Problem

Solve \(x^{2}-36=0\) by difference-of-squares factoring. Name the two square bases, write the conjugate factors, and solve both zero-product equations.

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What this problem is really about

The subtraction of two perfect squares signals conjugate factors with opposite signs. We’ll identify both square bases, apply the difference-of-squares identity, solve both zero-product branches, and check both candidates in the original squared equation so the negative branch is not lost.

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squares=x² and6²; factors=(x−6)(x+6)=0; zero-product branches=x−6=0,x+6=0; zeros=x=6,-6; check=both squares equal36

squares=x² and6²; factors=(x−6)(x+6)=0; zero-product branch=positive branch only; zero=x=6; check=positive root works

squares=x² and6²; factors=uses the same sign twice or leaves nested factors unsimplified; zero-product branches=incomplete; zeros=incomplete; expansion check=produces a middle term or unfinished form

Squares x² and 6²; factors (x − 6)(x − 6) = 0; zero-product branches both give x = 6; repeated zero 6; expansion x² − 12x + 36, not the original equation.

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Solve \(x^{2}-36=0\) by difference-of-squares factoring. Name the two square bases, write the conjugate factors, and solve both zero-product equations.

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