MathDB
Operation Complex Numbers

Source: 2023 AMC 12B #12

November 15, 2023
AMCAMC 12complex numbers2023 AMC2023 AMC 12B

Problem Statement

For complex numbers u=a+biu=a+bi and v=c+div=c+di, define the binary operation ⊗\otimes by u⊗v=ac+bdi.u\otimes v=ac+bdi. Suppose zz is a complex number such that z⊗z=z2+40z\otimes z=z^{2}+40. What is ∣z∣|z|?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 26<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 52<span class='latex-bold'>(A)</span>~\sqrt{10}\qquad<span class='latex-bold'>(B)</span>~3\sqrt{2}\qquad<span class='latex-bold'>(C)</span>~2\sqrt{6}\qquad<span class='latex-bold'>(D)</span>~6\qquad<span class='latex-bold'>(E)</span>~5\sqrt{2}
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