Find The Product. Show All Steps.\[$(4x - 7)(4x + 7)\$\]

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Introduction

In algebra, expanding expressions is a crucial skill that helps us simplify complex equations and solve problems. One common type of expression that requires expansion is the product of two binomials. In this article, we will focus on finding the product of the expression $(4x - 7)(4x + 7)$ using the distributive property and other algebraic techniques.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one binomial by each term in the other binomial. This property states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

We will use this property to expand the given expression $(4x - 7)(4x + 7)$.

Step 1: Apply the Distributive Property

To expand the expression $(4x - 7)(4x + 7)$, we will apply the distributive property by multiplying each term in the first binomial $(4x - 7)$ by each term in the second binomial $(4x + 7)$. This will give us:

$(4x - 7)(4x + 7) = (4x)(4x) + (4x)(7) - (7)(4x) - (7)(7)$

Step 2: Simplify the Expression

Now that we have applied the distributive property, we can simplify the expression by combining like terms. We will start by multiplying the terms:

$(4x)(4x) = 16x^2$

$(4x)(7) = 28x$

$-(7)(4x) = -28x$

$-(7)(7) = -49$

Step 3: Combine Like Terms

Now that we have multiplied the terms, we can combine like terms to simplify the expression further. We will combine the terms with the same variable and coefficient:

$16x^2 + 28x - 28x - 49$

Step 4: Simplify the Expression Further

Now that we have combined like terms, we can simplify the expression further by canceling out any terms that have the same variable and coefficient but opposite signs. In this case, we can cancel out the $28x$ terms:

$16x^2 - 49$

Conclusion

In this article, we have shown the step-by-step process of finding the product of the expression $(4x - 7)(4x + 7)$ using the distributive property and other algebraic techniques. We have applied the distributive property, simplified the expression, combined like terms, and simplified the expression further to arrive at the final answer. This process demonstrates the importance of understanding and applying algebraic techniques to solve complex problems.

Final Answer

The final answer to the expression $(4x - 7)(4x + 7)$ is:

$16x^2 - 49$

Additional Tips and Tricks

• When expanding expressions, it is essential to apply the distributive property carefully to avoid making mistakes.
• Combining like terms is a crucial step in simplifying expressions and arriving at the final answer.
• Algebraic techniques, such as the distributive property, can be used to solve a wide range of problems in mathematics and other fields.

Real-World Applications

The concept of expanding expressions is used in various real-world applications, such as:

• Calculating the area of a rectangle or a triangle
• Finding the volume of a cube or a rectangular prism
• Solving problems in physics, engineering, and other fields that involve algebraic equations

Common Mistakes to Avoid

• Failing to apply the distributive property correctly
• Not combining like terms properly
• Making mistakes when simplifying expressions

Conclusion

In conclusion, finding the product of the expression $(4x - 7)(4x + 7)$ requires a step-by-step approach that involves applying the distributive property, simplifying the expression, combining like terms, and simplifying the expression further. By following these steps and avoiding common mistakes, we can arrive at the final answer and solve complex problems in mathematics and other fields.

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Introduction

In our previous article, we showed the step-by-step process of finding the product of the expression $(4x - 7)(4x + 7)$ using the distributive property and other algebraic techniques. In this article, we will provide a Q&A guide to help you better understand the concept and address any questions or concerns you may have.

Q&A

Q: What is the distributive property, and how is it used to expand expressions?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one binomial by each term in the other binomial. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

We use the distributive property to expand expressions by multiplying each term in one binomial by each term in the other binomial.

Q: How do I apply the distributive property to expand the expression $(4x - 7)(4x + 7)$?

A: To apply the distributive property, we multiply each term in the first binomial $(4x - 7)$ by each term in the second binomial $(4x + 7)$. This will give us:

$(4x - 7)(4x + 7) = (4x)(4x) + (4x)(7) - (7)(4x) - (7)(7)$

Q: What are like terms, and how do I combine them?

A: Like terms are terms that have the same variable and coefficient but opposite signs. For example, $28x$ and $-28x$ are like terms. To combine like terms, we add or subtract the coefficients of the like terms.

Q: How do I simplify the expression $(4x - 7)(4x + 7)$?

A: To simplify the expression, we apply the distributive property, combine like terms, and simplify the expression further. We start by multiplying the terms:

$(4x)(4x) = 16x^2$

$(4x)(7) = 28x$

$-(7)(4x) = -28x$

$-(7)(7) = -49$

Then, we combine like terms:

$16x^2 + 28x - 28x - 49$

Finally, we simplify the expression further by canceling out any terms that have the same variable and coefficient but opposite signs:

$16x^2 - 49$

Q: What are some common mistakes to avoid when expanding expressions?

A: Some common mistakes to avoid when expanding expressions include:

• Failing to apply the distributive property correctly
• Not combining like terms properly
• Making mistakes when simplifying expressions

Q: How do I use the distributive property to expand expressions in real-world applications?

A: The distributive property is used in various real-world applications, such as:

• Calculating the area of a rectangle or a triangle
• Finding the volume of a cube or a rectangular prism
• Solving problems in physics, engineering, and other fields that involve algebraic equations

Conclusion

In this Q&A guide, we have addressed common questions and concerns about expanding expressions using the distributive property. By following the steps outlined in this guide, you can better understand the concept and apply it to solve complex problems in mathematics and other fields.

Additional Resources

• For more information on the distributive property and expanding expressions, see our previous article: "Find the Product: A Step-by-Step Guide to Expanding the Expression $(4x - 7)(4x + 7)$"
• For practice problems and exercises, see our algebra workbook: "Algebra: A Comprehensive Guide"

Final Tips and Tricks

• Always apply the distributive property carefully to avoid making mistakes.
• Combine like terms properly to simplify expressions.
• Use the distributive property to expand expressions in real-world applications.

By following these tips and tricks, you can become proficient in expanding expressions using the distributive property and apply it to solve complex problems in mathematics and other fields.