MathDB
Complex Numbers

Source: 2018 AMC 12A #22

February 8, 2018
AMCAMC 12AMC 12 Ageometryparallelogramcomplex numbers

Problem Statement

The solutions to the equations z2=4+415iz^2=4+4\sqrt{15}i and z2=2+23i,z^2=2+2\sqrt 3i, where i=1,i=\sqrt{-1}, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form pqrs,p\sqrt q-r\sqrt s, where p,p, q,q, r,r, and ss are positive integers and neither qq nor ss is divisible by the square of any prime number. What is p+q+r+s?p+q+r+s?
<spanclass=latexbold>(A)</span>20<spanclass=latexbold>(B)</span>21<spanclass=latexbold>(C)</span>22<spanclass=latexbold>(D)</span>23<spanclass=latexbold>(E)</span>24<span class='latex-bold'>(A) </span> 20 \qquad <span class='latex-bold'>(B) </span> 21 \qquad <span class='latex-bold'>(C) </span> 22 \qquad <span class='latex-bold'>(D) </span> 23 \qquad <span class='latex-bold'>(E) </span> 24