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Home » The Best TV for Gaming Secret: Upgrade Your Screen Today & Load Up Faster! - SUpost

The Best TV for Gaming Secret: Upgrade Your Screen Today & Load Up Faster! - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 The fully factored form is $ oxed{(3x - 5y - 4z)(3x - 5y + 4z)} $.Question: Compute the sum of the roots of the equation $ u^{3/2} - 4u + 3\sqrt{u} = 0 $, given that all roots are non-negative real numbers.

📌 Solution: Let $ v = \sqrt{u} $, so $ u = v^2 $. Substituting into the equation gives $ v^3 - 4v^2 + 3v = 0 $. Factoring yields $ v(v^2 - 4v + 3) = 0 $, which simplifies to $ v(v-1)(v-3) = 0 $. Thus, $ v = 0, 1, 3 $, corresponding to $ u = 0, 1, 9 $. The sum of the roots is $ 0 + 1 + 9 = 10 $. oxed{10}

📌 Question: Given the functional equation $ f(x^2 - 3) = 2x^4 - 12x^2 + 13 $, find $ f(x^2 + 1) $.

📌 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 $.

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