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Home » Calculate the total initial volume of the mixture: - Sterling Industries

Calculate the total initial volume of the mixture: - Sterling Industries

📅 April 17, 2026 👤 scraface

Apr 17, 2026

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📌 Question: Factor the expression $ x^4 - 16y^4 + 8x^2 + 64 $.

📌 Solution: Group the expression:

📌 x^4 - 16y^4 + 8x^2 + 64 = (x^4 + 8x^2 + 16) - (16y^4 - 64)

📌 Note that $ x^4 + 8x^2 + 16 = (x^2 + 4)^2 $ and $ 16y^4 - 64 = 16(y^4 - 4) = 16(y^2 - 2)(y^2 + 2) $.

📌 Let $ A = x^2 + 4 $, $ B = y^2 + 2 $, and $ C = y^2 - 2 $, then:

📌 A^2 - 16BC = (A - 4\sqrt{BC})(A + 4\sqrt{BC})

📌 But this factoring is not ideal over real numbers. Instead, observe that $ x^4 - 16y^4 = (x^2)^2 - (4y^2)^2 = (x^2 - 4y^2)(x^2 + 4y^2) $.

📌 So the full expression becomes:

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