Simplify The Expression: \[$(3x - 7)(2x + 4)\$\]

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common methods of simplifying expressions is by using the distributive property, which states that for any real numbers a, b, and c:

a(b + c) = ab + ac

In this article, we will use the distributive property to simplify the expression $(3x - 7)(2x + 4)$.

The Distributive Property

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The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one expression by each term in another expression. To apply the distributive property, we need to multiply each term in the first expression by each term in the second expression.

Multiplying Each Term

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Let's multiply each term in the first expression $(3x - 7)$ by each term in the second expression $(2x + 4)$.

• Multiply $3x$ by $2x$: $3x \cdot 2x = 6x^2$
• Multiply $3x$ by $4$: $3x \cdot 4 = 12x$
• Multiply $-7$ by $2x$: $-7 \cdot 2x = -14x$
• Multiply $-7$ by $4$: $-7 \cdot 4 = -28$

Combining Like Terms

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Now that we have multiplied each term, we need to combine like terms. Like terms are terms that have the same variable raised to the same power.

Combining Like Terms

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Let's combine the like terms in the expression:

• Combine the terms with $x^2$: $6x^2$
• Combine the terms with $x$: $12x - 14x = -2x$
• Combine the constant terms: $-28$

Simplifying the Expression

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Now that we have combined like terms, we can simplify the expression by writing it in the form of a single expression.

Simplifying the Expression

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The simplified expression is:

$6x^2 - 2x - 28$

Conclusion

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In this article, we used the distributive property to simplify the expression $(3x - 7)(2x + 4)$. We multiplied each term in the first expression by each term in the second expression and then combined like terms to simplify the expression. The simplified expression is $6x^2 - 2x - 28$.

Example Use Cases

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The distributive property is a fundamental concept in algebra that has many real-world applications. Here are a few example use cases:

• Simplifying expressions: The distributive property is used to simplify expressions in algebra, which is essential for solving equations and inequalities.
• Factoring expressions: The distributive property is used to factor expressions, which is essential for solving quadratic equations.
• Solving systems of equations: The distributive property is used to solve systems of equations, which is essential for solving problems in physics, engineering, and other fields.

Tips and Tricks

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Here are a few tips and tricks for simplifying expressions using the distributive property:

• Use the distributive property to multiply each term: When multiplying expressions, use the distributive property to multiply each term in one expression by each term in the other expression.
• Combine like terms: When combining like terms, make sure to combine the terms with the same variable raised to the same power.
• Simplify the expression: Once you have combined like terms, simplify the expression by writing it in the form of a single expression.

Common Mistakes

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Here are a few common mistakes to avoid when simplifying expressions using the distributive property:

• Not using the distributive property: Failing to use the distributive property can lead to incorrect results.
• Not combining like terms: Failing to combine like terms can lead to incorrect results.
• Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.

Final Thoughts

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In conclusion, the distributive property is a fundamental concept in algebra that is used to simplify expressions. By using the distributive property, we can multiply each term in one expression by each term in the other expression and then combine like terms to simplify the expression. The simplified expression is $6x^2 - 2x - 28$.

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Introduction

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In our previous article, we used the distributive property to simplify the expression $(3x - 7)(2x + 4)$. In this article, we will answer some frequently asked questions about simplifying expressions using the distributive property.

Q&A

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Q: What is the distributive property?

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A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one expression by each term in another expression.

Q: How do I apply the distributive property?

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A: To apply the distributive property, you need to multiply each term in one expression by each term in the other expression. For example, to simplify the expression $(3x - 7)(2x + 4)$, you would multiply each term in the first expression by each term in the second expression.

Q: What are like terms?

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A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $-3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

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A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, to combine the terms $2x$ and $-3x$, you would add their coefficients: $2x - 3x = -x$.

Q: What is the difference between the distributive property and factoring?

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A: The distributive property is used to expand expressions by multiplying each term in one expression by each term in another expression. Factoring is used to break down an expression into simpler expressions. For example, the expression $6x^2 - 2x - 28$ can be factored as $(3x - 7)(2x + 4)$.

Q: How do I simplify expressions using the distributive property?

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A: To simplify expressions using the distributive property, you need to follow these steps:

1. Multiply each term in one expression by each term in the other expression.
2. Combine like terms.
3. Simplify the expression by writing it in the form of a single expression.

Q: What are some common mistakes to avoid when simplifying expressions using the distributive property?

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A: Some common mistakes to avoid when simplifying expressions using the distributive property include:

• Not using the distributive property.
• Not combining like terms.
• Not simplifying the expression.

Example Problems

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Here are a few example problems to help you practice simplifying expressions using the distributive property:

Example 1

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Simplify the expression $(2x + 5)(3x - 2)$.

Solution

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To simplify the expression, we need to multiply each term in the first expression by each term in the second expression:

• Multiply $2x$ by $3x$: $2x \cdot 3x = 6x^2$
• Multiply $2x$ by $-2$: $2x \cdot -2 = -4x$
• Multiply $5$ by $3x$: $5 \cdot 3x = 15x$
• Multiply $5$ by $-2$: $5 \cdot -2 = -10$

Now, we need to combine like terms:

• Combine the terms with $x^2$: $6x^2$
• Combine the terms with $x$: $-4x + 15x = 11x$
• Combine the constant terms: $-10$

The simplified expression is $6x^2 + 11x - 10$.

Example 2

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Simplify the expression $(x - 3)(2x + 1)$.

Solution

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To simplify the expression, we need to multiply each term in the first expression by each term in the second expression:

• Multiply $x$ by $2x$: $x \cdot 2x = 2x^2$
• Multiply $x$ by $1$: $x \cdot 1 = x$
• Multiply $-3$ by $2x$: $-3 \cdot 2x = -6x$
• Multiply $-3$ by $1$: $-3 \cdot 1 = -3$

Now, we need to combine like terms:

• Combine the terms with $x^2$: $2x^2$
• Combine the terms with $x$: $x - 6x = -5x$
• Combine the constant terms: $-3$

The simplified expression is $2x^2 - 5x - 3$.

Conclusion

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In this article, we answered some frequently asked questions about simplifying expressions using the distributive property. We also provided example problems to help you practice simplifying expressions using the distributive property. Remember to follow the steps outlined in this article to simplify expressions using the distributive property.

Final Thoughts

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Simplifying expressions using the distributive property is an essential skill in algebra. By following the steps outlined in this article, you can simplify expressions and solve equations and inequalities. Remember to practice regularly to become proficient in simplifying expressions using the distributive property.