Graph The System Of Equations To Find The Solutions Of Y = X 3 + 6 X 2 − 40 X Y = 192 \begin{array}{l} y = X^3 + 6x^2 - 40x \\ y = 192 \end{array} Y = X 3 + 6 X 2 − 40 X Y = 192 ​ Select All Of The Solution(s) To The Equation:A. X = − 8 X = -8 X = − 8 B. X = − 4 X = -4 X = − 4 C. X = 0 X = 0 X = 0 D. $x =

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Introduction

Graphing systems of equations is a powerful tool for solving equations and finding the solutions. In this article, we will explore how to graph the system of equations to find the solutions of the given equation:

y=x3+6x2−40xy=192\begin{array}{l} y = x^3 + 6x^2 - 40x \\ y = 192 \end{array}

Understanding the Problem

The given equation is a system of two equations, where the first equation is a cubic equation and the second equation is a constant equation. We need to find the values of x that satisfy both equations.

Step 1: Graph the First Equation

To graph the first equation, we need to find the x-intercepts, which are the values of x where the graph crosses the x-axis. We can do this by setting y = 0 and solving for x.

0=x3+6x2−40x0 = x^3 + 6x^2 - 40x

We can factor out x from the equation:

0=x(x2+6x−40)0 = x(x^2 + 6x - 40)

Now, we can factor the quadratic expression:

0=x(x+10)(x−4)0 = x(x + 10)(x - 4)

This gives us three x-intercepts: x = 0, x = -10, and x = 4.

Step 2: Graph the Second Equation

The second equation is a horizontal line at y = 192. We can graph this line by drawing a horizontal line at y = 192.

Step 3: Find the Intersection Points

To find the solutions to the system of equations, we need to find the intersection points of the two graphs. We can do this by finding the x-values where the two graphs intersect.

From the graph, we can see that the two graphs intersect at x = -8 and x = 4.

Step 4: Check the Solutions

To check the solutions, we need to plug the x-values back into the original equations and verify that they are true.

For x = -8:

y=(−8)3+6(−8)2−40(−8)y = (-8)^3 + 6(-8)^2 - 40(-8)

y=−512+384+320y = -512 + 384 + 320

y=192y = 192

This confirms that x = -8 is a solution to the system of equations.

For x = 4:

y=(4)3+6(4)2−40(4)y = (4)^3 + 6(4)^2 - 40(4)

y=64+96−160y = 64 + 96 - 160

y=0y = 0

This confirms that x = 4 is not a solution to the system of equations.

Conclusion

In conclusion, we have graphed the system of equations and found the solutions to the equation:

y=x3+6x2−40xy=192\begin{array}{l} y = x^3 + 6x^2 - 40x \\ y = 192 \end{array}

The solutions to the system of equations are x = -8 and x = 0.

Discussion

This problem is a great example of how graphing systems of equations can be used to solve equations and find the solutions. By graphing the two equations and finding the intersection points, we were able to find the solutions to the system of equations.

Final Answer

The final answer is:

  • A. x=−8x = -8
  • C. x=0x = 0

Introduction

Graphing systems of equations is a powerful tool for solving equations and finding the solutions. In this article, we will explore some common questions and answers related to graphing systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other. Each equation in the system is called a component equation.

Q: How do I graph a system of equations?

A: To graph a system of equations, you need to graph each component equation separately and then find the intersection points of the two graphs.

Q: What is the intersection point of two graphs?

A: The intersection point of two graphs is the point where the two graphs meet. This point represents the solution to the system of equations.

Q: How do I find the intersection points of two graphs?

A: To find the intersection points of two graphs, you need to set the two equations equal to each other and solve for the variable.

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

A: A linear equation is an equation in which the highest power of the variable is 1. A nonlinear equation is an equation in which the highest power of the variable is greater than 1.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find the x-intercept and the y-intercept of the equation. The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

Q: How do I graph a nonlinear equation?

A: To graph a nonlinear equation, you need to find the x-intercepts and the y-intercepts of the equation. The x-intercepts are the points where the graph crosses the x-axis, and the y-intercepts are the points where the graph crosses the y-axis.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a system of equations in which each component equation is a linear equation. A system of nonlinear equations is a system of equations in which each component equation is a nonlinear equation.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the intersection points of the two graphs. You can do this by setting the two equations equal to each other and solving for the variable.

Q: How do I solve a system of nonlinear equations?

A: To solve a system of nonlinear equations, you need to find the intersection points of the two graphs. You can do this by setting the two equations equal to each other and solving for the variable.

Q: What is the importance of graphing systems of equations?

A: Graphing systems of equations is an important tool for solving equations and finding the solutions. It allows you to visualize the relationships between the variables and to find the intersection points of the two graphs.

Conclusion

In conclusion, graphing systems of equations is a powerful tool for solving equations and finding the solutions. By understanding the basics of graphing systems of equations, you can solve a wide range of problems and find the solutions to complex equations.

Final Answer

The final answer is:

  • A system of equations is a set of two or more equations that are related to each other.
  • To graph a system of equations, you need to graph each component equation separately and then find the intersection points of the two graphs.
  • The intersection point of two graphs is the point where the two graphs meet.
  • To find the intersection points of two graphs, you need to set the two equations equal to each other and solve for the variable.
  • A linear equation is an equation in which the highest power of the variable is 1.
  • A nonlinear equation is an equation in which the highest power of the variable is greater than 1.
  • A system of linear equations is a system of equations in which each component equation is a linear equation.
  • A system of nonlinear equations is a system of equations in which each component equation is a nonlinear equation.
  • To solve a system of linear equations, you need to find the intersection points of the two graphs.
  • To solve a system of nonlinear equations, you need to find the intersection points of the two graphs.
  • Graphing systems of equations is an important tool for solving equations and finding the solutions.