Understanding Extraneous Solutions in Mathematics: Definition, Examples, and Importance

extraneous solution

An extraneous solution in mathematics refers to a solution that arises during the process of solving an equation or a problem but does not satisfy the original problem

An extraneous solution in mathematics refers to a solution that arises during the process of solving an equation or a problem but does not satisfy the original problem. In other words, it is a solution that is not valid or meaningful in the given context.

This situation commonly occurs when solving equations involving radicals, logarithms, or rational expressions. It is important to identify and discard extraneous solutions because they lead to incorrect or inconsistent conclusions.

To better understand this concept, let’s consider an example:

Problem: Solve the equation √(x+4) = 2

Solution:

Step 1: Square both sides of the equation to eliminate the square root:

√(x+4)^2 = 2^2

x + 4 = 4

Step 2: Subtract 4 from both sides:

x = 0

So given the equation, x = 0 seems to be the solution. However, we need to check whether this solution is valid by plugging it back into the original equation:

√(0+4) = 2

√(4) = 2

2 = 2

The equation is true, so x = 0 is indeed a valid solution.

However, now let’s consider a slightly modified version of the problem:

Problem: Solve the equation √(x+4) = -2

Solution:

Step 1: Square both sides of the equation:

√(x+4)^2 = (-2)^2

x + 4 = 4

Step 2: Subtract 4 from both sides:

x = 0

Again, this produces x = 0 as a solution. But before concluding, we need to check if x = 0 is valid by plugging it into the original equation:

√(0+4) = -2

√(4) = -2

2 = -2

This equation is not true, which means that x = 0 is an extraneous solution. The solution x = 0 does not satisfy the original problem √(x+4) = -2 because the square root of any number cannot be negative when dealing with real numbers. Therefore, we discard x = 0 as a valid solution to the original problem.

In conclusion, identifying extraneous solutions is crucial when solving equations involving certain mathematical concepts. It is important to verify found solutions by substituting them back into the original equation to ensure their validity and relevance in the given problem context.

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