How come $q = \frac{2p+2}{p+2}$ turns into $q^2 - 2 = \frac{2(p^2-2)}{(p+2)^2}$ I've tried factoring, square both sides, but I cant see it. this sample belongs to Rudin Analysis book
2026-04-07 04:59:33.1775537973
Analysis math irrational proof
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Well, $$q^2 = \frac{4p^2 + 8p + 4}{p^2 + 4p + 4}$$ $$q^2 - 2 = \frac{4p^2 + 8p + 4 - (2p^2 + 8p + 8)}{p^2 + 4p + 4} = \frac{2p^2 - 4}{p^2 + 4p + 4} = \frac{2(p^2 - 2)}{(p+2)^2}.$$