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Home » "Star Wars Rebels Cast Secrets: Who’s Actually Leading the Resistance? - Leaselab

"Star Wars Rebels Cast Secrets: Who’s Actually Leading the Resistance? - Leaselab

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 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: $oxed{\infty}$

📌 Question: Find the center of the hyperbola $ 9x^2 - 18x - 16y^2 - 64y = 144 $.

📌 Solution: Group terms: $ 9(x^2 - 2x) - 16(y^2 + 4y) = 144 $. Complete the square: $ 9[(x - 1)^2 - 1] - 16[(y + 2)^2 - 4] = 144 $. Expand: $ 9(x - 1)^2 - 9 - 16(y + 2)^2 + 64 = 144 $. Simplify: $ 9(x - 1)^2 - 16(y + 2)^2 = 89 $. The center is at $ (1, -2) $. Final answer: $oxed

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