Use A Graphing Calculator And A System Of Equations To Find The Roots Of The Equation. X 4 − 4 X 3 = 6 X 2 − 12 X X^4 - 4x^3 = 6x^2 - 12x X 4 − 4 X 3 = 6 X 2 − 12 X From Least To Greatest, What Are The Integral Roots Of The Equation? { \square$}$ And { \square$}$

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Introduction

In this article, we will explore the process of solving a fourth-degree polynomial equation using a graphing calculator and a system of equations. The given equation is $x^4 - 4x^3 = 6x^2 - 12x$. Our goal is to find the integral roots of the equation from least to greatest.

Step 1: Rearrange the Equation

To begin solving the equation, we need to rearrange it to set it equal to zero. This will allow us to use the graphing calculator to find the roots.

$$x^4 - 4x^3 - 6x^2 + 12x = 0$$

Step 2: Factor the Equation

Before using the graphing calculator, let's try to factor the equation. We can start by factoring out the greatest common factor (GCF), which is $x$.

$$x(x^3 - 4x^2 - 6x + 12) = 0$$

Step 3: Use the Graphing Calculator

Now that we have factored the equation, we can use the graphing calculator to find the roots. We will graph the function $y = x^3 - 4x^2 - 6x + 12$ and find the x-intercepts.

Graphing the Function

To graph the function, we need to enter it into the graphing calculator. We can do this by pressing the "Y=" button and entering the function.

y = x^3 - 4x^2 - 6x + 12

Finding the x-Intercepts

To find the x-intercepts, we need to press the "2nd" button and then the "TRACE" button. This will allow us to find the x-intercepts of the graph.

Step 4: Find the Integral Roots

Once we have found the x-intercepts, we can use the graphing calculator to find the integral roots of the equation. We will round the x-intercepts to the nearest integer.

Finding the Integral Roots

To find the integral roots, we need to press the "2nd" button and then the "CALC" button. This will allow us to find the x-intercepts of the graph.

Step 5: Write the Final Answer

Now that we have found the integral roots, we can write the final answer.

Final Answer

The integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$ from least to greatest are $\boxed{-2}$ and $\boxed{3}$.

Conclusion

In this article, we have explored the process of solving a fourth-degree polynomial equation using a graphing calculator and a system of equations. We have rearranged the equation, factored it, and used the graphing calculator to find the roots. Finally, we have found the integral roots of the equation from least to greatest.

System of Equations

To solve the equation $x^4 - 4x^3 = 6x^2 - 12x$, we can also use a system of equations. We can rewrite the equation as a system of two equations:

$$x^4 - 4x^3 = 0$$

$$6x^2 - 12x = 0$$

Solving the System of Equations

To solve the system of equations, we can use the substitution method. We can solve the first equation for $x^3$ and substitute it into the second equation.

$$x^3 = 4x^2$$

$$6x^2 - 12x = 0$$

Simplifying the Equation

We can simplify the equation by dividing both sides by $x^2$.

$$6 - 12 = 0$$

Finding the Roots

We can find the roots of the equation by setting the expression equal to zero.

$$6 - 12 = 0$$

$$-6 = 0$$

Conclusion

Unfortunately, we cannot find the roots of the equation using the system of equations. However, we can use the graphing calculator to find the roots.

Graphing Calculator

To solve the equation $x^4 - 4x^3 = 6x^2 - 12x$, we can also use a graphing calculator. We can graph the function $y = x^4 - 4x^3 - 6x^2 + 12x$ and find the x-intercepts.

Graphing the Function

To graph the function, we need to enter it into the graphing calculator. We can do this by pressing the "Y=" button and entering the function.

y = x^4 - 4x^3 - 6x^2 + 12x

Finding the x-Intercepts

To find the x-intercepts, we need to press the "2nd" button and then the "TRACE" button. This will allow us to find the x-intercepts of the graph.

Conclusion

In this article, we have explored the process of solving a fourth-degree polynomial equation using a graphing calculator and a system of equations. We have rearranged the equation, factored it, and used the graphing calculator to find the roots. Finally, we have found the integral roots of the equation from least to greatest.

Final Answer

The integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$ from least to greatest are $\boxed{-2}$ and $\boxed{3}$.

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Introduction

In our previous article, we explored the process of solving a fourth-degree polynomial equation using a graphing calculator and a system of equations. We rearranged the equation, factored it, and used the graphing calculator to find the roots. Finally, we found the integral roots of the equation from least to greatest.

Q&A

Here are some frequently asked questions about solving the equation $x^4 - 4x^3 = 6x^2 - 12x$ using a graphing calculator and a system of equations.

Q: What is the first step in solving the equation $x^4 - 4x^3 = 6x^2 - 12x$?

A: The first step in solving the equation is to rearrange it to set it equal to zero.

Q: How do I factor the equation $x^4 - 4x^3 = 6x^2 - 12x$?

A: To factor the equation, we can start by factoring out the greatest common factor (GCF), which is $x$.

Q: What is the purpose of using a graphing calculator to solve the equation $x^4 - 4x^3 = 6x^2 - 12x$?

A: The purpose of using a graphing calculator is to find the roots of the equation. We can graph the function $y = x^4 - 4x^3 - 6x^2 + 12x$ and find the x-intercepts.

Q: How do I find the integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$?

A: To find the integral roots, we need to round the x-intercepts to the nearest integer.

Q: Can I use a system of equations to solve the equation $x^4 - 4x^3 = 6x^2 - 12x$?

A: Yes, we can use a system of equations to solve the equation. We can rewrite the equation as a system of two equations and use the substitution method to solve it.

Q: What are the integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$ from least to greatest?

A: The integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$ from least to greatest are $\boxed{-2}$ and $\boxed{3}$.

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $x^4 - 4x^3 = 6x^2 - 12x$ using a graphing calculator and a system of equations. We have explained the process of rearranging the equation, factoring it, and using the graphing calculator to find the roots. Finally, we have found the integral roots of the equation from least to greatest.

Final Answer

The integral roots of the equation $x^4 - 4x^3 = 6x^2 - 12x$ from least to greatest are $\boxed{-2}$ and $\boxed{3}$.

Additional Resources

If you have any additional questions or need further clarification on the process of solving the equation $x^4 - 4x^3 = 6x^2 - 12x$ using a graphing calculator and a system of equations, please refer to the following resources:

• Graphing Calculator Tutorial
• System of Equations Tutorial
• Mathematics Online Community

We hope this article has been helpful in answering your questions about solving the equation $x^4 - 4x^3 = 6x^2 - 12x$ using a graphing calculator and a system of equations.