Which is the Best Description of an Extraneous Solution?

An extraneous solution is a value derived during the process of solving an equation that appears to be a valid answer but, upon verification, does not satisfy the original equation. Essentially, it is a root obtained from a modified version of the equation that is not a true root of the initial equation, primarily because it was excluded from the domain of the original equation.

Understanding Extraneous Solutions

Extraneous solutions are a common occurrence in algebra, particularly when performing operations that might alter the original equation's domain or introduce new solutions that weren't present initially. They are not mistakes in calculation but rather a byproduct of the solution process itself.

Key Characteristics of an Extraneous Solution:

• Origin: It arises as a valid root of a transformed version of the original equation.
• Validity: It fails to satisfy the original equation when substituted back into it.
• Domain Issue: Often, it falls outside the permissible domain of the original equation, meaning certain operations (like square roots of negative numbers or division by zero) would make the original expression undefined.

Why Do Extraneous Solutions Occur?

Extraneous solutions typically emerge when solving equations that involve:

• Radical Equations: Squaring both sides of an equation, while necessary to eliminate a square root, can introduce solutions. For example, squaring both x = 2 and x = -2 both yield x^2 = 4. If the original equation was √x = -2, squaring gives x = 4, but √4 = 2, not -2.
• Rational Equations: Multiplying by a variable expression to clear denominators can introduce solutions that make the original denominator zero, rendering the expression undefined.
• Logarithmic Equations: The argument of a logarithm must always be positive. Operations during solving can lead to solutions where the argument becomes zero or negative.
• Absolute Value Equations: When solving equations involving absolute values, one often considers positive and negative cases, which can sometimes lead to extra solutions.

Identifying and Avoiding Extraneous Solutions

The most critical step in dealing with extraneous solutions is verification.

1. Solve the Equation: Proceed with the algebraic steps to find all potential solutions.
2. Check All Potential Solutions: Substitute each potential solution back into the original equation.
   • If the substitution results in a true statement, the solution is valid.
   • If the substitution results in a false statement or an undefined expression, the solution is extraneous.

Example:
Consider the equation: √(x + 2) = x

1. Solve:

   • Square both sides: (√(x + 2))^2 = x^2
   • x + 2 = x^2
   • Rearrange into a quadratic equation: x^2 - x - 2 = 0
   • Factor: (x - 2)(x + 1) = 0
   • Potential solutions: x = 2 or x = -1

2. Check:

   • For x = 2:
     • Substitute into √(x + 2) = x: √(2 + 2) = 2
     • √4 = 2
     • 2 = 2 (This is true, so x = 2 is a valid solution).
   • For x = -1:
     • Substitute into √(x + 2) = x: √(-1 + 2) = -1
     • √1 = -1
     • 1 = -1 (This is false, so x = -1 is an extraneous solution).

Therefore, the only valid solution to √(x + 2) = x is x = 2.

Comparing Valid vs. Extraneous Solutions

Understanding the fundamental difference helps reinforce why checking solutions is paramount.

In summary, an extraneous solution is a mathematical artifact of the solving process, a number that appears to be a solution but ultimately violates the conditions or domain of the original equation. Always verify your solutions to distinguish true solutions from extraneous ones.