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Home » \Rightarrow \frac9d^22 = 14a^2 - ToelettAPP

\Rightarrow \frac9d^22 = 14a^2 - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: Use identity $ p^2 + q^2 = (p + q)^2 - 2pq $. Substituting: $ 10 - 24i = (3 + 4i)^2 - 2pq $. Compute $ (3 + 4i)^2 = 9 + 24i - 16 = -7 + 24i $. Then $ 10 - 24i = -7 + 24i - 2pq $. Rearranging: $ 17 - 48i = -2pq \Rightarrow pq = \frac{-17 + 48i}{2} $. \boxed{-\dfrac{17}{2} + 24i}

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

📌 Solution: Assume $ f $ is quadratic: $ f(x) = ax^2 + bx + c $. Substitute into equation: $ a(x + y)^2 + b(x + y) + c = ax^2 + bx + c + ay^2 + by + c + 2xy $. Expand left: $ ax^2 + 2axy + ay^2 + bx + by + c $. Right: $ ax^2 + ay^2 + bx + by + 2c + 2xy $. Equate coefficients: $ 2a = 2 \Rightarrow a = 1 $, and $ 2c = c \Rightarrow c = 0 $. Thus, $ f(x) = x^2 + bx $. Check: $ (x + y)^2 + b(x + y) = x^2 + y^2 + 2xy + bx + by $, which matches. Any $ b $ works, so infinitely many solutions. \boxed{\infty}<think1. A train travels at a constant speed of 90 miles per hour. If it travels for 2.5 hours, how far does the train travel?

📌 The distance traveled by the train is given by the formula:

📌 Distance = Speed × Time

📌 Distance = 90 miles/hour × 2.5 hours = 225 miles

📌 Perimeter = 2(Length + Width)

📌 = 2(3w + w)

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= \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056}
A = 84
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