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Home » From Stagnant Puddles to Lush Greenery: The Powerful Impact of Tsuyu Explained! - Londonproperty

From Stagnant Puddles to Lush Greenery: The Powerful Impact of Tsuyu Explained! - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Combine like terms: $ 8a^3 - 27b^3 $. The expanded form is $ oxed{8a^3 - 27b^3} $.

📌 Question: Two biotechnology strains grow according to $ y = 3x + 2 $ and $ y = -2x + 12 $. Find their intersection point.

📌 Solution: Set $ 3x + 2 = -2x + 12 $. Solve: $ 5x = 10 $, so $ x = 2 $. Substitute into $ y = 3(2) + 2 = 8 $. The intersection is $ oxed{(2, 8)} $.Question: Let $ g(x) $ be a polynomial such that $ g(2x + 3) = 4x^2 + 12x + 5 $. Find $ g(x^2 - 1) $.

📌 Solution: Let $ u = 2x + 3 $. Then $ x = rac{u - 3}{2} $. Substitute into $ g(u) = 4\left(rac{u - 3}{2}

📌 ight)^2 + 12\left(rac{u - 3}{2}

📌 ight) + 5 $. Simplify: $ g(u) = (u - 3)^2 + 6(u - 3) + 5 = u^2 - 6u + 9 + 6u - 18 + 5 = u^2 - 4 $. Thus, $ g(x^2 - 1) = (x^2 - 1)^2 - 4 = x^4 - 2x^2 + 1 - 4 = x^4 - 2x^2 - 3 $. Final answer: $oxed{x^4 - 2x^2 - 3}$

📌 Question: For all real numbers $ x $ and $ y $, find the number of functions $ f : \mathbb{R} o \mathbb{R} $ such that $ f(x + y) = f(x) + f(y) + 2xy $.

📌 Solution: Assume $ f $ is quadratic: $ f(x) = ax^2 + bx + c $. Substitute into the equation: $ a(x + y)^2 + b(x + y) + c = ax^2 + bx + c + ay^2 + by + c + 2xy $. Expand and compare coefficients: $ ax^2 + 2axy + ay^2 + bx + by + c = ax^2 + ay^2 + bx + by + 2c + 2xy $. Matching terms: $ 2a = 2 \Rightarrow a = 1 $, and $ 2c = c \Rightarrow c = 0 $. Thus, $ f(x) = x^2 + bx $. Any real $ b $ satisfies the equation, so there are infinitely many solutions. Final answer: $oxed{\infty}$

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