Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 11 - Quadratic Functions and Equations - 11.3 Studying Solutions of Quadratic Equations - 11.3 Exercise Set - Page 718: 51



Work Step by Step

$1-\frac{\sqrt{21}}{3}\text{ and }1+\frac{\sqrt{21}}{3}$ It can be expressed as: $x=1+\frac{\sqrt{21}}{3}$ and $x=1-\frac{\sqrt{21}}{3}$ $\begin{align} & x-\left( 1+\frac{\sqrt{21}}{3} \right)=0 \\ & \left( x-1 \right)-\frac{\sqrt{21}}{3}=0 \end{align}$ Or, $\begin{align} & x-\left( 1-\frac{\sqrt{21}}{3} \right)=0 \\ & \left( x-1 \right)+\frac{\sqrt{21}}{3}=0 \\ \end{align}$ Apply the zero-product property $\left( \left( x-1 \right)-\frac{\sqrt{21}}{3} \right)\left( \left( x-1 \right)+\frac{\sqrt{21}}{3} \right)=0$ $\begin{align} & \left( \left( x-1 \right)-\frac{\sqrt{21}}{3} \right)\left( \left( x-1 \right)+\frac{\sqrt{21}}{3} \right)=0 \\ & \left( x-1 \right)\left( x-1 \right)+\frac{\sqrt{21}}{3}\left( x-1 \right)-\frac{\sqrt{21}}{3}\left( x-1 \right)-{{\left( \frac{\sqrt{21}}{3} \right)}^{2}}=0 \end{align}$ Combine like terms: ${{\left( x-1 \right)}^{2}}-{{\left( \frac{\sqrt{21}}{3} \right)}^{2}}=0$ Apply the formula, ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ $\begin{align} & {{x}^{2}}+1-2x-\left( \frac{21}{9} \right)=0 \\ & {{x}^{2}}-2x+1-\frac{21}{9}=0 \\ & {{x}^{2}}-2x+\frac{9-21}{9}=0 \\ & {{x}^{2}}-2x-\frac{12}{9}=0 \end{align}$ Multiply the equation by $9$ to clear the fraction, $\begin{align} & 9\left( {{x}^{2}}-2x-\frac{12}{9} \right)=0 \\ & 9{{x}^{2}}-18x-12=0 \end{align}$
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