How to Factor With the Zero Product Rule

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The zero product rule is a mathematical rule pertaining to algebraic formulas. The rule states that if, using the variables "a" and "b," if ab = 0 then either a or b (or both) must equal zero. This is because zero can only result from a multiplication between itself and another number. Quadratic formulas can be solved using the zero product rule through factoring and setting each side of the factor to zero, then solving.

  • Use the equation x^2 - 2x = 8 for this example. Set the entire equation equal to 0 by subtracting 8 from each side, leaving you with x^2 - 2x - 8 = 0.

  • Factor x^2 - 2x - 8 = 0. Begin the setup with an x leading each factor so that it looks like this: (x )(x ). Add in a "2" to the first grouping and a "4" to the second: (x 2)(x 4). Finish the factoring by adding the correct signs: (x + 2)(x - 4). Work out the factoring to ensure that it does, indeed, equal x^2 - 2x - 8 = 0.

  • Use the zero product rule to set both groups of the factor to 0: (x + 2) = 0 and (x - 4) = 0. Solve each new equation for x, which will equal out to x = -2 and x = 4. Plug these solutions, one at a time, back into the original equation to confirm that they do work out correctly.

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