## Intermediate Algebra (12th Edition)

$x=\left\{ -6,-3 \right\}$
$\bf{\text{Solution Outline:}}$ Express the given equation, $x^2+9x+18=0 ,$ 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} (x+3)(x+6)=0 .\end{array} Equating each factor to zero (Zero-Factor Property), then \begin{array}{l}\require{cancel} x+3=0 \text{ OR } x+6=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+6=0 \\\\ x=-6 .\end{array} Hence, the solutions are $x=\left\{ -6,-3 \right\} .$ $\bf{\text{Supplementary Solution:}}$ To factor the expression, $x^2+9x+18 ,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$ in the quadratic expression $x^2+bx+c.$ Then, express the factored form as $(x+m_1)(x+m_2).$ In the given expression, the value of $c$ is $18$ and the value of $b$ is $9 .$ The possible pairs of integers whose product is $c$ are \begin{array}{l}\require{cancel} \{ 1,18 \}, \{ 2,9 \}, \{ 3,6 \}, \{ -1,-18 \}, \{ -2,-9 \}, \{ -3,-6 \} .\end{array} Among these pairs, the one that gives a sum of $b$ is $\{ 3,6 \}.$ Hence, the factored form of the expression above is \begin{array}{l}\require{cancel} (x+3)(x+6) .\end{array}