Select The Best Answer For The Question.1. Simplify $(3n - 2m)^2 =?$A. $9n^2 - 12mn - 4m^2$ B. \$9n^2 + 12mn + 4m^2$[/tex\] C. $6n^2 - 12mn - 4m^2$ D. $9n^2 - 12mn + 4m^2$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will focus on simplifying the given expression (3nβˆ’2m)2(3n - 2m)^2 and explore the different options provided. We will break down the solution step by step, using algebraic techniques to arrive at the correct answer.

Understanding the Expression

The given expression is a quadratic expression in the form of (aβˆ’b)2(a-b)^2, where a=3na = 3n and b=2mb = 2m. To simplify this expression, we need to apply the formula for expanding a quadratic expression in the form of (aβˆ’b)2(a-b)^2.

Expanding the Quadratic Expression

The formula for expanding a quadratic expression in the form of (aβˆ’b)2(a-b)^2 is:

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

In this case, a=3na = 3n and b=2mb = 2m. Substituting these values into the formula, we get:

(3nβˆ’2m)2=(3n)2βˆ’2(3n)(2m)+(2m)2(3n - 2m)^2 = (3n)^2 - 2(3n)(2m) + (2m)^2

Simplifying the Expression

Now, let's simplify the expression step by step:

(3n)2=9n2(3n)^2 = 9n^2

2(3n)(2m)=12mn2(3n)(2m) = 12mn

(2m)2=4m2(2m)^2 = 4m^2

Substituting these simplified values back into the expression, we get:

(3nβˆ’2m)2=9n2βˆ’12mn+4m2(3n - 2m)^2 = 9n^2 - 12mn + 4m^2

Comparing the Options

Now, let's compare the simplified expression with the options provided:

A. $9n^2 - 12mn - 4m^2$

B. $9n^2 + 12mn + 4m^2$

C. $6n^2 - 12mn - 4m^2$

D. $9n^2 - 12mn + 4m^2$

Based on our simplification, we can see that the correct answer is:

D. $9n^2 - 12mn + 4m^2$

Conclusion

In this article, we simplified the given expression (3nβˆ’2m)2(3n - 2m)^2 using algebraic techniques. We applied the formula for expanding a quadratic expression in the form of (aβˆ’b)2(a-b)^2 and simplified the expression step by step. By comparing the simplified expression with the options provided, we arrived at the correct answer, which is option D.

Tips and Tricks

  • When simplifying algebraic expressions, always start by identifying the type of expression and applying the relevant formula.
  • Use algebraic techniques such as expanding and simplifying to arrive at the correct answer.
  • Compare the simplified expression with the options provided to ensure that you have arrived at the correct answer.

Practice Problems

  • Simplify the expression (2x+3y)2(2x + 3y)^2.
  • Simplify the expression (4xβˆ’2y)2(4x - 2y)^2.
  • Simplify the expression (x+2y)2(x + 2y)^2.

Q: What is the formula for expanding a quadratic expression in the form of (a-b)^2?

A: The formula for expanding a quadratic expression in the form of (a-b)^2 is:

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

Q: How do I simplify a quadratic expression in the form of (a-b)^2?

A: To simplify a quadratic expression in the form of (a-b)^2, you need to apply the formula and simplify the resulting expression. Here's a step-by-step guide:

  1. Identify the values of a and b.
  2. Substitute the values of a and b into the formula.
  3. Simplify the resulting expression by combining like terms.

Q: What is the difference between (a-b)^2 and (a+b)^2?

A: The main difference between (a-b)^2 and (a+b)^2 is the sign of the middle term. In (a-b)^2, the middle term is -2ab, while in (a+b)^2, the middle term is 2ab.

Q: How do I expand a quadratic expression in the form of (a+b)^2?

A: To expand a quadratic expression in the form of (a+b)^2, you can use the formula:

(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2

Q: What is the difference between (a-b)^2 and (a-b)(a+b)?

A: The main difference between (a-b)^2 and (a-b)(a+b) is the way the expression is expanded. (a-b)^2 is expanded using the formula (a-b)^2 = a^2 - 2ab + b^2, while (a-b)(a+b) is expanded using the distributive property.

Q: How do I simplify a quadratic expression in the form of (a-b)(a+b)?

A: To simplify a quadratic expression in the form of (a-b)(a+b), you can use the distributive property to expand the expression and then simplify the resulting expression.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Forgetting to combine like terms
  • Making errors when expanding or simplifying expressions
  • Not checking the final answer for accuracy

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems, such as those found in algebra textbooks or online resources. You can also try simplifying expressions on your own and then checking your answers with a calculator or online tool.

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

A: Simplifying algebraic expressions has many real-world applications, including:

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling real-world phenomena, such as population growth or financial transactions

By practicing simplifying algebraic expressions and understanding the formulas and techniques involved, you can become more confident and proficient in solving algebraic problems.