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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 #### \(\frac{4}{5}\)Question: An angel investor models the growth of two startups' valuations with $ f(x) = x^2 - 3x + m $ and $ g(x) = x^2 - 3x + 5m $. If $ f(2) = g(2) $, determine $ m $.

📌 Solution: Set $ f(2) = g(2) $. Compute $ f(2) = 4 - 6 + m = -2 + m $ and $ g(2) = 4 - 6 + 5m = -2 + 5m $. Equate: $ -2 + m = -2 + 5m $. Simplify: $ 0 = 4m $, so $ m = \boxed{0} $.

📌 Question: A chemist designs a catalyst where the reaction efficiency is modeled by $ h(x) = x^2 - 4x + m $. If the efficiency at $ x = 3 $ equals the efficiency of a second catalyst $ k(x) = x^2 - 4x + 2m $ at $ x = 1 $, find $ m $.

📌 Solution: Compute $ h(3) = 9 - 12 + m = -3 + m $ and $ k(1) = 1 - 4 + 2m = -3 + 2m $. Set equal: $ -3 + m = -3 + 2m $. Simplify: $ 0 = m $, so $ m = \boxed{0} $.

📌 Question: If $ a + b = 7 $ and $ a^2 + b^2 = 35 $, find $ a^3 + b^3 $.

📌 Solution: Use identity $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $. First find $ ab $: $ (a + b)^2 = a^2 + 2ab + b^2 \Rightarrow 49 = 35 + 2ab \Rightarrow ab = 7 $. Then $ a^3 + b^3 = 343 - 3 \cdot 7 \cdot 7 = 343 - 147 = \boxed{196} $.

📌 Question: Find the intersection point of the lines $ 2x + y = 10 $ and $ x - y = 2 $.

📌 Solution: Solve the system. From the second equation, $ x = y + 2 $. Substitute into the first: $ 2(y + 2) + y = 10 \Rightarrow 2y + 4 + y = 10 \Rightarrow 3y = 6 \Rightarrow y = 2 $. Then $ x = 2 + 2 = 4 $. The intersection is $ \boxed{(4, 2)} $.

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