Create a system of equations

Oct 13, 2025 by windailys.com 113 views

Introduction

In mathematics, a polynomial equation is an equation in which the variable(s) appear in the form of a polynomial. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Polynomial equations can be solved using various methods, including factoring, the quadratic formula, and graphing. In this article, we will explore two methods to find the root of the polynomial equation $$2(x-2)(x+3)=18$$: using a graphing calculator and a system of equations.

Method 1: Using a Graphing Calculator

A graphing calculator is a powerful tool that can be used to visualize and solve polynomial equations. To find the root of the equation $$2(x-2)(x+3)=18$$ using a graphing calculator, follow these steps:

Enter the equation: Enter the equation $$2(x-2)(x+3)=18$$ into the graphing calculator. You can do this by typing the equation into the calculator's input field or by using the calculator's built-in equation editor.
Graph the equation: Graph the equation by pressing the "Graph" button or by using the calculator's built-in graphing function.
Find the root: Find the root of the equation by looking for the point on the graph where the equation equals zero. This point represents the root of the equation.

Using a graphing calculator, we can see that the graph of the equation $$2(x-2)(x+3)=18$$ intersects the x-axis at the point (3, 0). This means that the root of the equation is x = 3.

Method 2: Using a System of Equations

Another method to find the root of the polynomial equation $$2(x-2)(x+3)=18$$ is to use a system of equations. A system of equations is a set of two or more equations that are solved simultaneously. To find the root of the equation using a system of equations, follow these steps:

Simplify the equation: Simplify the equation $$2(x-2)(x+3)=18$$ by expanding the left-hand side of the equation.
Create a system of equations: Create a system of equations by setting the simplified equation equal to zero and solving for x.
Solve the system: Solve the system of equations using algebraic methods or by using a graphing calculator.

Simplifying the equation $$2(x-2)(x+3)=18$$, we get:

2(x^2 + x - 6) = 18

Expanding the left-hand side of the equation, we get:

2x^2 + 2x - 12 = 18

Subtracting 18 from both sides of the equation, we get:

2x^2 + 2x - 30 = 0

Dividing both sides of the equation by 2, we get:

x^2 + x - 15 = 0

This is a quadratic equation that can be factored as:

(x + 5)(x - 3) = 0

Setting each factor equal to zero, we get:

x + 5 = 0 \quad \text{or} \quad x - 3 = 0

Solving for x, we get:

x = -5 \quad \text{or} \quad x = 3

Since the root of the equation must be a real number, we can discard the solution x = -5. Therefore, the root of the equation is x = 3.

Conclusion

In this article, we explored two methods to find the root of the polynomial equation $$2(x-2)(x+3)=18$$: using a graphing calculator and a system of equations. Using a graphing calculator, we found that the root of the equation is x = 3. Using a system of equations, we also found that the root of the equation is x = 3. Therefore, the correct answer is D. 3.

Frequently Asked Questions

What is the root of the polynomial equation $$2(x-2)(x+3)=18$$? The root of the polynomial equation $$2(x-2)(x+3)=18$$ is x = 3.
How can I find the root of a polynomial equation? You can find the root of a polynomial equation using a graphing calculator or by using a system of equations.
What is a system of equations? A system of equations is a set of two or more equations that are solved simultaneously.

References

Introduction

In our previous article, we explored two methods to find the root of the polynomial equation $$2(x-2)(x+3)=18$$: using a graphing calculator and a system of equations. In this article, we will answer some frequently asked questions about polynomial equations and graphing calculators.

Q&A

Q: What is a polynomial equation?

A: A polynomial equation is an equation in which the variable(s) appear in the form of a polynomial. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents.

Q: How can I find the root of a polynomial equation?

A: You can find the root of a polynomial equation using a graphing calculator or by using a system of equations. To use a graphing calculator, enter the equation into the calculator and graph it. The point on the graph where the equation equals zero represents the root of the equation. To use a system of equations, simplify the equation and solve for x.

Q: What is a graphing calculator?

A: A graphing calculator is a powerful tool that can be used to visualize and solve polynomial equations. It can graph equations and find the roots of the equations.

Q: How do I use a graphing calculator to find the root of a polynomial equation?

A: To use a graphing calculator to find the root of a polynomial equation, follow these steps:

Enter the equation into the calculator.
Graph the equation.
Find the point on the graph where the equation equals zero. This point represents the root of the equation.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously. To find the root of a polynomial equation using a system of equations, simplify the equation and solve for x.

Q: How do I use a system of equations to find the root of a polynomial equation?

A: To use a system of equations to find the root of a polynomial equation, follow these steps:

Simplify the equation.
Create a system of equations by setting the simplified equation equal to zero and solving for x.
Solve the system of equations using algebraic methods or by using a graphing calculator.

Q: What is the difference between a polynomial equation and a linear equation?

A: A polynomial equation is an equation in which the variable(s) appear in the form of a polynomial. A linear equation is an equation in which the variable(s) appear in the form of a linear expression. For example, the equation $$x^2 + 2x - 3 = 0$$ is a polynomial equation, while the equation $$2x - 3 = 0$$ is a linear equation.

Q: How can I determine if a polynomial equation has a real root?

A: To determine if a polynomial equation has a real root, you can use the quadratic formula or the discriminant. The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where a, b, and c are the coefficients of the polynomial equation. The discriminant is $$b^2 - 4ac$$. If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Conclusion

In this article, we answered some frequently asked questions about polynomial equations and graphing calculators. We hope that this article has been helpful in understanding how to find the root of a polynomial equation using a graphing calculator and a system of equations.

Frequently Asked Questions

What is a polynomial equation? A polynomial equation is an equation in which the variable(s) appear in the form of a polynomial.
How can I find the root of a polynomial equation? You can find the root of a polynomial equation using a graphing calculator or by using a system of equations.
What is a graphing calculator? A graphing calculator is a powerful tool that can be used to visualize and solve polynomial equations.
How do I use a graphing calculator to find the root of a polynomial equation? To use a graphing calculator to find the root of a polynomial equation, enter the equation into the calculator, graph the equation, and find the point on the graph where the equation equals zero.