Question
What is an extraneous solution of a rational equation?
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Answered by
GPT-4o mini
An extraneous solution of a rational equation is a solution that appears to satisfy the equation after manipulation, such as simplifying or solving, but does not actually satisfy the original equation. This can occur particularly in rational equations when both sides of the equation are multiplied by a variable or expression that may be zero or is not allowed in the original equation's domain.
For example, consider the rational equation:
\[
\frac{1}{x} = \frac{2}{x + 1}
\]
If we multiply both sides by \(x(x + 1)\) to eliminate the denominators, we might accidentally introduce solutions that are not valid. Solving this transformed equation might yield values for \(x\) that lead to division by zero when substituted back into the original equation.
To find extraneous solutions, one should always check potential solutions by substituting them back into the original equation and ensuring that they do not lead to undefined expressions (like division by zero). Any solution that does not satisfy the original equation, even if it was obtained through the manipulation process, is considered extraneous.
For example, consider the rational equation:
\[
\frac{1}{x} = \frac{2}{x + 1}
\]
If we multiply both sides by \(x(x + 1)\) to eliminate the denominators, we might accidentally introduce solutions that are not valid. Solving this transformed equation might yield values for \(x\) that lead to division by zero when substituted back into the original equation.
To find extraneous solutions, one should always check potential solutions by substituting them back into the original equation and ensuring that they do not lead to undefined expressions (like division by zero). Any solution that does not satisfy the original equation, even if it was obtained through the manipulation process, is considered extraneous.