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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: Compute the square of $ \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $.

📌 Solution: Let $ x = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} $. We compute $ x^2 $:

📌 x^2 = \left( \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} \right)^2 = (7 + 2\sqrt{10}) + (7 - 2\sqrt{10}) + 2\sqrt{(7 + 2\sqrt{10})(7 - 2\sqrt{10})}

📌 x^2 = 14 + 2\sqrt{7^2 - (2\sqrt{10})^2} = 14 + 2\sqrt{49 - 40} = 14 + 2\sqrt{9} = 14 + 2 \times 3 = 14 + 6 = 20

📌 Thus, $ x^2 = \boxed{20} $.

📌 Question: Find the largest value of $ x $ that satisfies $ |2x - 5| + |x + 3| = 12 $.

📌 Solution: We analyze the expression by considering the critical points where expressions inside absolute values change sign: $ x = \frac{5}{2} $ and $ x = -3 $. We consider three intervals:

📌 $ x < -3 $:

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