Concept

Extraneous Solution

Solving equations by using inverse operations may lead to solutions that do not satisfy the original equation. These solutions are called extraneous solutions, and they occur primarily in radical equations when the radical is eliminated, in rational equations, or in logarithmic equations. Below, an example radical equation is shown.

2 = x + 2 x - 1

By using inverse operations and Properties of Equality, the solutions to this equation are found to be

x = 1

and

x = 5 .

However, they produce different outcomes when substituted into the original equation.

	Substitute	Simplify
$x = 1$	$2 = ? 1 + 2 (1) - 1$	$2 = 2$ $✓$
$x = 5$	$2 = ? 5 + 2 (5) - 1$	$2 \neq = 8$ $\times$

For the example equation, $x = 1$ is a valid solution but $x = 5$ does not result in a true statement. Therefore, it is an extraneous solution.

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