Algebra 2 (1st Edition)

Published by McDougal Littell
ISBN 10: 0618595414
ISBN 13: 978-0-61859-541-9

Chapter 4 Quadratic Functions and Factoring - 4.6 Perform Operations with Complex Numbers - 4.6 Exercises - Skill Practice - Page 280: 63


$\displaystyle \frac{ac-bd}{c^{2}+d^{2}}+\frac{ad+bc}{c^{2}+d^{2}}i$

Work Step by Step

$\displaystyle \frac{a+bi}{c-di}\qquad$ ...rationalize by multiplying both the numerator and denominator with $c+di$. $=\displaystyle \frac{(a+bi)(c+di)}{(c-di)(c+di)}$ $\qquad$ ...use the FOIL method in the numerator and the difference of squares: $(a-b)(a+b)=a^{2}-b^{2}$ in the denominator. $=\displaystyle \frac{ac+adi+bci+bdi^{2}}{c^{2}-(di)^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$). $=\displaystyle \frac{ac+adi+bci-bd}{c^{2}+d^{2}}\qquad$ the real and imaginary parts in the numerator. $=\displaystyle \frac{ac-bd+(ad+bc)i}{c^{2}+d^{2}}\qquad$ ...write in standard form $a+bi$ $=\displaystyle \frac{ac-bd}{c^{2}+d^{2}}+\frac{ad+bc}{c^{2}+d^{2}}i$
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