Intermediate Algebra (12th Edition)

$x=\left\{ -3,3 \right\}$
$\bf{\text{Solution Outline:}}$ Express the given equation, $-3x^2+27=0 ,$ in factored form. Then, use the Zero-Factor Property by equating each factor to zero. Finally, solve each equation. $\bf{\text{Solution Details:}}$ The factored form of the equation above is \begin{array}{l}\require{cancel} -3(x^2-9)=0 \\\\ -3(x+3)(x-3)=0 \\\\ \dfrac{-3(x+3)(x-3)}{-3}=\dfrac{0}{-3} \\\\ (x+3)(x-3)=0 .\end{array} Equating each factor to zero (Zero-Factor Property), then \begin{array}{l}\require{cancel} x+3=0 \text{ OR } x-3=0 .\end{array} Using the properties of equality to solve each of the equation above results to \begin{array}{l}\require{cancel} x+3=0 \\\\ x=-3 \\\\\text{ OR }\\\\ x-3=0 \\\\ x=3 .\end{array} Hence, the solutions are $x=\left\{ -3,3 \right\} .$ $\bf{\text{Supplementary Solution/s:}}$ Factoring the negative $GCF= -3 ,$ the expression, $-3x^2+27 ,$ above is equivalent to \begin{array}{l}\require{cancel} -3 \left( \dfrac{-3x^2}{-3}+\dfrac{27}{-3} \right) \\\\= -3 \left( x^2-9 \right) .\end{array} Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} -3 \left( x+3 \right) \left( x-3 \right) .\end{array}