Algebra 2 Common Core

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

Chapter 4 - Quadratic Functions and Equations - Chapter Review - Page 271: 58

Answer

$x=1-i\sqrt{3}$ OR $x=1+i\sqrt{3}$

Work Step by Step

In the form $x^2+bx=c,$ the given equation, $ x^2-2x+4=0 ,$ is equivalent to \begin{align*} x^2-2x+4-4=0-4 \\ x^2-2x=-4 .\end{align*} Adding $\left( \dfrac{b}{2} \right)^2$ on both sides to complete the square of the left side results to \begin{align*} x^2-2x+\left(\dfrac{-2}{2} \right)^2&=-4+\left(\dfrac{-2}{2} \right)^2 \\\\ x^2-2x+\left( -1 \right)^2&=-4+\left(-1\right)^2 \\\\ x^2-2x+1&=-4+1 \\\\ (x-1)^2&=-3 .\end{align*} Taking the square root of both sides (Square Root Property) and then solving for the variable, the equation above is equivalent to \begin{align*} x-1&=\pm\sqrt{-3} \\\\ x-1&=\pm i\sqrt{3} \\\\ x&=1\pm i\sqrt{3} .\end{align*} Hence, the solutions are $x=1-i\sqrt{3}$ OR $x=1+i\sqrt{3}$.
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