What Is The Root Of The Polynomial Equation $x(x-2)(x+3)=18$?Use A Graphing Calculator And A System Of Equations.A. -3 B. 0 C. 2 D. 3

Introduction

Solving polynomial equations can be a challenging task, especially when dealing with cubic equations. In this article, we will explore two methods to find the root of the polynomial equation $x(x-2)(x+3)=18$. The first method involves using a graphing calculator to visualize the equation and find its roots. The second method involves setting up a system of equations to solve for the root.

Method 1: Using a Graphing Calculator

A graphing calculator is a powerful tool that can help us visualize the equation and find its roots. To use a graphing calculator, we need to enter the equation into the calculator and then use the "graph" function to visualize the equation.

Step 1: Enter the Equation

To enter the equation into the graphing calculator, we need to follow these steps:

• Press the "Y=" button to enter the equation.
• Type in the equation $x(x-2)(x+3)=18$.
• Press the "Enter" button to enter the equation.

Step 2: Graph the Equation

Once we have entered the equation, we can use the "graph" function to visualize the equation. To do this, we need to follow these steps:

• Press the "Graph" button to graph the equation.
• Use the "Zoom" function to zoom in on the graph.
• Use the "Trace" function to find the roots of the equation.

Step 3: Find the Roots

Using the graphing calculator, we can find the roots of the equation by using the "Trace" function. To do this, we need to follow these steps:

• Press the "Trace" button to find the roots of the equation.
• Use the "X" function to find the x-coordinates of the roots.
• Use the "Y" function to find the y-coordinates of the roots.

Method 2: Using a System of Equations

Another method to find the root of the polynomial equation is to set up a system of equations. This method involves setting up two equations, one for the x-coordinates and one for the y-coordinates.

Step 1: Set Up the System of Equations

To set up the system of equations, we need to follow these steps:

• Set up the equation $x(x-2)(x+3)=18$.
• Set up the equation $y=x(x-2)(x+3)$.

Step 2: Solve the System of Equations

Once we have set up the system of equations, we can solve for the root by using the "solve" function. To do this, we need to follow these steps:

• Press the "Solve" button to solve the system of equations.
• Use the "X" function to find the x-coordinates of the root.
• Use the "Y" function to find the y-coordinates of the root.

Conclusion

In this article, we have explored two methods to find the root of the polynomial equation $x(x-2)(x+3)=18$. The first method involves using a graphing calculator to visualize the equation and find its roots. The second method involves setting up a system of equations to solve for the root. Both methods are effective and can be used to find the root of the equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3. This can be verified by plugging x = 3 into the equation and solving for y.

Step 1: Plug x = 3 into the Equation

To verify the root, we need to plug x = 3 into the equation and solve for y.

$$x(x-2)(x+3)=18$$

$$3(3-2)(3+3)=18$$

$$3(1)(6)=18$$

$$18=18$$

Step 2: Solve for y

Since the equation is true, we can solve for y.

$$y=x(x-2)(x+3)$$

$$y=3(3-2)(3+3)$$

$$y=3(1)(6)$$

$$y=18$$

Step 3: Verify the Root

Since the equation is true, we can verify that x = 3 is the root of the polynomial equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3. This can be verified by plugging x = 3 into the equation and solving for y.

Step 1: Plug x = 3 into the Equation

To verify the root, we need to plug x = 3 into the equation and solve for y.

$$x(x-2)(x+3)=18$$

$$3(3-2)(3+3)=18$$

$$3(1)(6)=18$$

$$18=18$$

Step 2: Solve for y

Since the equation is true, we can solve for y.

$$y=x(x-2)(x+3)$$

$$y=3(3-2)(3+3)$$

$$y=3(1)(6)$$

$$y=18$$

Step 3: Verify the Root

Since the equation is true, we can verify that x = 3 is the root of the polynomial equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3. This can be verified by plugging x = 3 into the equation and solving for y.

Step 1: Plug x = 3 into the Equation

To verify the root, we need to plug x = 3 into the equation and solve for y.

$$x(x-2)(x+3)=18$$

$$3(3-2)(3+3)=18$$

$$3(1)(6)=18$$

$$18=18$$

Step 2: Solve for y

Since the equation is true, we can solve for y.

$$y=x(x-2)(x+3)$$

$$y=3(3-2)(3+3)$$

$$y=3(1)(6)$$

$$y=18$$

Step 3: Verify the Root

Since the equation is true, we can verify that x = 3 is the root of the polynomial equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3. This can be verified by plugging x = 3 into the equation and solving for y.

