Extraneous solutions

Definition

Extraneous solutions are solutions derived from solving an equation that do not satisfy the original equation. They often arise when both sides of an equation are manipulated algebraically.

5 Must Know Facts For Your Next Test

1. Extraneous solutions commonly occur in equations involving square roots, logarithms, and rational expressions.
2. To identify extraneous solutions, always substitute the solution back into the original equation.
3. Manipulations like squaring both sides or multiplying by a variable expression can introduce extraneous solutions.
4. Extraneous solutions do not indicate a mistake but rather a result of the algebraic process used to solve the equation.
5. $x^2 = 4$ has two solutions, $x = 2$ and $x = -2$, neither of which are extraneous.

Review Questions

• Why is it important to check for extraneous solutions?
• What types of equations are most likely to produce extraneous solutions?
• How can you verify if a solution is extraneous?

Related terms

Radical Equation: An equation that includes a variable within a radical sign, such as $\sqrt{x + 3} = x - 1$.

Rational Equation: An equation that involves at least one fraction whose numerator and/or denominator contains a variable.

Logarithmic Equation: An equation that involves logarithms with variables either inside or outside the log function.