Solve The Following System Of Equations And Show All Work.${ \begin{array}{l} y = -x^2 + 12x + 6 \ y = X^2 - 5x + 18 \end{array} }$

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two quadratic equations. We will use algebraic methods to find the solutions and show all the work.

The System of Equations

The given system of equations is:

$$\begin{array}{l} y = -x^2 + 12x + 6 \\ y = x^2 - 5x + 18 \end{array}$$

To solve this system, we need to find the values of x and y that satisfy both equations.

Step 1: Set the Equations Equal to Each Other

Since both equations are equal to y, we can set them equal to each other:

$$-x^2 + 12x + 6 = x^2 - 5x + 18$$

Step 2: Simplify the Equation

Combine like terms and simplify the equation:

$$-x^2 - x^2 + 12x + 5x + 6 - 18 = 0$$

$$-2x^2 + 17x - 12 = 0$$

Step 3: Factor the Quadratic Equation

Unfortunately, this quadratic equation does not factor easily. We will use the quadratic formula to find the solutions.

The Quadratic Formula

The quadratic formula is:

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

In this case, a = -2, b = 17, and c = -12.

Step 4: Plug in the Values

Plug in the values into the quadratic formula:

$$x = \frac{-17 \pm \sqrt{17^2 - 4(-2)(-12)}}{2(-2)}$$

$$x = \frac{-17 \pm \sqrt{289 - 96}}{-4}$$

$$x = \frac{-17 \pm \sqrt{193}}{-4}$$

Step 5: Simplify the Solutions

Simplify the solutions:

$$x = \frac{-17 + \sqrt{193}}{-4}$$

$$x = \frac{-17 - \sqrt{193}}{-4}$$

$$x = \frac{17 - \sqrt{193}}{4}$$

$$x = \frac{17 + \sqrt{193}}{4}$$

Step 6: Find the Corresponding y-Values

Now that we have the x-values, we can find the corresponding y-values by plugging them into one of the original equations.

Let's use the first equation:

$$y = -x^2 + 12x + 6$$

Plug in the x-values:

$$y = -\left(\frac{-17 + \sqrt{193}}{-4}\right)^2 + 12\left(\frac{-17 + \sqrt{193}}{-4}\right) + 6$$

$$y = -\left(\frac{17 - \sqrt{193}}{4}\right)^2 + 12\left(\frac{17 - \sqrt{193}}{4}\right) + 6$$

$$y = -\left(\frac{17 + \sqrt{193}}{4}\right)^2 + 12\left(\frac{17 + \sqrt{193}}{4}\right) + 6$$

$$y = -\left(\frac{17 - \sqrt{193}}{4}\right)^2 + 12\left(\frac{17 - \sqrt{193}}{4}\right) + 6$$

Step 7: Simplify the y-Values

Simplify the y-values:

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} - \frac{3(17 - \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} - \frac{3(17 - \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} + \frac{3(17 + \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} + \frac{3(17 - \sqrt{193})}{2} + 6$$

Conclusion

In this article, we solved a system of two quadratic equations using algebraic methods. We set the equations equal to each other, simplified the resulting equation, factored it, and used the quadratic formula to find the solutions. We then found the corresponding y-values by plugging the x-values into one of the original equations. The solutions to the system of equations are:

$$x = \frac{-17 + \sqrt{193}}{-4}$$

$$x = \frac{-17 - \sqrt{193}}{-4}$$

$$x = \frac{17 - \sqrt{193}}{4}$$

$$x = \frac{17 + \sqrt{193}}{4}$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} - \frac{3(17 - \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} + \frac{3(17 + \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} + \frac{3(17 - \sqrt{193})}{2} + 6$$

$$y = \frac{289 - 2(17 - \sqrt{193})(\sqrt{193}) + 193}{16} - \frac{3(17 + \sqrt{193})}{2} + 6$$

These solutions represent the points of intersection between the two curves.

References

• [1] "Quadratic Formula." Math Open Reference, mathopenref.com/quadratic_formula.html.
• [2] "Solving Systems of Equations." Khan Academy, khanacademy.org/math/algebra/solving-systems-of-equations.

Note

Introduction

In our previous article, we solved a system of two quadratic equations using algebraic methods. We set the equations equal to each other, simplified the resulting equation, factored it, and used the quadratic formula to find the solutions. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I know if a system of equations has a solution?

A system of equations has a solution if the two equations are consistent, meaning that they have at least one point in common.

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

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. A system of nonlinear equations is a set of two or more nonlinear equations that are solved simultaneously to find the values of the variables.

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

To solve a system of linear equations, you can use the following methods:

• Substitution Method: Substitute one equation into the other equation to solve for one variable.
• Elimination Method: Add or subtract the equations to eliminate one variable.
• Graphical Method: Graph the two equations on a coordinate plane and find the point of intersection.

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

To solve a system of nonlinear equations, you can use the following methods:

• Algebraic Method: Use algebraic techniques such as substitution, elimination, and factoring to solve the system.
• Numerical Method: Use numerical techniques such as the Newton-Raphson method to approximate the solution.
• Graphical Method: Graph the two equations on a coordinate plane and find the point of intersection.

Q: What is the quadratic formula?

The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

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

To use the quadratic formula to solve a system of equations, you need to:

• Set the equations equal to each other: Set the two equations equal to each other to get a single equation.
• Simplify the equation: Simplify the resulting equation to get a quadratic equation.
• Use the quadratic formula: Use the quadratic formula to solve the quadratic equation.

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

Some common mistakes to avoid when solving systems of equations include:

• Not checking for consistency: Make sure that the two equations are consistent before solving the system.
• Not using the correct method: Choose the correct method to solve the system, such as substitution, elimination, or graphical method.
• Not checking for extraneous solutions: Check for extraneous solutions, such as solutions that do not satisfy both equations.

Conclusion

Solving systems of equations can be a challenging task, but with the right techniques and methods, it can be done. In this article, we answered some frequently asked questions about solving systems of equations, including what a system of equations is, how to know if a system has a solution, and how to use the quadratic formula to solve a system. We also discussed some common mistakes to avoid when solving systems of equations. By following these tips and techniques, you can become proficient in solving systems of equations and tackle even the most challenging problems.

References

• [1] "Solving Systems of Equations." Khan Academy, khanacademy.org/math/algebra/solving-systems-of-equations.
• [2] "Quadratic Formula." Math Open Reference, mathopenref.com/quadratic_formula.html.
• [3] "Systems of Equations." Math Is Fun, mathisfun.com/algebra/systems-equations.html.

Note

This article is for educational purposes only. The solutions to the system of equations are complex and may not be easily interpretable.