Chapter 4.7, Problem 8OE

To determine

To solve the equation.

Answer to Problem 8OE

The values of $z=-3$

Explanation of Solution

Given information:

  $z^2+6z+9=0$

Calculation:

The given equation already has 0 as one side, and the other side is already factored. Simply set each factored equal to 0 and solve.

The equationcan be calculated as

  $\begin{array}{l}{z}^2+6z+9=0\\ z^2+\left(3z+3z\right)+9=0\left[\text{write 6z}=\left(3z+3z\right)\right]\\ z^2+3z+3z+9=0\\ z\left(z+3\right)+3\left(z+3\right)=0\left[\begin{array}{l}\text{the first and second term has a common factor,}z\text{ and}\\ \text{the third and fourth term has a common factor,3}\end{array}\right]\\ \left(z+3\right)\left(z+3\right)=0\end{array}$

Therefore,

  $\begin{array}{l}\left(z+3\right)=0\text{ or }\left(z+3\right)=0\left[\text{use the zero-product property}\right]\\ z=-3\text{ or }z=-3\\ \\ \text{check: }f\left(-3\right)={\left(-3\right)}^2+6\left(-3\right)+9=9-18+9=18-18=0\end{array}$

Hence,

The values of $z=-3$