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Home » u = rac2 \pm \sqrt-42 = rac2 \pm 2i2 = 1 \pm i - Abu Waleed Tea

u = rac2 \pm \sqrt-42 = rac2 \pm 2i2 = 1 \pm i - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Using the quadratic formula \( x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = -6 \), and \( c = 2 \):

📌 x = rac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = rac{6 \pm \sqrt{36 - 24}}{6} = rac{6 \pm \sqrt{12}}{6}

📌 Simplify \( \sqrt{12} = 2\sqrt{3} \):

📌 x = rac{6 \pm 2\sqrt{3}}{6} = 1 \pm rac{\sqrt{3}}{3}

📌 Thus, the critical points are:

📌 oxed{1 + rac{\sqrt{3}}{3}, 1 - rac{\sqrt{3}}{3}}

📌 Question:** A quantum machine learning algorithm designer models the probability amplitude for a quantum state transition with the function \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \). Verify if \( x = 1 \) is a root of multiplicity greater than 1.

📌 Solution:** To verify if \( x = 1 \) is a root of multiplicity greater than 1 for \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), we first check if \( p(1) = 0 \).

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