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What causes a solution to a rational equation to be an extraneous solution? When there is more than one solution, one of the solutions is extraneous. If a solution results in zero when substituted into the denominator of the equation, the solution is extraneous. If a solution results in a negative number when substituted into the denominator of the equation, the solution is extraneous. When a solution is a fraction, the solution is extraneous.

Ask by Ayala Harris. in the United States
Mar 11,2025

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If a solution makes the denominator zero when substituted into the equation, it is extraneous.

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- A rational equation is an equation that involves fractions with polynomials in the numerator and denominator. - An extraneous solution is a solution that is obtained during the solving process but does not satisfy the original equation. - The key issue in rational equations is division by zero. For instance, if we have the rational expression \[ \frac{P(x)}{Q(x)}, \] then \( Q(x) \) must not be zero. - If substituting a candidate solution into \( Q(x) \) results in \[ Q(x) = 0, \] that candidate must be discarded because it would make the original expression undefined. - Therefore, the correct statement is: **If a solution results in zero when substituted into the denominator of the equation, the solution is extraneous.**

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Extra Insights

Extraneous solutions often arise when we manipulate the equation, especially during the process of clearing denominators or squaring both sides. This can introduce solutions that don't actually satisfy the original equation. It’s like finding the elusive treasure only to discover it’s just fool’s gold! For example, consider the equation \( \frac{1}{x-2} = 0 \). If you multiply both sides by \( x-2 \), you might think the solution is \( x = 2 \). However, substituting back into the original equation leads to division by zero, making \( x = 2 \) an extraneous solution, proving that not every answer you find is a true gem!

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