When an Apparent Solution Does Not Satisfy the Original Equation It Is Called
In the world of mathematics, equations play a crucial role in problem-solving and understanding various phenomena. Solving equations involves finding values that satisfy the equation, making it true. However, there are instances when an apparent solution does not satisfy the original equation, leading to interesting mathematical concepts and further exploration. This phenomenon is known as an extraneous solution or a spurious solution. In this article, we will delve into the concept of extraneous solutions, understand why they occur, and explore some frequently asked questions related to this topic.
What is an Extraneous Solution?
An extraneous solution refers to a solution that does not satisfy the original equation, despite satisfying intermediate steps or equations during the problem-solving process. In simpler terms, it is a solution that may seem correct at first glance but fails to fulfill the requirements of the original equation.
Extraneous solutions often arise when manipulating equations involving radicals or logarithms. These equations may involve squaring both sides, taking square roots, or applying logarithmic functions, which can introduce extraneous solutions into the equation.
Why Do Extraneous Solutions Occur?
Extraneous solutions occur due to the nature of the mathematical operations involved in solving equations. Let’s consider an example to understand this better.
Suppose we have the equation √x = 2. To solve this equation, we square both sides, resulting in x = 4. However, when we substitute x = 4 back into the original equation, we find that √4 is not equal to 2. Instead, it is equal to ±2, as both positive and negative values satisfy the equation.
In this case, squaring both sides introduced an extraneous solution. The original equation has only one solution, x = 4, but the squared equation has two solutions, x = 4 and x = -4. The extraneous solution, x = -4, does not satisfy the original equation and is therefore considered invalid.
This phenomenon occurs because squaring both sides of an equation introduces additional solutions that do not necessarily satisfy the original equation. It is crucial to check the solutions obtained by substituting them back into the original equation to ensure their validity.
Frequently Asked Questions (FAQs)
Q1. How can we identify an extraneous solution?
A: To identify an extraneous solution, substitute the obtained solution(s) back into the original equation. If the solution satisfies the original equation, it is valid. However, if it does not, it is an extraneous solution.
Q2. Are extraneous solutions common?
A: Extraneous solutions are relatively common, especially when dealing with equations involving radicals or logarithms. However, not all equations have extraneous solutions.
Q3. Can extraneous solutions be avoided?
A: Extraneous solutions can be avoided by being cautious while manipulating equations. It is essential to check the validity of solutions obtained by substituting them back into the original equation.
Q4. Are extraneous solutions considered errors?
A: Extraneous solutions are not considered errors but rather a natural occurrence in certain mathematical operations. They provide valuable insights into the properties of equations and mathematical concepts.
Q5. Are extraneous solutions always introduced by squaring both sides of an equation?
A: Extraneous solutions are commonly introduced by squaring both sides of an equation, but they can also arise from other mathematical operations such as taking square roots or applying logarithmic functions.
Extraneous solutions, also known as spurious solutions, are solutions that appear to satisfy intermediate equations but fail to satisfy the original equation. They occur due to the nature of mathematical operations involved in solving equations, particularly when manipulating radicals or logarithms. It is important to identify and discard extraneous solutions by substituting them back into the original equation. Understanding the concept of extraneous solutions enhances mathematical problem-solving skills and provides a deeper insight into the intricacies of equations.