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 272: 82

Answer

$x=\left\{ 2-i\sqrt{6}, 2+i\sqrt{6} \right\} $

Work Step by Step

Using $ax^2+bx+c=0,$ the given equation, \begin{align*} -x^2+4x&=10 \\ -x^2+4x-10&=0 ,\end{align*} has $a= -1 ,$ $b= 4 ,$ and $c= -10 .$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then \begin{align*}\require{cancel} x&=\dfrac{-4\pm\sqrt{4^2-4(-1)(-10)}}{2(-1)} \\\\&= \dfrac{-4\pm\sqrt{16-40}}{-2} \\\\&= \dfrac{-4\pm\sqrt{-24}}{-2} \\\\&= \dfrac{-4\pm\sqrt{-1}\cdot\sqrt{4}\cdot\sqrt{6}}{-2} \\\\&= \dfrac{-4\pm\sqrt{-1}\cdot2\sqrt{6}}{-2} \\\\&= \dfrac{-4\pm i\cdot2\sqrt{6}}{-2} &\text{ (use $i=\sqrt{-1}$)} \\\\&= \dfrac{-4\pm 2i\sqrt{6}}{-2} \\\\&= \dfrac{\cancel{-4}^2\pm \cancel{2}^{-1}i\sqrt{6}}{\cancel{-2}^1} &\text{ (divide by $-2$)} \\\\&= 2\pm i\sqrt{6} .\end{align*} The solutions are $ x=\left\{ 2-i\sqrt{6}, 2+i\sqrt{6} \right\} .$
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