Extraneous Solutions

Algebra 1 · Axiom Academy

Why clearing the denominators of a rational equation can hand you a "solution" that secretly breaks the original. 1. A Solution the Equation Never Allowed Every rational equation has a hidden rule: a denominator can never be zero. The values that would make a denominator zero are excluded from the start — they are not in the equation's domain. When we clear the fractions, we drop that rule, so the cleared equation can produce a value that the original forbids. 2. Where It Comes From: Clearing the LCD Take a concrete rational equation. The denominators are x-2 and x^2-4=(x-2)(x+2) , so the LCD is (x-2)(x+2) . Multiplying every term by it clears the fractions — but watch which factors cancel. The cleared equation hands us a single candidate, x=2 . But cancelling the (x-2) factor is exactly where the original's rule got thrown away. The candidate it produced is the very value the original forbids. 3. Send the Candidate Back to the Original A candidate is only a claim . The original equation is the judge. Substitute x=2 into and look at the denominators first. Both x-2 and x^2-4 become 0 . The expression is undefined, so x=2 is extraneous . There is no valid value left — this equation has no solution . 4. When One Survives and One Doesn't Extraneous roots don't mean every candidate is doomed — they mean you must test each one. Here the cleared equation is a quadratic, so it produces two candidates, and they meet different fates.

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