Before solving an equation, how can you recognize that there may be extraneous solutions?
A There is no way of knowing if an equation may lead to extraneous solutions before solving
B If there is an expression that requires a non-reversible operation to solve for x, such as a square root
C If you know you have to use an non-reversible operation as the first step
D If there is an expression that requires a non-reversible operation, whether or not x is within that expression
The correct answer and explanation is:
The correct answer is D: If there is an expression that requires a non-reversible operation, whether or not x is within that expression.
Extraneous solutions are solutions that arise when solving an equation, but do not actually satisfy the original equation. They can appear when certain operations, such as squaring both sides of an equation or taking square roots, are used. These operations are not reversible, meaning that they can introduce solutions that do not work in the context of the original equation.
For example, when you square both sides of an equation, the process may introduce extra solutions because squaring is a non-reversible operation. This is especially true when dealing with equations that involve square roots or absolute value functions. In the case of square roots, squaring both sides could result in solutions that, when substituted back into the original equation, don’t satisfy the equation. For example, solving the equation x=−2\sqrt{x} = -2 would suggest squaring both sides, but no real number squared gives a negative result, which signals an extraneous solution.
It is crucial to recognize operations that could introduce extraneous solutions before solving, such as squaring both sides, taking square roots, or dealing with absolute values. By understanding where these non-reversible operations are used, you can be cautious and check potential solutions against the original equation to verify they are valid.
In summary, when you use non-reversible operations like squaring, you should be aware that the solutions found might not be valid for the original equation. Therefore, checking for extraneous solutions by substituting them back into the original equation is a necessary step in solving such equations.