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Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6 + 2ab, then what is a^3 + b^3?

 May 1, 2024
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\(a^2 + b^2 = 6 + 2ab\\ a^2 + 2ab + b^2 = 6 + 4ab\\ (a + b)^2 = 6 + 4ab\)

But a + b = 4, so

\(4^2 = 6 + 4 ab\\ ab = \dfrac 52\)

 

Then consequently \(a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 4^3 -3\cdot \dfrac52 \cdot 4 = 34\).

 May 1, 2024

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