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2023 AMC 12B Problems/Problem 12

Revision as of 16:46, 15 November 2023 by Failure.net (talk | contribs) (→‎Problem)

Problem

For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation

$u \cdot v = ac + bdi$

Suppose $z$ is a complex number such that $z\cdot z = z^{2}+40$. What is $|z|$?

$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}$

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