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Home » The function can be expressed in factored form based on the roots: - Malaeb

The function can be expressed in factored form based on the roots: - Malaeb

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: Let $ t = x^2 - 3 $, so $ x^2 = t + 3 $. Substitute into the equation: $ f(t) = 2(t + 3)^2 - 12(t + 3) + 13 $. Expanding: $ 2(t^2 + 6t + 9) - 12t - 36 + 13 = 2t^2 + 12t + 18 - 12t - 36 + 13 = 2t^2 - 5 $. Thus, $ f(x^2 + 1) = 2(x^2 + 1)^2 - 5 = 2x^4 + 4x^2 + 2 - 5 = 2x^4 + 4x^2 - 3 $. oxed{2x^4 + 4x^2 - 3}

📌 Question: The diagonal of a rectangle is 25. What is the maximum possible area of the rectangle?

📌 Solution: Let the sides be $ a $ and $ b $. By the Pythagorean theorem, $ a^2 + b^2 = 25^2 = 625 $. The area $ A = ab $. To maximize $ A $, use the identity $ (a + b)^2 = a^2 + 2ab + b^2 $, but it is more direct to note that for fixed $ a^2 + b^2 $, $ ab $ is maximized when $ a = b $. Thus, $ 2a^2 = 625 \Rightarrow a = rac{25}{\sqrt{2}} $. The maximum area is $ A = \left(rac{25}{\sqrt{2}}

📌 ight)^2 = rac{625}{2} = 312.5 $. oxed{\dfrac{625}{2}}

📌 Question: Find the number of functions $ f : \mathbb{R} o \mathbb{R} $ such that $ f(a + b) = f(a) + f(b) + ab $ for all real $ a, b $.

📌 Solution: Assume $ f $ is quadratic. Let $ f(x) = px^2 + qx + r $. Substitute into the equation:

📌 Left-hand side: $ f(a + b) = p(a + b)^2 + q(a + b) + r = p(a^2 + 2ab + b^2) + q(a + b) + r $.

📌 Right-hand side: $ f(a) + f(b) + ab = p(a^2 + b^2) + q(a + b) + 2r + ab $.

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