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

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Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into simpler components. In this article, we will focus on factoring the expression x2βˆ’49x^2 - 49. This expression can be factored using various techniques, and we will explore each option to determine the correct answer.

Understanding the Expression

The given expression is x2βˆ’49x^2 - 49. This is a quadratic expression, which means it can be factored into the product of two binomials. The expression can be rewritten as (x)2βˆ’(7)2(x)^2 - (7)^2, where we have identified the two terms as perfect squares.

Factoring the Expression

To factor the expression, we can use the difference of squares formula, which states that a2βˆ’b2=(a+b)(aβˆ’b)a^2 - b^2 = (a + b)(a - b). In this case, we have a=xa = x and b=7b = 7. Applying the formula, we get:

x2βˆ’49=(x+7)(xβˆ’7)x^2 - 49 = (x + 7)(x - 7)

This is the factored form of the expression.

Evaluating the Options

Now that we have factored the expression, let's evaluate the options provided:

  • A. (xβˆ’49)(xβˆ’1)(x-49)(x-1): This option is incorrect because it does not match the factored form we obtained.
  • B. (x+49)(xβˆ’1)(x+49)(x-1): This option is also incorrect because it does not match the factored form we obtained.
  • C. (x+7)(xβˆ’7)(x+7)(x-7): This option matches the factored form we obtained, making it the correct answer.
  • D. (xβˆ’7)(xβˆ’7)(x-7)(x-7): This option is incorrect because it does not match the factored form we obtained.

Conclusion

In conclusion, the correct answer is option C, (x+7)(xβˆ’7)(x+7)(x-7). This is the factored form of the expression x2βˆ’49x^2 - 49, which can be obtained using the difference of squares formula.

Additional Tips and Tricks

When factoring expressions, it's essential to recognize the difference of squares pattern. This pattern is a common occurrence in algebra, and being able to identify it can help you factor expressions quickly and efficiently. Additionally, make sure to apply the formula correctly, and don't be afraid to check your work by multiplying the factors together to ensure that you get the original expression.

Common Mistakes to Avoid

When factoring expressions, there are several common mistakes to avoid:

  • Not recognizing the difference of squares pattern
  • Applying the formula incorrectly
  • Not checking your work
  • Not considering alternative factoring methods

By avoiding these common mistakes, you can ensure that you factor expressions correctly and efficiently.

Real-World Applications

Factoring expressions has numerous real-world applications in various fields, including:

  • Physics: Factoring expressions is essential in physics, particularly when working with equations of motion and energy.
  • Engineering: Factoring expressions is used in engineering to solve problems involving circuits, mechanics, and thermodynamics.
  • Computer Science: Factoring expressions is used in computer science to solve problems involving algorithms and data structures.

Conclusion

Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into simpler components. 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 problems.

Q: What is factoring an expression?

A: Factoring an expression involves breaking it down into simpler components, such as the product of two or more binomials. This is done by identifying the common factors or patterns within the expression.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Difference of squares: This involves factoring an expression of the form a2βˆ’b2a^2 - b^2 into (a+b)(aβˆ’b)(a + b)(a - b).
  • Sum of squares: This involves factoring an expression of the form a2+b2a^2 + b^2 into (a+bi)(aβˆ’bi)(a + bi)(a - bi).
  • Grouping: This involves factoring an expression by grouping terms that have common factors.
  • Factoring out a greatest common factor (GCF): This involves factoring out the largest factor that divides all terms in the expression.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the following steps:

  1. Identify the two terms in the expression.
  2. Look for a common factor or pattern within the two terms.
  3. If the two terms have a common factor, factor it out.
  4. If the two terms have a difference of squares pattern, use the difference of squares formula to factor the expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

a2βˆ’b2=(a+b)(aβˆ’b)a^2 - b^2 = (a + b)(a - b)

This formula can be used to factor expressions of the form a2βˆ’b2a^2 - b^2.

Q: How do I factor an expression with a GCF?

A: To factor an expression with a GCF, you can use the following steps:

  1. Identify the GCF of the terms in the expression.
  2. Factor the GCF out of each term.
  3. Write the factored form of the expression.

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

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

  • Not recognizing the difference of squares pattern
  • Applying the formula incorrectly
  • Not checking your work
  • Not considering alternative factoring methods

Q: How do I check my work when factoring expressions?

A: To check your work when factoring expressions, you can use the following steps:

  1. Multiply the factors together to get the original expression.
  2. Simplify the expression to ensure that it matches the original expression.
  3. Check that the factored form is correct.

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

A: Factoring expressions has numerous real-world applications in various fields, including:

  • Physics: Factoring expressions is essential in physics, particularly when working with equations of motion and energy.
  • Engineering: Factoring expressions is used in engineering to solve problems involving circuits, mechanics, and thermodynamics.
  • Computer Science: Factoring expressions is used in computer science to solve problems involving algorithms and data structures.

Conclusion

In conclusion, factoring expressions is a fundamental concept in algebra that involves breaking down an expression into simpler components. By understanding the different types of factoring and how to apply them, you can solve problems involving expressions and equations. Remember to check your work and consider alternative factoring methods to ensure that you get the correct answer.