Which Expression Is Equivalent To $64 - 9x^2$?A. $(8)^2 - (3x)^2$ B. \$(32)^2 + (-3x)^2$[/tex\] C. $(32)^2 - (3x)^2$ D. $(8)^2 + (-3x)^2$

Introduction

In mathematics, algebraic expressions are used to represent mathematical statements. These expressions can be manipulated using various mathematical operations such as addition, subtraction, multiplication, and division. In this article, we will explore which expression is equivalent to $64 - 9x^2$.

Understanding the Given Expression

The given expression is $64 - 9x^2$. This expression consists of two terms: a constant term $64$ and a variable term $-9x^2$. The variable term is a quadratic expression, which is a polynomial of degree two.

Breaking Down the Expression

To determine which expression is equivalent to $64 - 9x^2$, we need to break down the expression into its constituent parts. We can rewrite the expression as:

$$64 - 9x^2 = (8)^2 - (3x)^2$$

This expression can be rewritten using the difference of squares formula, which states that:

$$a^2 - b^2 = (a + b)(a - b)$$

Using this formula, we can rewrite the expression as:

$$(8)^2 - (3x)^2 = (8 + 3x)(8 - 3x)$$

Analyzing the Options

Now that we have broken down the expression, let's analyze the options:

A. $(8)^2 - (3x)^2$

B. $(32)^2 + (-3x)^2$

C. $(32)^2 - (3x)^2$

D. $(8)^2 + (-3x)^2$

Option A: $(8)^2 - (3x)^2$

Option A is the same as the expression we broke down earlier. This option is equivalent to $64 - 9x^2$.

Option B: $(32)^2 + (-3x)^2$

Option B is not equivalent to $64 - 9x^2$. The expression $(32)^2 + (-3x)^2$ is a sum of squares, not a difference of squares.

Option C: $(32)^2 - (3x)^2$

Option C is not equivalent to $64 - 9x^2$. The expression $(32)^2 - (3x)^2$ is a difference of squares, but it is not the same as the expression we broke down earlier.

Option D: $(8)^2 + (-3x)^2$

Option D is not equivalent to $64 - 9x^2$. The expression $(8)^2 + (-3x)^2$ is a sum of squares, not a difference of squares.

Conclusion

In conclusion, the expression that is equivalent to $64 - 9x^2$ is:

$$(8)^2 - (3x)^2$$

This expression can be rewritten using the difference of squares formula, which states that:

$$a^2 - b^2 = (a + b)(a - b)$$

Using this formula, we can rewrite the expression as:

$$(8)^2 - (3x)^2 = (8 + 3x)(8 - 3x)$$

Therefore, the correct answer is option A: $(8)^2 - (3x)^2$.

Final Answer

Introduction

In our previous article, we explored which expression is equivalent to $64 - 9x^2$. We broke down the expression and analyzed the options to determine the correct answer. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be used to rewrite expressions in the form of a difference of squares.

Q: How can I apply the difference of squares formula to the expression $64 - 9x^2$?

A: To apply the difference of squares formula to the expression $64 - 9x^2$, we can rewrite it as:

$$(8)^2 - (3x)^2 = (8 + 3x)(8 - 3x)$$

This is done by recognizing that $64 = (8)^2$ and $9x^2 = (3x)^2$.

Q: What is the significance of the difference of squares formula in algebra?

A: The difference of squares formula is a fundamental concept in algebra that allows us to rewrite expressions in a more simplified form. It is used to factorize expressions and solve equations.

Q: Can the difference of squares formula be used to rewrite expressions with negative coefficients?

A: Yes, the difference of squares formula can be used to rewrite expressions with negative coefficients. For example:

$$-a^2 + b^2 = -(a^2 - b^2) = -(a + b)(a - b)$$

Q: How can I determine if an expression can be rewritten using the difference of squares formula?

A: To determine if an expression can be rewritten using the difference of squares formula, we need to check if it is in the form of $a^2 - b^2$. If it is, then we can apply the formula to rewrite it.

Q: What are some common applications of the difference of squares formula?

A: The difference of squares formula has many applications in algebra, including:

• Factoring expressions
• Solving equations
• Simplifying expressions
• Rewriting expressions in a more simplified form

Conclusion

In conclusion, the difference of squares formula is a powerful tool in algebra that allows us to rewrite expressions in a more simplified form. By applying this formula, we can factorize expressions, solve equations, and simplify expressions. We hope that this Q&A section has provided additional information and clarified any doubts.

Final Answer

The final answer is option A: $(8)^2 - (3x)^2$.