Simplify The Following Expression:${ (4y + 7)(y - 2) }$ { \text{What Is The Simplified Expression In The Form } Ay^2 + By + C? \}

Mar 13, 2025 by ADMIN 134 views

Simplify the Expression: (4y + 7)(y - 2)

Introduction

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 allows us to multiply each term in one expression by each term in another expression. In this article, we will simplify the expression (4y + 7)(y - 2) and express it in the form ay^2 + by + c.

The Distributive Property

The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be extended to more than two terms, and it is the key to simplifying expressions like the one we are dealing with.

Simplifying the Expression

To simplify the expression (4y + 7)(y - 2), we will use the distributive property to multiply each term in the first expression by each term in the second expression.

(4y + 7)(y - 2) = 4y(y - 2) + 7(y - 2)

Multiply Each Term

Now, we will multiply each term in the first expression by each term in the second expression.

4y(y - 2) = 4y^2 - 8y 7(y - 2) = 7y - 14

Combine Like Terms

Now that we have multiplied each term, we can combine like terms to simplify the expression.

4y^2 - 8y + 7y - 14 = 4y^2 - y - 14

The Simplified Expression

The simplified expression is 4y^2 - y - 14, which is in the form ay^2 + by + c.

Conclusion

In this article, we simplified the expression (4y + 7)(y - 2) using the distributive property and combined like terms to express it in the form ay^2 + by + c. This is an important skill in algebra that helps us solve equations and inequalities.

Example Problems

Here are a few example problems that you can try to practice simplifying expressions:

(3x + 2)(x - 1)
(2y - 3)(y + 4)
(x + 1)(x - 5)

Tips and Tricks

Here are a few tips and tricks that can help you simplify expressions:

Use the distributive property to multiply each term in one expression by each term in another expression.
Combine like terms to simplify the expression.
Check your work by plugging in values for the variables to make sure the expression is true.

Common Mistakes

Here are a few common mistakes that you should avoid when simplifying expressions:

Failing to use the distributive property to multiply each term in one expression by each term in another expression.
Failing to combine like terms to simplify the expression.
Making errors when multiplying or combining terms.

Real-World Applications

Simplifying expressions has many real-world applications, including:

Solving equations and inequalities in physics and engineering.
Modeling population growth and decline in biology.
Analyzing data in statistics and data science.

Final Thoughts

Simplifying expressions is an important skill in algebra that helps us solve equations and inequalities. By using the distributive property and combining like terms, we can simplify expressions and express them in the form ay^2 + by + c. With practice and patience, you can become proficient in simplifying expressions and apply this skill to real-world problems.

Introduction

In our previous article, we simplified the expression (4y + 7)(y - 2) using the distributive property and combined like terms to express it in the form ay^2 + by + c. In this article, we will answer some frequently asked questions about simplifying expressions and provide additional examples and tips.

Q&A

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, the following equation holds:

a(b + c) = ab + ac

This property can be extended to more than two terms, and it is the key to simplifying expressions like the one we are dealing with.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to multiply each term in one expression by each term in another expression. For example, to simplify the expression (4y + 7)(y - 2), you would multiply each term in the first expression by each term in the second expression.

Q: What is the difference between multiplying and combining like terms?

A: Multiplying and combining like terms are two different operations. Multiplying involves multiplying each term in one expression by each term in another expression, while combining like terms involves adding or subtracting terms that have the same variable and exponent.

Q: How do I know when to multiply and when to combine like terms?

A: You should multiply when you are dealing with two or more expressions that are being multiplied together, and you should combine like terms when you are simplifying an expression that has multiple terms.

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

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

Failing to use the distributive property to multiply each term in one expression by each term in another expression.
Failing to combine like terms to simplify the expression.
Making errors when multiplying or combining terms.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through example problems, such as the ones listed below:

(3x + 2)(x - 1)
(2y - 3)(y + 4)
(x + 1)(x - 5)

You can also try simplifying expressions on your own, using the distributive property and combining like terms as needed.

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

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

Solving equations and inequalities in physics and engineering.
Modeling population growth and decline in biology.
Analyzing data in statistics and data science.