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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 Solution: Group and complete the square for $ x $ and $ y $.

📌 For $ x $: $ 9(x^2 - 4x) = 9[(x - 2)^2 - 4] $.

📌 For $ y $: $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $.

📌 Substitute back: $ 9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36 $. Simplify: $ 9(x (x - 2)^2 - 4) - 4(y - 3)^2 = 36 $. Divide by 36: $ \frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1 $. The center is at $ (2, 3) $. The center is $\boxed{(2, 3)}$.

📌 Question: For all real numbers $ x $, $ y $, and $ z $, find the number of functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying $ f(x + y) + f(z) = f(x + z) + f(y) $.

📌 Solution: Rearrange the equation: $ f(x + y) - f(x + z) = f(y) - f(z) $. Let $ x = 0 $: $ f

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