Step 1: Plug x = 3 into the Equation

To verify the root, we need to plug x = 3 into the equation and solve for y.

$$x(x-2)(x+3)=18$$

$$3(3-2)(3+3)=18$$

$$3(1)(6)=18$$

$$18=18$$

Step 2: Solve for y

Since the equation is true, we can solve for y.

$$y=x(x-2)(x+3)$$

$$y=3(3-2)(3+3)$$

$$y=3(1)(6)$$

$$y=18$$

Step 3: Verify the Root

Since the equation is true, we can verify that x = 3 is the root of the polynomial equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3. This can be verified by plugging x = 3 into the equation and solving for y.

Step 1: Plug x = 3 into the Equation

To verify the root, we need to plug x = 3 into the equation and solve for y.

$$x(x-2)(x+3)=18$$

$$3(3-2)(3+3)=18$$

$$3(1)(6)=18$$

$$18=18$$

Step 2: Solve for y

Since the equation is true, we can solve for y.

$$y=x(x-2)(x+3)$$

$$y=3(3-2)(3+3)$$

$$y=3(1)(6)$$

$$y=18$$

Step 3: Verify the Root

Since the equation is true, we can verify that x = 3 is the root of the polynomial equation.

Final Answer

The final answer to the problem is:

• The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Discussion

The root of the polynomial equation $x(x-2)(x+3)=

Introduction

In our previous article, we explored two methods to find the root of the polynomial equation $x(x-2)(x+3)=18$. In this article, we will answer some frequently asked questions about the root of the polynomial equation.

Q1: What is the root of the polynomial equation $x(x-2)(x+3)=18$?

A1: The root of the polynomial equation $x(x-2)(x+3)=18$ is x = 3.

Q2: How do I verify the root of the polynomial equation?

A2: To verify the root, you can plug x = 3 into the equation and solve for y. If the equation is true, then x = 3 is the root of the polynomial equation.

Q3: What is the significance of the root of the polynomial equation?

A3: The root of the polynomial equation represents the value of x that satisfies the equation. In this case, the root x = 3 is the value of x that makes the equation true.

Q4: Can I use a graphing calculator to find the root of the polynomial equation?

A4: Yes, you can use a graphing calculator to find the root of the polynomial equation. Simply enter the equation into the calculator and use the "graph" function to visualize the equation. Then, use the "trace" function to find the roots of the equation.

Q5: Can I use a system of equations to find the root of the polynomial equation?

A5: Yes, you can use a system of equations to find the root of the polynomial equation. Simply set up two equations, one for the x-coordinates and one for the y-coordinates. Then, solve the system of equations to find the root.

Q6: What is the difference between the two methods of finding the root of the polynomial equation?

A6: The two methods of finding the root of the polynomial equation are the graphing calculator method and the system of equations method. The graphing calculator method involves using a graphing calculator to visualize the equation and find its roots. The system of equations method involves setting up a system of equations to solve for the root.

Q7: Which method is more accurate?

A7: Both methods are accurate, but the graphing calculator method is more visual and can help you understand the equation better. The system of equations method is more mathematical and can help you solve the equation more precisely.

Q8: Can I use other methods to find the root of the polynomial equation?

A8: Yes, you can use other methods to find the root of the polynomial equation. Some other methods include using a numerical method, such as the Newton-Raphson method, or using a computer algebra system, such as Mathematica.

Q9: What are some common mistakes to avoid when finding the root of the polynomial equation?

A9: Some common mistakes to avoid when finding the root of the polynomial equation include:

• Not checking the equation for extraneous solutions
• Not using a graphing calculator or system of equations to verify the root
• Not checking the root for accuracy
• Not using a numerical method or computer algebra system to solve the equation

Q10: What are some real-world applications of finding the root of the polynomial equation?

A10: Some real-world applications of finding the root of the polynomial equation include:

• Modeling population growth
• Modeling economic systems
• Modeling physical systems, such as springs and pendulums
• Solving optimization problems

Conclusion

In this article, we have answered some frequently asked questions about the root of the polynomial equation $x(x-2)(x+3)=18$. We have discussed the two methods of finding the root, the graphing calculator method and the system of equations method, and have provided some common mistakes to avoid and real-world applications of finding the root of the polynomial equation.