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Home » Question: Expand the product $ (2a - 3b)(4a^2 + 6ab + 9b^2) $. - ClickBalance

Question: Expand the product $ (2a - 3b)(4a^2 + 6ab + 9b^2) $. - ClickBalance

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Solution: To find the point on the line \(y = 2x + 3\) closest to \((1, -1)\), we minimize the distance between \((x, 2x + 3)\) and \((1, -1)\). The squared distance is:

📌 D = (x - 1)^2 + (2x + 3 + 1)^2 = (x - 1)^2 + (2x + 4)^2

📌 Expanding:

📌 D = (x^2 - 2x + 1) + (4x^2 + 16x + 16) = 5x^2 + 14x + 17

📌 \(D' = 10x + 14\). Set to zero:

📌 \(10x + 14 = 0 \Rightarrow x = -rac{7}{5}\)

📌 Then \(y = 2(-rac{7}{5}) + 3 = -rac{14}{5} + rac{15}{5} = rac{1}{5}\)

📌 Thus, the closest point is \(\left(-rac{7}{5}, rac{1}{5}

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