Which Equation Is The Inverse Of $2(x-2)^2=8(7+y)$?A. − 2 ( X − 2 ) 2 = − 8 ( 7 + Y -2(x-2)^2=-8(7+y − 2 ( X − 2 ) 2 = − 8 ( 7 + Y ]B. Y = 1 4 X 2 − X − 6 Y=\frac{1}{4} X^2-x-6 Y = 4 1 ​ X 2 − X − 6 C. Y = − 2 ± 28 + 4 X Y=-2 \pm \sqrt{28+4 X} Y = − 2 ± 28 + 4 X ​ D. Y = 2 ± 28 + 4 X Y=2 \pm \sqrt{28+4 X} Y = 2 ± 28 + 4 X ​

Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. In the context of quadratic equations, finding the inverse involves solving for the variable that is not isolated on one side of the equation. In this article, we will explore how to find the inverse of a quadratic equation and apply this concept to the given equation $2(x-2)^2=8(7+y)$.

Understanding the Concept of Inverse Functions

Before we dive into solving the given equation, let's briefly review the concept of inverse functions. An inverse function is a function that undoes the action of another function. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This means that applying the inverse function to the output of the original function will return the original input.

Step 1: Simplify the Given Equation

To find the inverse of the given equation, we first need to simplify it. Let's start by expanding the squared term on the left-hand side:

$$2(x-2)^2=8(7+y)$$

Expanding the squared term, we get:

$$2(x^2-4x+4)=8(7+y)$$

Simplifying further, we get:

$$2x^2-8x+8=56+8y$$

Step 2: Isolate the Variable y

Next, we need to isolate the variable y on one side of the equation. To do this, we can start by subtracting 56 from both sides of the equation:

$$2x^2-8x-48=8y$$

Now, we can divide both sides of the equation by 8 to isolate y:

$$y=\frac{1}{8}(2x^2-8x-48)$$

Step 3: Simplify the Expression

To simplify the expression, we can start by factoring out the common factor of 2 from the first two terms:

$$y=\frac{1}{8}(2x^2-8x-48)$$

$$y=\frac{1}{8}(2(x^2-4x-24))$$

Simplifying further, we get:

$$y=\frac{1}{4}(x^2-4x-24)$$

Step 4: Expand and Simplify

To expand and simplify the expression, we can start by multiplying the fraction by the expression inside the parentheses:

$$y=\frac{1}{4}(x^2-4x-24)$$

$$y=\frac{1}{4}x^2-\frac{1}{4}(4x)-\frac{1}{4}(24)$$

Simplifying further, we get:

$$y=\frac{1}{4}x^2-x-6$$

Conclusion

In conclusion, the inverse of the given equation $2(x-2)^2=8(7+y)$ is:

$$y=\frac{1}{4}x^2-x-6$$

This is option B in the given multiple-choice question.

Comparison with Other Options

Let's compare our solution with the other options:

A. $-2(x-2)^2=-8(7+y)$

This option is not the inverse of the given equation.

B. $y=\frac{1}{4}x^2-x-6$

This is the correct solution.

C. $y=-2 \pm \sqrt{28+4 x}$

This option is not the inverse of the given equation.

D. $y=2 \pm \sqrt{28+4 x}$

This option is not the inverse of the given equation.

Final Answer

$$

Introduction

In our previous article, we explored how to find the inverse of a quadratic equation. In this article, we will provide a Q&A guide to help you better understand the concept of inverse functions and how to apply it to quadratic equations.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Q: Why is it important to find the inverse of a quadratic equation?

A: Finding the inverse of a quadratic equation is important because it allows us to solve for the variable that is not isolated on one side of the equation. This can be useful in a variety of applications, such as solving systems of equations or modeling real-world phenomena.

Q: How do I find the inverse of a quadratic equation?

A: To find the inverse of a quadratic equation, you can follow these steps:

1. Simplify the equation by expanding and combining like terms.
2. Isolate the variable y on one side of the equation.
3. Simplify the expression by factoring out common factors or combining like terms.
4. Expand and simplify the expression to get the final answer.

Q: What are some common mistakes to avoid when finding the inverse of a quadratic equation?

A: Some common mistakes to avoid when finding the inverse of a quadratic equation include:

• Not simplifying the equation enough before isolating the variable y.
• Not factoring out common factors or combining like terms.
• Not expanding and simplifying the expression enough to get the final answer.

Q: Can I use a calculator to find the inverse of a quadratic equation?

A: Yes, you can use a calculator to find the inverse of a quadratic equation. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: How do I know if I have found the correct inverse of a quadratic equation?

A: To check if you have found the correct inverse of a quadratic equation, you can plug the original equation into the inverse equation and see if it simplifies to the original equation. If it does, then you have found the correct inverse.

Q: Can I use the inverse of a quadratic equation to solve systems of equations?

A: Yes, you can use the inverse of a quadratic equation to solve systems of equations. By finding the inverse of one of the equations, you can isolate the variable and substitute it into the other equation to solve for the remaining variable.

Q: Are there any real-world applications of finding the inverse of a quadratic equation?

A: Yes, there are many real-world applications of finding the inverse of a quadratic equation. For example, you can use it to model the motion of an object under the influence of gravity, or to solve problems involving optimization and maximization.

Conclusion

In conclusion, finding the inverse of a quadratic equation is an important concept in mathematics that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can find the correct inverse of a quadratic equation and use it to solve a variety of problems.

Common Quadratic Equations and Their Inverses

Here are some common quadratic equations and their inverses:

• $x^2 + 4x + 4 = 0$ -> $x = -2 \pm \sqrt{3}$
• $x^2 - 6x + 9 = 0$ -> $x = 3 \pm \sqrt{2}$
• $x^2 + 2x + 1 = 0$ -> $x = -1 \pm \sqrt{2}$

Practice Problems

Here are some practice problems to help you practice finding the inverse of a quadratic equation:

• Find the inverse of the equation $x^2 + 2x + 1 = 0$.
• Find the inverse of the equation $x^2 - 6x + 9 = 0$.
• Find the inverse of the equation $x^2 + 4x + 4 = 0$.

Answer Key

Here are the answers to the practice problems:

• $x = -1 \pm \sqrt{2}$
• $x = 3 \pm \sqrt{2}$
• $x = -2 \pm \sqrt{3}$