Factor The Expression Below.$25x^2 - 49$A. \[$(5x + 7)(5x - 7)\$\]B. \[$(5x - 7)(5x - 7)\$\]C. \[$(25x - 7)(x - 7)\$\]D. \[$(25x + 7)(x - 7)\$\]

Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In this article, we will focus on factoring the expression $25x^2 - 49$. Factoring this expression will help us understand the underlying structure of the expression and make it easier to solve equations and inequalities that involve this expression.

Understanding the Expression

Before we dive into factoring the expression, let's take a closer look at what we're dealing with. The expression $25x^2 - 49$ is a quadratic expression, which means it is a polynomial of degree two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 25$, $b = 0$, and $c = -49$.

Factoring the Expression

To factor the expression $25x^2 - 49$, we need to find two binomials whose product is equal to the given expression. A binomial is an expression that consists of two terms, such as $5x + 7$ or $x - 7$. We can start by looking for two binomials whose product is equal to $25x^2 - 49$.

One way to approach this is to look for two binomials whose product is equal to the first term, $25x^2$. We can start by listing the factors of $25x^2$:

• $1 \times 25x^2$
• $5x \times 5x$
• $25x \times x$

We can also look for two binomials whose product is equal to the second term, $-49$. We can start by listing the factors of $-49$:

• $1 \times -49$
• $7 \times -7$

Now, let's try to combine these factors to form two binomials whose product is equal to $25x^2 - 49$. We can start by combining the factors of $25x^2$ and $-49$:

• $(5x + 7)(5x - 7)$
• $(25x + 7)(x - 7)$
• $(25x - 7)(x + 7)$

We can see that the first option, $(5x + 7)(5x - 7)$, is a perfect square trinomial, which means it can be factored as a single binomial squared. This is because the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.

Simplifying the Expression

Now that we have factored the expression $25x^2 - 49$ as $(5x + 7)(5x - 7)$, we can simplify it further by combining like terms. We can start by multiplying the two binomials:

$(5x + 7)(5x - 7) = 25x^2 - 35x + 35x - 49$

We can see that the middle terms cancel each other out, leaving us with:

$25x^2 - 49$

This is the same expression we started with, which means that our factoring was correct.

Conclusion

In this article, we have factored the expression $25x^2 - 49$ as $(5x + 7)(5x - 7)$. We have also shown that this factoring is correct by simplifying the expression further and verifying that it is equal to the original expression. Factoring expressions is an important concept in algebra that can help us solve equations and inequalities more easily. By understanding how to factor expressions, we can gain a deeper understanding of the underlying structure of the expression and make it easier to work with.

Answer

The correct answer is:

A. {(5x + 7)(5x - 7)$}$

This is the only option that correctly factors the expression $25x^2 - 49$.

Additional Examples

Here are a few additional examples of factoring expressions:

• $x^2 - 16 = (x + 4)(x - 4)$
• $x^2 - 9 = (x + 3)(x - 3)$
• $x^2 - 25 = (x + 5)(x - 5)$

These examples illustrate the concept of factoring expressions and how it can be used to simplify complex expressions.

Tips and Tricks

Here are a few tips and tricks for factoring expressions:

• Look for two binomials whose product is equal to the given expression.
• Use the distributive property to multiply the two binomials.
• Simplify the expression by combining like terms.
• Verify that the factoring is correct by simplifying the expression further.

Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In this article, we will provide a Q&A guide to help you understand the concept of factoring expressions and how to apply it to solve equations and inequalities.

Q: What is factoring an expression?

A: Factoring an expression involves expressing it as a product of simpler expressions. For example, factoring the expression $x^2 + 5x + 6$ involves expressing it as $(x + 3)(x + 2)$.

Q: Why is factoring important?

A: Factoring is important because it can help us solve equations and inequalities more easily. By factoring an expression, we can identify its underlying structure and make it easier to work with.

Q: How do I factor an expression?

A: To factor an expression, you need to look for two binomials whose product is equal to the given expression. You can use the distributive property to multiply the two binomials and simplify the expression.

Q: What are some common factoring techniques?

A: Some common factoring techniques include:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring quadratic expressions
• Factoring polynomial expressions

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to look for two binomials whose product is equal to the given expression. You can use the formula $ax^2 + bx + c = (x + r)(x + s)$, where $r$ and $s$ are the roots of the quadratic equation.

Q: How do I factor a polynomial expression?

A: To factor a polynomial expression, you need to look for two binomials whose product is equal to the given expression. You can use the distributive property to multiply the two binomials and simplify the expression.

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

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

• Not looking for the greatest common factor (GCF)
• Not using the distributive property to multiply the binomials
• Not simplifying the expression
• Not verifying that the factoring is correct

Q: How do I verify that the factoring is correct?

A: To verify that the factoring is correct, you need to simplify the expression and check that it is equal to the original expression.

Q: What are some real-world applications of factoring expressions?

A: Some real-world applications of factoring expressions include:

• Solving equations and inequalities in physics and engineering
• Modeling population growth and decline in biology
• Analyzing data in statistics and data analysis

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of factoring expressions and how to apply it to solve equations and inequalities. By following these tips and techniques, you can become more proficient in factoring expressions and solve problems more easily.

Additional Resources

Here are some additional resources to help you learn more about factoring expressions:

• Khan Academy: Factoring Expressions
• Mathway: Factoring Expressions
• Wolfram Alpha: Factoring Expressions

By using these resources, you can learn more about factoring expressions and practice your skills.

Practice Problems

Here are some practice problems to help you practice factoring expressions:

• Factor the expression $x^2 + 5x + 6$
• Factor the expression $x^2 - 7x + 12$
• Factor the expression $x^2 + 2x - 15$

By solving these problems, you can practice your skills and become more proficient in factoring expressions.