Factor The Expression Below.${x^2-14x+49}$A. { (x+7)(x+7)$}$ B. { (x-1)(x-49)$}$ C. { (x-7)(x-7)$}$ D. { (x+1)(x+49)$}$

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 $x^2-14x+49$ and explore the different methods and techniques used to factor quadratic expressions.

Understanding the Expression

Before we dive into factoring the expression, let's take a closer look at its structure. The given expression is a quadratic expression in the form of $ax^2+bx+c$, where $a=1$, $b=-14$, and $c=49$. Our goal is to factor this expression into a product of two binomials.

The Difference of Squares Formula

One of the most common methods used to factor quadratic expressions is the difference of squares formula. The difference of squares formula states that:

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

This formula can be applied to factor expressions of the form $x^2-14x+49$, where the first term is a perfect square and the last term is the square of a binomial.

Factoring the Expression

Let's apply the difference of squares formula to factor the expression $x^2-14x+49$. We can rewrite the expression as:

$$x^2-14x+49=(x-7)^2$$

Using the difference of squares formula, we can factor the expression as:

$$(x-7)(x-7)$$

This is the factored form of the expression $x^2-14x+49$.

Alternative Methods

While the difference of squares formula is a powerful tool for factoring quadratic expressions, there are other methods that can be used to factor expressions of the form $x^2-14x+49$. Some of these methods include:

• Factoring by grouping: This method involves grouping the terms of the expression into pairs and factoring out common factors.
• Factoring by substitution: This method involves substituting a variable or expression into the original expression and factoring the resulting expression.

Conclusion

In this article, we have explored the concept of factoring quadratic expressions and applied the difference of squares formula to factor the expression $x^2-14x+49$. We have also discussed alternative methods for factoring expressions of the form $x^2-14x+49$. By understanding the different methods and techniques used to factor quadratic expressions, we can develop a deeper appreciation for the beauty and power of algebra.

Answer

The correct answer is:

• C. {(x-7)(x-7)$}$

This is the factored form of the expression $x^2-14x+49$.

Discussion

• What is the difference of squares formula?
  • The difference of squares formula is a mathematical formula that states that $a^2-b^2=(a+b)(a-b)$.
• How do you factor a quadratic expression?
  • To factor a quadratic expression, you can use the difference of squares formula, factoring by grouping, or factoring by substitution.
• What is the factored form of the expression $x^2-14x+49$?
  • The factored form of the expression $x^2-14x+49$ is $(x-7)(x-7)$.

Additional Resources

• Algebra textbooks: For a comprehensive understanding of algebra, including factoring quadratic expressions, consult an algebra textbook.
• Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and exercises on factoring quadratic expressions.
• Practice problems: Practice factoring quadratic expressions with online resources or worksheets to develop your skills and build your confidence.
  Factoring Quadratic Expressions: A Q&A Guide =====================================================

Introduction

Factoring quadratic 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 comprehensive Q&A guide on factoring quadratic expressions, covering common questions and topics.

Q&A

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states that $a^2-b^2=(a+b)(a-b)$. This formula can be applied to factor expressions of the form $x^2-14x+49$, where the first term is a perfect square and the last term is the square of a binomial.

Q: How do you factor a quadratic expression?

A: To factor a quadratic expression, you can use the difference of squares formula, factoring by grouping, or factoring by substitution. The choice of method depends on the structure of the expression and the desired outcome.

Q: What is the factored form of the expression $x^2-14x+49$?

A: The factored form of the expression $x^2-14x+49$ is $(x-7)(x-7)$.

Q: Can you provide examples of factoring quadratic expressions?

A: Yes, here are some examples of factoring quadratic expressions:

• Example 1: Factor the expression $x^2+6x+8$.
  • Solution: $(x+2)(x+4)$
• Example 2: Factor the expression $x^2-9x+20$.
  • Solution: $(x-4)(x-5)$
• Example 3: Factor the expression $x^2+5x+6$.
  • Solution: $(x+2)(x+3)$

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

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

• Not checking for perfect squares: Make sure to check if the first term is a perfect square before applying the difference of squares formula.
• Not factoring out common factors: Make sure to factor out common factors from the expression before applying the difference of squares formula.
• Not using the correct method: Choose the correct method for factoring the expression based on its structure and the desired outcome.

Q: Can you provide tips for factoring quadratic expressions?

A: Yes, here are some tips for factoring quadratic expressions:

• Use the difference of squares formula: The difference of squares formula is a powerful tool for factoring quadratic expressions.
• Check for perfect squares: Make sure to check if the first term is a perfect square before applying the difference of squares formula.
• Factor out common factors: Make sure to factor out common factors from the expression before applying the difference of squares formula.
• Practice, practice, practice: Practice factoring quadratic expressions with online resources or worksheets to develop your skills and build your confidence.

Conclusion

In this article, we have provided a comprehensive Q&A guide on factoring quadratic expressions, covering common questions and topics. By understanding the different methods and techniques used to factor quadratic expressions, we can develop a deeper appreciation for the beauty and power of algebra.

Additional Resources

• Algebra textbooks: For a comprehensive understanding of algebra, including factoring quadratic expressions, consult an algebra textbook.
• Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and exercises on factoring quadratic expressions.
• Practice problems: Practice factoring quadratic expressions with online resources or worksheets to develop your skills and build your confidence.

Discussion

• What is the difference of squares formula?
  • The difference of squares formula is a mathematical formula that states that $a^2-b^2=(a+b)(a-b)$.
• How do you factor a quadratic expression?
  • To factor a quadratic expression, you can use the difference of squares formula, factoring by grouping, or factoring by substitution.
• What is the factored form of the expression $x^2-14x+49$?
  • The factored form of the expression $x^2-14x+49$ is $(x-7)(x-7)$.

Common Mistakes

• Not checking for perfect squares: Make sure to check if the first term is a perfect square before applying the difference of squares formula.
• Not factoring out common factors: Make sure to factor out common factors from the expression before applying the difference of squares formula.
• Not using the correct method: Choose the correct method for factoring the expression based on its structure and the desired outcome.

Tips and Tricks

• Use the difference of squares formula: The difference of squares formula is a powerful tool for factoring quadratic expressions.
• Check for perfect squares: Make sure to check if the first term is a perfect square before applying the difference of squares formula.
• Factor out common factors: Make sure to factor out common factors from the expression before applying the difference of squares formula.
• Practice, practice, practice: Practice factoring quadratic expressions with online resources or worksheets to develop your skills and build your confidence.