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An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation.

Table of Contents Show

What are the solutions to the equation?
Which of the solutions are extraneous?
What causes a solution to a rational equation to be an extraneous solution?
What are the solutions to the equation w 2w 3 4 w?

Example 1:

Solve for x , 1x − 2+1x + 2=4(x − 2)(x + 2) .

1x − 2+1x + 2=4(x − 2)(x + 2)

(x − 2)(x + 2)(x − 2)+(x − 2)(x + 2)(x + 2)=4(x − 2)(x + 2)(x − 2)(x + 2)

(x−2)+(x+2)=4

2x=4

x=2

But 2 is excluded from the domain of the original equation because it would make the denominator of one of the fractions zero--and division by zero is not allowed! . Therefore, it cannot be a root of the original equation. So, 2 is an extraneous solution. So, the equation has no solutions.

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

What are the solutions to the equation?

A solution of an equation is any value of the variable that satisfies the equality, that is, it makes the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation the same value. To solve an equation is to find the solution(s) for that equation.

Which of the solutions are extraneous?

Extraneous solutions are values that we get when solving equations that aren't really solutions to the equation.

What causes a solution to a rational equation to be an extraneous solution?

For rational equations, extraneous solutions are values that cause any denominator in the original problem to be 0. Of course, when we have 0 in the denominator we have an expression that is undefined.