Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 4 - Quadratic Functions and Equations - 4-8 Complex Numbers - Practice and Problem-Solving Exercises - Page 253: 42

Answer

$x=1-i \text{ OR } x=1+i$

Work Step by Step

In the given equation, \begin{align*} x^2-2x+2=0 ,\end{align*} $a= 1 ,$ $b= -2 ,$ and $c= 2 .$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then \begin{align*}\require{cancel}x&= \dfrac{-(-2)\pm\sqrt{(-2)^2-4(1)(2)}}{2(1)} \\\\&= \dfrac{2\pm\sqrt{4-8}}{2} \\\\&= \dfrac{2\pm\sqrt{-4}}{2} \\\\&= \dfrac{2\pm\sqrt{-1}\cdot\sqrt{4}}{2} \\\\&= \dfrac{2\pm\sqrt{-1}\cdot2}{2} \\\\&= \dfrac{2\pm i\cdot2}{2} &\text{ (use $i=\sqrt{-1}$)} \\\\&= \dfrac{2\pm 2i}{2} \\\\&= \dfrac{\cancel2^1\pm \cancel2^1i}{\cancel2^1} &\text{ (divide by $2$)} \\\\&= 1\pm i .\end{align*} The solutions are $ x=1-i \text{ and } x=1+i .$
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