What Is The Root Of The Polynomial Equation $x(x-2)(x+3)=18$? Use A Graphing Calculator And A System Of Equations To Find The Roots.

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Introduction

Polynomial equations are a fundamental concept in algebra, and solving them is a crucial skill for mathematicians and scientists. In this article, we will explore the root of the polynomial equation $x(x-2)(x+3)=18$ using a graphing calculator and a system of equations.

Understanding the Polynomial Equation

The given polynomial equation is $x(x-2)(x+3)=18$. To find the roots of this equation, we need to understand the concept of roots and how to solve polynomial equations.

What are Roots?

Roots of a polynomial equation are the values of x that satisfy the equation. In other words, when we substitute a root into the equation, the equation becomes true. For example, if we have the equation $x^2 - 4 = 0$, the roots are $x = 2$ and $x = -2$.

Solving Polynomial Equations

There are several methods to solve polynomial equations, including factoring, quadratic formula, and graphing. In this article, we will use a graphing calculator and a system of equations to find the roots of the given polynomial equation.

Using a Graphing Calculator

A graphing calculator is a powerful tool that can help us visualize the graph of a polynomial equation and find its roots. To use a graphing calculator, we need to enter the equation and set the calculator to graphing mode.

Step 1: Enter the Equation

Enter the equation $x(x-2)(x+3)=18$ into the graphing calculator.

Step 2: Set the Calculator to Graphing Mode

Set the calculator to graphing mode by pressing the "Graph" button.

Step 3: Graph the Equation

Graph the equation by pressing the "Graph" button.

Step 4: Find the Roots

Find the roots of the equation by looking at the graph. The roots are the x-intercepts of the graph.

Using a System of Equations

A system of equations is a set of two or more equations that are solved simultaneously. We can use a system of equations to find the roots of the given polynomial equation.

Step 1: Write the Equation as a System of Equations

Write the equation $x(x-2)(x+3)=18$ as a system of equations:

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

$x^3 - x^2 - 6x - 18 = 0$

Step 2: Solve the System of Equations

Solve the system of equations by using the quadratic formula or factoring.

Step 3: Find the Roots

Find the roots of the equation by solving the system of equations.

Conclusion

In this article, we used a graphing calculator and a system of equations to find the roots of the polynomial equation $x(x-2)(x+3)=18$. We learned that the roots of a polynomial equation are the values of x that satisfy the equation, and we used a graphing calculator and a system of equations to find the roots of the given equation.

The Final Answer

The final answer is:

$x = 6, x = -3, x = 2$

Note

The final answer is a set of three roots, which are the values of x that satisfy the equation.

Graphing Calculator Method

To use a graphing calculator, follow these steps:

1. Enter the equation $x(x-2)(x+3)=18$ into the graphing calculator.
2. Set the calculator to graphing mode by pressing the "Graph" button.
3. Graph the equation by pressing the "Graph" button.
4. Find the roots of the equation by looking at the graph. The roots are the x-intercepts of the graph.

System of Equations Method

To use a system of equations, follow these steps:

1. Write the equation $x(x-2)(x+3)=18$ as a system of equations: $x(x-2)(x+3) = 18$ $x^3 - x^2 - 6x - 18 = 0$
2. Solve the system of equations by using the quadratic formula or factoring.
3. Find the roots of the equation by solving the system of equations.

Quadratic Formula Method

To use the quadratic formula, follow these steps:

1. Write the equation $x^3 - x^2 - 6x - 18 = 0$ in the form $ax^2 + bx + c = 0$.
2. Plug the values of a, b, and c into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
3. Simplify the expression to find the roots of the equation.

Factoring Method

To use factoring, follow these steps:

1. Factor the equation $x^3 - x^2 - 6x - 18 = 0$ into the form $(x - r_1)(x - r_2)(x - r_3) = 0$.
2. Set each factor equal to zero and solve for x.
3. Find the roots of the equation by solving for x.

Solving the Equation

To solve the equation $x(x-2)(x+3)=18$, follow these steps:

1. Use a graphing calculator to graph the equation and find the roots.
2. Use a system of equations to solve the equation and find the roots.
3. Use the quadratic formula or factoring to solve the equation and find the roots.

Final Answer

The final answer is:

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

A: The polynomial equation $x(x-2)(x+3)=18$ is a cubic equation that can be solved using various methods, including graphing, systems of equations, and factoring.

Q: How do I use a graphing calculator to solve the equation?

A: To use a graphing calculator, follow these steps:

1. Enter the equation $x(x-2)(x+3)=18$ into the graphing calculator.
2. Set the calculator to graphing mode by pressing the "Graph" button.
3. Graph the equation by pressing the "Graph" button.
4. Find the roots of the equation by looking at the graph. The roots are the x-intercepts of the graph.

Q: How do I use a system of equations to solve the equation?

A: To use a system of equations, follow these steps:

1. Write the equation $x(x-2)(x+3)=18$ as a system of equations: $x(x-2)(x+3) = 18$ $x^3 - x^2 - 6x - 18 = 0$
2. Solve the system of equations by using the quadratic formula or factoring.
3. Find the roots of the equation by solving the system of equations.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula to solve the equation?

A: To use the quadratic formula, follow these steps:

1. Write the equation $x^3 - x^2 - 6x - 18 = 0$ in the form $ax^2 + bx + c = 0$.
2. Plug the values of a, b, and c into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
3. Simplify the expression to find the roots of the equation.

Q: What is factoring?

A: Factoring is a mathematical technique that can be used to solve polynomial equations. Factoring involves expressing a polynomial as a product of simpler polynomials.

Q: How do I use factoring to solve the equation?

A: To use factoring, follow these steps:

1. Factor the equation $x^3 - x^2 - 6x - 18 = 0$ into the form $(x - r_1)(x - r_2)(x - r_3) = 0$.
2. Set each factor equal to zero and solve for x.
3. Find the roots of the equation by solving for x.

Q: What are the roots of the equation $x(x-2)(x+3)=18$?

A: The roots of the equation $x(x-2)(x+3)=18$ are $x = 6, x = -3, x = 2$.

Q: How do I verify the roots of the equation?

A: To verify the roots of the equation, follow these steps:

1. Plug each root into the original equation and simplify.
2. Check if the simplified expression is equal to zero.
3. If the simplified expression is equal to zero, then the root is correct.

Q: What are some common mistakes to avoid when solving polynomial equations?

A: Some common mistakes to avoid when solving polynomial equations include:

• Not checking if the equation is a quadratic equation or a cubic equation.
• Not using the correct method to solve the equation (e.g. graphing, systems of equations, factoring).
• Not verifying the roots of the equation.
• Not checking if the simplified expression is equal to zero.

Conclusion

Solving polynomial equations can be a challenging task, but with the right tools and techniques, it can be done. In this article, we discussed how to use a graphing calculator, a system of equations, and factoring to solve the polynomial equation $x(x-2)(x+3)=18$. We also discussed some common mistakes to avoid when solving polynomial equations.