What is an extraneous solution?

An extraneous solution is a solution that emerges during the process of solving a mathematical problem, but is not a valid solution to the original problem. It's essentially a "false" solution.

Here's a breakdown:

• Definition: An extraneous solution (also known as an extraneous root) is a value that satisfies the transformed equation(s) in the solution process, but does not satisfy the original equation.

• How they arise: Extraneous solutions commonly arise when performing operations that are not reversible, such as:

  • Squaring both sides of an equation. This can introduce solutions that satisfy the squared equation but not the original. For example, solving sqrt(x) = -2 by squaring both sides gives x=4. But sqrt(4) is +2 and is not -2.
  • Multiplying both sides of an equation by an expression that could be zero. This might lead to solutions where that expression equals zero, and those values may not satisfy the original equation.
  • Taking logarithms. Logarithms are only defined for positive arguments, so a solution that makes the argument of a logarithm negative or zero is extraneous.
  • Working with rational equations. When you clear denominators by multiplying both sides of the equation, you might introduce solutions that make the original denominator zero, leading to an invalid solution.

• Identifying Extraneous Solutions: To identify extraneous solutions, it is crucial to check all potential solutions by substituting them back into the original equation. If the solution makes the original equation false, it is an extraneous solution and must be discarded.

• Example: Consider the equation √(x + 2) = x. Squaring both sides, we get x + 2 = x². Rearranging, we have x² - x - 2 = 0. Factoring, we get (x - 2)(x + 1) = 0. This gives us potential solutions x = 2 and x = -1.

  Checking x = 2 in the original equation: √(2 + 2) = √4 = 2. So, x = 2 is a valid solution. Checking x = -1 in the original equation: √(-1 + 2) = √1 = 1, which is not -1. So, x = -1 is an extraneous solution.

Therefore, in this example, x = 2 is the only valid solution. Understanding and identifying <a href="https://www.wikiwhat.page/kavramlar/Extraneous%20Solution">Extraneous Solution</a> is crucial for correctly solving equations.