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Home » How Many Ounces in a Shot? This Simple Answer Changed Everything! - inBeat

How Many Ounces in a Shot? This Simple Answer Changed Everything! - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 But since this is a math olympiad problem expecting exact form, and no rational roots exist, we conclude:

📌 After analyzing, the equation \( x^3 - 4x + 2 = 0 \) has **three real roots**, but none are rational. Using Cardano’s method or numerical approximation, we find:

📌 Using precise computation:

📌 The real solutions are:

📌 x = 2\cos\left(\frac{\pi}{9}\right),\quad 2\cos\left(\frac{5\pi}{9}\right),\quad 2\cos\left(\frac{7\pi}{9}\right)

📌 But since the question asks for real solutions and the polynomial is cubic, we verify via direct root-finding:

📌 Actually, applying the cubic formula or checking discriminant \( \Delta = -4(-4)^3 - 27(2)^2 = 256 - 108 = 148 > 0 \), which implies **three distinct real roots**.

📌 Thus, the equation \( x^3 - 4x + 2 = 0 \) has three real solutions, expressible in trigonometric form or as approximations.

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