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

$9{{x}^{2}}-18x-12=0$
$1-\frac{\sqrt{21}}{3}\text{ and }1+\frac{\sqrt{21}}{3}$ 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}