Problem

Solve \(9x^{2}-25=0\) by rewriting each term as a square. Factor the difference into conjugates, set each factor to zero, and give both zeros exactly.

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squares=(3x)² and5²; factors=(3x−5)(3x+5)=0; branches=3x−5=0,3x+5=0; zeros=-5,5; substitution check=fails when leading coefficients or factor signs are ignored

squares=(3x)² and5²; factors=(3x−5)(3x+5)=0; zero-product branch=positive branch only; zero=x=5/3; check=positive root works

squares=(3x)² and5²; factors=(3x−5)(3x+5)=0; branches=3x=±5; zeros=x=5/3,-5/3; check=9·25/9−25=0

squares=(3x)² and5²; 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

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Solve \(9x^{2}-25=0\) by rewriting each term as a square. Factor the difference into conjugates, set each factor to zero, and give both zeros exactly.

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