Q. 32 - Precalculus Enhanced with Graphing Utilities

Question

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.
$x^3+2x^2-9x-18=0$

Step-by-Step Solution

Answer

The required solutions are $x=-2,\hspace{0.17em}x=-3,\hspace{0.17em}x=3$. The solutions of the equation are given below:

Step 1. Given information.

We have:

$x^3+2x^2-9x-18=0$

Step 2. Solve the equation algebraically.

The equation is $x^3+2x^2-9x-18=0$.

Take the common out from the groups: 

$$\begin{gathered} x^2(x+2)-9(x+2)=0 \\ \left(x^2-9\right)\left(x+2\right)=0 \end{gathered}$$

Apply the difference of two squares formula: $x^2-y^2=\left(x-y\right)\left(x+y\right)$

$$\begin{gathered} \left(x^2-3^2\right)\left(x+2\right)=0 \\ \left(x-3\right)\left(x+3\right)\left(x+2\right)=0 \\ x=-2,-3,3 \end{gathered}$$

Step 3. Verify the solution using a graphing utility.

The solution of the equation are $x=-2,\hspace{0.17em}x=-3,\hspace{0.17em}x=3$.

Draw the graph of the equation $x^3+2x^2-9x-18=0$ using the graphing utility: