Why is it important to check for extraneous solutions after solving an equation?
• How can squaring both sides of an equation lead to extraneous solutions?
• What steps should you take to verify if a solution is extraneous?

Related terms

Radical Equation: An equation that contains a variable within a radical, typically a square root.

Absolute Value Equation: An equation that involves the absolute value function, which represents the distance of a number from zero on the number line.

Quadratic Equation: $$ax^2 + bx + c = 0$$ where $$a \neq 0$$. Solutions are found using factoring, completing the square, or the quadratic formula.