Consider The Functions $f$ And $g$ Below:$\[ \begin{array}{l} f(x) = -9x^2 - 7x + 12 \\ g(x) = 3x^2 - 4x - 15 \end{array} \\]Find $f(x) - G(x$\].A. \[$-12x^2 - 11x - 3\$\] B. \[$-12x^2 - 3x + 27\$\]

Introduction

In this article, we will explore the concept of finding the difference of two quadratic functions. Quadratic functions are a fundamental concept in algebra and are used to model various real-world phenomena. The difference of two quadratic functions is a crucial operation that can be used to simplify complex expressions and solve equations.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is:

$$f(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

In this article, we will consider two quadratic functions:

$$f(x) = -9x^2 - 7x + 12$$

$$g(x) = 3x^2 - 4x - 15$$

Finding the Difference of Two Quadratic Functions

To find the difference of two quadratic functions, we need to subtract the second function from the first function. This can be done by subtracting the corresponding terms of the two functions.

Let's start by subtracting the two functions:

$$f(x) - g(x) = (-9x^2 - 7x + 12) - (3x^2 - 4x - 15)$$

Using the distributive property, we can rewrite the expression as:

$$f(x) - g(x) = -9x^2 - 7x + 12 - 3x^2 + 4x + 15$$

Now, let's combine like terms:

$$f(x) - g(x) = (-9x^2 - 3x^2) + (-7x + 4x) + (12 + 15)$$

Simplifying further, we get:

$$f(x) - g(x) = -12x^2 - 3x + 27$$

Conclusion

In this article, we have shown how to find the difference of two quadratic functions. By subtracting the corresponding terms of the two functions and combining like terms, we arrived at the final expression:

$$f(x) - g(x) = -12x^2 - 3x + 27$$

This result is consistent with option B.

Discussion

The difference of two quadratic functions is an important operation that can be used to simplify complex expressions and solve equations. By understanding how to find the difference of two quadratic functions, we can apply this knowledge to various real-world problems.

Example Problems

1. Find the difference of the following two quadratic functions:

$$f(x) = 2x^2 + 5x - 3$$

$$g(x) = x^2 - 2x + 1$$

2. Find the difference of the following two quadratic functions:

$$f(x) = -4x^2 + 3x - 2$$

$$g(x) = 2x^2 - 5x + 1$$

Practice Problems

1. Find the difference of the following two quadratic functions:

$$f(x) = x^2 + 2x - 1$$

$$g(x) = 2x^2 - 3x + 2$$

2. Find the difference of the following two quadratic functions:

$$f(x) = -3x^2 + 2x - 1$$

$$g(x) = x^2 - 4x + 2$$

Answer Key

1. $f(x) - g(x) = x^2 + 5x - 4$
2. $f(x) - g(x) = -4x^2 + 6x - 3$

Final Thoughts

Introduction

In our previous article, we explored the concept of finding the difference of two quadratic functions. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples.

Q&A

Q: What is the difference of two quadratic functions?

A: The difference of two quadratic functions is the result of subtracting one quadratic function from another. This can be done by subtracting the corresponding terms of the two functions.

Q: How do I find the difference of two quadratic functions?

A: To find the difference of two quadratic functions, you need to subtract the second function from the first function. This can be done by subtracting the corresponding terms of the two functions.

Q: What is the formula for finding the difference of two quadratic functions?

A: The formula for finding the difference of two quadratic functions is:

$$f(x) - g(x) = (ax^2 + bx + c) - (dx^2 + ex + f)$$

where $a$, $b$, $c$, $d$, $e$, and $f$ are constants, and $x$ is the variable.

Q: Can I use the distributive property to simplify the expression?

A: Yes, you can use the distributive property to simplify the expression. This involves multiplying each term of the first function by the negative of the corresponding term of the second function.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the final expression for the difference of two quadratic functions?

A: The final expression for the difference of two quadratic functions is:

$$f(x) - g(x) = (a - d)x^2 + (b - e)x + (c - f)$$

Q: Can I use this formula to find the difference of two quadratic functions with different variables?

A: No, this formula is only applicable to quadratic functions with the same variable.

Q: How do I apply this formula to real-world problems?

A: To apply this formula to real-world problems, you need to identify the quadratic functions involved and substitute the corresponding values into the formula.

Example Problems

1. Find the difference of the following two quadratic functions:

$$f(x) = 2x^2 + 5x - 3$$

$$g(x) = x^2 - 2x + 1$$

2. Find the difference of the following two quadratic functions:

$$f(x) = -4x^2 + 3x - 2$$

$$g(x) = 2x^2 - 5x + 1$$

Practice Problems

1. Find the difference of the following two quadratic functions:

$$f(x) = x^2 + 2x - 1$$

$$g(x) = 2x^2 - 3x + 2$$

2. Find the difference of the following two quadratic functions:

$$f(x) = -3x^2 + 2x - 1$$

$$g(x) = x^2 - 4x + 2$$

Answer Key

1. $f(x) - g(x) = x^2 + 7x - 4$
2. $f(x) - g(x) = -6x^2 + 8x - 3$

Final Thoughts

In conclusion, finding the difference of two quadratic functions is a crucial operation that can be used to simplify complex expressions and solve equations. By understanding how to find the difference of two quadratic functions, we can apply this knowledge to various real-world problems.