Which Expression Is Equivalent To:$\[ 2\left(-5a^2 + 6a + 2\right) - 4\left(3a^2 - 4a - 5\right) \\]

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression: $2\left(-5a^2 + 6a + 2\right) - 4\left(3a^2 - 4a - 5\right)$. We will break down the expression into manageable parts, apply the distributive property, and combine like terms to arrive at the final simplified expression.

Understanding the Expression

The given expression is a combination of two terms, each enclosed in parentheses. The first term is $2\left(-5a^2 + 6a + 2\right)$, and the second term is $-4\left(3a^2 - 4a - 5\right)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Applying the Distributive Property

To apply the distributive property, we need to multiply each term inside the parentheses by the corresponding coefficient outside the parentheses. For the first term, we have:

$$2\left(-5a^2 + 6a + 2\right) = 2(-5a^2) + 2(6a) + 2(2)$$

Using the distributive property, we can rewrite this expression as:

$$-10a^2 + 12a + 4$$

Similarly, for the second term, we have:

$$-4\left(3a^2 - 4a - 5\right) = -4(3a^2) + -4(-4a) + -4(-5)$$

Using the distributive property, we can rewrite this expression as:

$$-12a^2 + 16a + 20$$

Combining Like Terms

Now that we have applied the distributive property, we can combine like terms to simplify the expression further. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $a$ raised to the power of 2, and two terms with the variable $a$ raised to the power of 1.

We can combine the like terms as follows:

$$-10a^2 + 12a + 4 - 12a^2 + 16a + 20$$

Combining the like terms, we get:

$$-22a^2 + 28a + 24$$

Conclusion

In this article, we have simplified the given algebraic expression using the distributive property and combining like terms. We have shown that the expression $2\left(-5a^2 + 6a + 2\right) - 4\left(3a^2 - 4a - 5\right)$ can be simplified to $-22a^2 + 28a + 24$. This process demonstrates the importance of applying the distributive property and combining like terms in simplifying algebraic expressions.

Tips and Tricks

• When simplifying algebraic expressions, it is essential to apply the distributive property to each term inside the parentheses.
• Combining like terms is a crucial step in simplifying algebraic expressions. Make sure to identify and combine like terms carefully.
• When working with algebraic expressions, it is essential to use parentheses to group terms correctly.

Common Mistakes

• Failing to apply the distributive property to each term inside the parentheses.
• Not combining like terms correctly.
• Not using parentheses to group terms correctly.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, algebraic expressions are used to describe the motion of objects, while in engineering, they are used to design and optimize systems. In economics, algebraic expressions are used to model economic systems and make predictions about future trends.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression: $2\left(-5a^2 + 6a + 2\right) - 4\left(3a^2 - 4a - 5\right)$. We broke down the expression into manageable parts, applied the distributive property, and combined like terms to arrive at the final simplified expression. In this article, we will answer some common questions related to simplifying 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 means that we can multiply each term inside the parentheses by the corresponding coefficient outside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the corresponding coefficient outside the parentheses. For example, if we have $2\left(-5a^2 + 6a + 2\right)$, we would multiply each term inside the parentheses by 2, resulting in $-10a^2 + 12a + 4$.

Q: What are like terms?

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

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the like terms. For example, if we have $2a + 3a$, we would combine the like terms by adding the coefficients, resulting in $5a$.

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

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

• Failing to apply the distributive property to each term inside the parentheses.
• Not combining like terms correctly.
• Not using parentheses to group terms correctly.

Q: How do I know when to use parentheses?

A: You should use parentheses to group terms correctly when you have multiple operations or when you want to indicate the order of operations. For example, if we have $2 + 3 \times 4$, we would use parentheses to group the terms correctly, resulting in $(2 + 3) \times 4$.

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

A: Simplifying algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, algebraic expressions are used to describe the motion of objects, while in engineering, they are used to design and optimize systems. In economics, algebraic expressions are used to model economic systems and make predictions about future trends.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises in your math textbook or online resources. You can also try simplifying expressions on your own and then checking your work with a calculator or online tool.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can simplify complex expressions and arrive at the final answer. This process demonstrates the importance of algebraic expressions in real-world applications and highlights the need for careful attention to detail when working with these expressions.