Solving Rational Equations

Calculus Readiness · Axiom Academy

LESSON Solving Rational Equations Clear every fraction by multiplying through by the LCD, solve the polynomial that's left — then reject the fake answers that break a denominator. 1. Multiply Through by the LCD Take . The denominators are x and 2 , so the LCD is 2x . Multiply every term by 2x and watch each fraction cancel — the equation sheds its denominators and becomes a clean polynomial. 2. Factor First to Find the LCD Before you can clear fractions you need the right LCD — and a denominator like x^2 - 4 is hiding its factors. Factor every denominator first: x^2 - 4 = (x-2)(x+2) . Now the shared pieces are visible, and the LCD is just the product of the distinct factors. Break composites into linear factors: x^2-4=(x-2)(x+2) . Shared factors now line up. The LCD is (x-2)(x+2) — not (x-2)(x+2)(x^2-4) . Don't double-count a shared factor. Each factor that can hit zero is forbidden: here and . Write them down now. clears to x+2+2(x-2)=8 , giving (valid). Skip the factoring and you might use (x-2)(x+2)(x^2-4) as your "LCD" — far too big, with repeated factors that make the algebra a mess. Factoring reveals that x^2-4 already contains both (x-2) and (x+2) , so the true LCD is small and clean. 3. Reject the Extraneous Solutions Solve . The LCD is x-3 , so multiply through: x^2 = 9 , giving two candidates x = 3 and x = -3 . But x-3 is a denominator — so x = 3 is forbidden . Watch each candidate get tested against that excluded value.

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