Find The Product. ( − 12 − N ) 2 = (-12-n)^2 = ( − 12 − N ) 2 =

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Introduction

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In mathematics, the product of two squared expressions is a fundamental concept that is used extensively in various mathematical operations. The expression $(-12-n)^2$ represents the square of a binomial, where $n$ is a variable. In this article, we will explore the concept of finding the product of two squared expressions and provide a step-by-step solution to the given problem.

Understanding the Concept of Squaring a Binomial

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To find the product of two squared expressions, we need to understand the concept of squaring a binomial. A binomial is an algebraic expression consisting of two terms, such as $a + b$ or $a - b$. When we square a binomial, we multiply the binomial by itself, resulting in a quadratic expression.

For example, consider the binomial $a + b$. When we square it, we get:

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

Similarly, when we square the binomial $a - b$, we get:

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

Applying the Concept to the Given Problem

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Now, let's apply the concept of squaring a binomial to the given problem:

$$(-12-n)^2 = ?$$

To find the product of the squared expression, we need to expand the binomial and simplify the resulting expression.

Expanding the Binomial

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Using the formula for squaring a binomial, we can expand the given expression as follows:

$$(-12-n)^2 = (-12)^2 + 2(-12)(-n) + (-n)^2$$

Simplifying the Expression

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Now, let's simplify the expression by evaluating the terms:

$$(-12)^2 = 144$$

$$2(-12)(-n) = 24n$$

$$(-n)^2 = n^2$$

Substituting these values back into the expression, we get:

$$(-12-n)^2 = 144 + 24n + n^2$$

Finding the Product of Two Squared Expressions

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Now that we have simplified the expression, we can find the product of two squared expressions by multiplying the two expressions together.

For example, consider the two expressions:

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

$$(c + d)^2 = c^2 + 2cd + d^2$$

To find the product of these two expressions, we can multiply them together as follows:

$$(a + b)^2 (c + d)^2 = (a^2 + 2ab + b^2) (c^2 + 2cd + d^2)$$

Expanding the Product

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Using the distributive property, we can expand the product as follows:

$$(a^2 + 2ab + b^2) (c^2 + 2cd + d^2) = a^2c^2 + 2a^2cd + a^2d^2 + 2ab^2c + 4abcd + 2ab^2d + b^2c^2 + 2b^2cd + b^2d^2$$

Simplifying the Product

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Now, let's simplify the product by combining like terms:

$$a^2c^2 + 2a^2cd + a^2d^2 + 2ab^2c + 4abcd + 2ab^2d + b^2c^2 + 2b^2cd + b^2d^2$$

$$= a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 + 2a^2cd + 2ab^2c + 2b^2cd + 4abcd + 2ab^2d$$

Conclusion

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In this article, we have explored the concept of finding the product of two squared expressions. We have applied the concept to the given problem and simplified the resulting expression. We have also found the product of two squared expressions by multiplying the two expressions together and simplifying the resulting expression.

The final answer to the given problem is:

$$(-12-n)^2 = 144 + 24n + n^2$$

We hope that this article has provided a clear understanding of the concept of finding the product of two squared expressions and has helped readers to develop their problem-solving skills.

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Introduction

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In our previous article, we explored the concept of finding the product of two squared expressions. We applied the concept to the given problem and simplified the resulting expression. In this article, we will provide a Q&A section to help readers understand the concept better and to address any questions they may have.

Q&A

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Q: What is the product of two squared expressions?

A: The product of two squared expressions is the result of multiplying two squared expressions together. For example, consider the two expressions:

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

$$(c + d)^2 = c^2 + 2cd + d^2$$

The product of these two expressions is:

$$(a + b)^2 (c + d)^2 = (a^2 + 2ab + b^2) (c^2 + 2cd + d^2)$$

Q: How do I simplify the product of two squared expressions?

A: To simplify the product of two squared expressions, you can use the distributive property to expand the product and then combine like terms. For example, consider the product:

$$(a + b)^2 (c + d)^2 = (a^2 + 2ab + b^2) (c^2 + 2cd + d^2)$$

Using the distributive property, we can expand the product as follows:

$$(a^2 + 2ab + b^2) (c^2 + 2cd + d^2) = a^2c^2 + 2a^2cd + a^2d^2 + 2ab^2c + 4abcd + 2ab^2d + b^2c^2 + 2b^2cd + b^2d^2$$

Then, we can combine like terms to simplify the product:

$$a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 + 2a^2cd + 2ab^2c + 2b^2cd + 4abcd + 2ab^2d$$

Q: What is the final answer to the given problem?

A: The final answer to the given problem is:

$$(-12-n)^2 = 144 + 24n + n^2$$

Q: How do I apply the concept of finding the product of two squared expressions to real-world problems?

A: The concept of finding the product of two squared expressions can be applied to various real-world problems, such as:

• Finding the area of a rectangle with a variable length and width
• Finding the volume of a rectangular prism with a variable length, width, and height
• Finding the surface area of a sphere with a variable radius

To apply the concept to real-world problems, you can use the formula for the product of two squared expressions and substitute the values of the variables into the formula.

Q: What are some common mistakes to avoid when finding the product of two squared expressions?

A: Some common mistakes to avoid when finding the product of two squared expressions include:

• Not using the distributive property to expand the product
• Not combining like terms to simplify the product
• Not checking the final answer for errors

By avoiding these common mistakes, you can ensure that your answer is accurate and complete.

Conclusion

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In this article, we have provided a Q&A section to help readers understand the concept of finding the product of two squared expressions. We have addressed common questions and provided examples to illustrate the concept. We hope that this article has been helpful in clarifying the concept and providing a better understanding of the subject.