Multiply \[$(x-6)(4x+3)\$\].A. \[$4x^2 - 24x - 18\$\] B. \[$4x^2 - 18\$\] C. \[$4x^2 - 21x - 18\$\] D. \[$4x^2 - 3x - 18\$\]

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Introduction

In algebra, multiplying expressions is a fundamental operation that helps us simplify complex equations and solve problems. In this article, we will focus on multiplying two binomial expressions, (xβˆ’6)(4x+3){(x-6)(4x+3)}, and explore the different methods to achieve this. We will also examine the various answer choices and determine the correct solution.

Understanding the Problem

The given problem involves multiplying two binomial expressions, (xβˆ’6)(4x+3){(x-6)(4x+3)}. To solve this, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b+c) = ab + ac. We will use this property to expand the given expression and simplify it.

Method 1: Using the Distributive Property

To multiply the given expressions, we will use the distributive property. We will multiply each term in the first expression, (xβˆ’6){(x-6)}, by each term in the second expression, (4x+3){(4x+3)}.

(xβˆ’6)(4x+3)=x(4x+3)βˆ’6(4x+3){(x-6)(4x+3) = x(4x+3) - 6(4x+3)}

Now, we will apply the distributive property to each term:

x(4x+3)=4x2+3x{x(4x+3) = 4x^2 + 3x}

βˆ’6(4x+3)=βˆ’24xβˆ’18{-6(4x+3) = -24x - 18}

Combining the two results, we get:

(xβˆ’6)(4x+3)=4x2+3xβˆ’24xβˆ’18{(x-6)(4x+3) = 4x^2 + 3x - 24x - 18}

Simplifying the expression, we get:

(xβˆ’6)(4x+3)=4x2βˆ’21xβˆ’18{(x-6)(4x+3) = 4x^2 - 21x - 18}

Method 2: Using FOIL Method

The FOIL method is a shortcut for multiplying two binomial expressions. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms.

Using the FOIL method, we multiply the first terms in each expression, then the outer terms, then the inner terms, and finally the last terms.

(xβˆ’6)(4x+3)=x(4x)+x(3)βˆ’6(4x)βˆ’6(3){(x-6)(4x+3) = x(4x) + x(3) - 6(4x) - 6(3)}

Simplifying the expression, we get:

(xβˆ’6)(4x+3)=4x2+3xβˆ’24xβˆ’18{(x-6)(4x+3) = 4x^2 + 3x - 24x - 18}

Combining like terms, we get:

(xβˆ’6)(4x+3)=4x2βˆ’21xβˆ’18{(x-6)(4x+3) = 4x^2 - 21x - 18}

Conclusion

In this article, we explored two methods for multiplying the binomial expressions (xβˆ’6)(4x+3){(x-6)(4x+3)}. Using the distributive property and the FOIL method, we arrived at the same solution: 4x2βˆ’21xβˆ’18{4x^2 - 21x - 18}. This result is consistent with the answer choice C.

Answer Choice Analysis

Let's analyze the answer choices to determine which one is correct.

A. 4x2βˆ’24xβˆ’18{4x^2 - 24x - 18}

This answer choice is incorrect because it lacks the term βˆ’21x{-21x}.

B. 4x2βˆ’18{4x^2 - 18}

This answer choice is incorrect because it lacks the term βˆ’21x{-21x} and the constant term βˆ’18{-18}.

C. 4x2βˆ’21xβˆ’18{4x^2 - 21x - 18}

This answer choice is correct because it matches the solution we obtained using the distributive property and the FOIL method.

D. 4x2βˆ’3xβˆ’18{4x^2 - 3x - 18}

This answer choice is incorrect because it lacks the term βˆ’21x{-21x}.

Final Answer

The correct answer is C. 4x2βˆ’21xβˆ’18{4x^2 - 21x - 18}. This result is consistent with the solution we obtained using the distributive property and the FOIL method.

Additional Tips and Resources

  • To practice multiplying binomial expressions, try using the distributive property and the FOIL method.
  • Use online resources, such as Khan Academy or Mathway, to explore more examples and practice problems.
  • For a more in-depth understanding of algebraic expressions, consult a textbook or online resource, such as Algebra I by Michael Artin or Algebra by Michael Artin.
    Multiplying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of multiplying binomial expressions using the distributive property and the FOIL method. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in multiplying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, a(b+c) = ab + ac. This property allows us to expand and simplify complex expressions by multiplying each term in one expression by each term in another expression.

Q: How do I apply the distributive property to multiply binomial expressions?

A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression. For example, to multiply (x-6) and (4x+3), you would multiply x by 4x, x by 3, -6 by 4x, and -6 by 3.

Q: What is the FOIL method?

A: The FOIL method is a shortcut for multiplying two binomial expressions. FOIL stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms. Using the FOIL method, you multiply the first terms in each expression, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I use the FOIL method to multiply binomial expressions?

A: To use the FOIL method, you need to multiply the first terms in each expression, then the outer terms, then the inner terms, and finally the last terms. For example, to multiply (x-6) and (4x+3), you would multiply x by 4x, x by 3, -6 by 4x, and -6 by 3.

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

A: Some common mistakes to avoid when multiplying binomial expressions include:

  • Forgetting to multiply each term in one expression by each term in the other expression
  • Not combining like terms
  • Not simplifying the expression

Q: How do I simplify an expression after multiplying binomial expressions?

A: To simplify an expression after multiplying binomial expressions, you need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

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

A: Multiplying binomial expressions has many real-world applications, including:

  • Solving quadratic equations
  • Finding the area and perimeter of shapes
  • Modeling population growth and decay
  • Analyzing data and making predictions

Q: How can I practice multiplying binomial expressions?

A: You can practice multiplying binomial expressions by:

  • Using online resources, such as Khan Academy or Mathway
  • Working with a tutor or teacher
  • Practicing with worksheets and exercises
  • Using real-world examples and applications

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in multiplying algebraic expressions. We covered topics such as the distributive property, the FOIL method, common mistakes to avoid, and real-world applications. By practicing and applying these concepts, you can become more confident and proficient in multiplying binomial expressions.

Additional Resources

  • Khan Academy: Multiplying Binomial Expressions
  • Mathway: Multiplying Binomial Expressions
  • Algebra I by Michael Artin: Chapter 3, Multiplying Binomial Expressions
  • Algebra by Michael Artin: Chapter 3, Multiplying Binomial Expressions