Clearing the denominators
SAT glossary · Math
Definition
Multiplying both sides by the least common denominator to reduce a rational equation to a polynomial one. Like squaring, it manufactures extraneous roots — multiplying by (x-3) is multiplying by zero when x=3, and multiplying by zero makes any equation true — so the final check is not optional.
Clearing the denominators in plain English
When an equation has fractions with variables in the denominator, 'clearing the denominators' means multiplying every term by whatever expression will make those denominators disappear, turning a messy fraction equation into a simpler polynomial one. The catch is that this move can secretly create fake solutions that don't actually work in the original equation — because if you multiply both sides by something like (x-3), and it turns out x really does equal 3, you just multiplied everything by ZERO, and multiplying by zero can make basically any equation look true. That's why, after clearing denominators and solving, you always have to plug your answer back into the ORIGINAL equation to make sure it didn't sneak in a value that breaks a denominator — skipping that check is not optional.
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