What is an extraneous solution?
A A solution that is the result of squaring or multiplying by a variable expression
B An invalid solution that arises through the process of solving an equation
C An extra solution to an equation
D A solution that is invalid
The correct answer and explanation is:
The correct answer is B) An invalid solution that arises through the process of solving an equation.
An extraneous solution is a solution that, while it may appear to satisfy an equation during the process of solving, does not actually satisfy the original equation when substituted back. This typically occurs when operations that are not reversible are applied to both sides of an equation. Operations like squaring both sides of an equation or multiplying both sides by a variable expression are prime examples where extraneous solutions can emerge.
For instance, consider the equation x=−3\sqrt{x} = -3. If you square both sides to eliminate the square root, you get x=9x = 9. However, when you substitute x=9x = 9 back into the original equation, you find that 9=3\sqrt{9} = 3, not -3. Therefore, x=9x = 9 is an extraneous solution, as it does not satisfy the original equation.
Extraneous solutions often arise in equations involving square roots, absolute values, logarithms, or rational expressions. These operations can introduce solutions that do not hold true for the original equation. That’s why it’s essential to check all solutions by substituting them back into the original equation after solving.
In some cases, the presence of extraneous solutions can be minimized by carefully considering the domain of the equation and the operations applied to both sides. For example, when squaring both sides of an equation, the domain must be considered, as negative values may introduce invalid results.