Perform The Indicated Operation And Simplify Your Answer. { (7x - 8) + (-5x^2 - 7x - 3) = \square$}$

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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 how to simplify a given algebraic expression by performing the indicated operation. We will use the expression (7x−8)+(−5x2−7x−3)=□{(7x - 8) + (-5x^2 - 7x - 3) = \square} as an example and break down the steps involved in simplifying it.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. The expression consists of two terms: 7x−8{7x - 8} and −5x2−7x−3{-5x^2 - 7x - 3}. The first term is a linear expression, while the second term is a quadratic expression. To simplify the expression, we need to combine like terms.

Combining Like Terms

Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: 7x{7x} and −7x{-7x}. We can combine these terms by adding their coefficients. The coefficient of 7x{7x} is 7, and the coefficient of −7x{-7x} is -7. When we add these coefficients, we get:

7x−7x=0{7x - 7x = 0}

So, the term 7x−7x{7x - 7x} simplifies to 0.

Simplifying the Expression

Now that we have combined the like terms, we can simplify the expression by combining the remaining terms. We have two terms left: −8{-8} and −3{-3}. We can combine these terms by adding their coefficients. The coefficient of −8{-8} is -8, and the coefficient of −3{-3} is -3. When we add these coefficients, we get:

−8−3=−11{-8 - 3 = -11}

So, the expression (7x−8)+(−5x2−7x−3){(7x - 8) + (-5x^2 - 7x - 3)} simplifies to:

−5x2−11{-5x^2 - 11}

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By combining like terms and performing the indicated operation, we can simplify complex expressions and make them easier to work with. In this article, we used the expression (7x−8)+(−5x2−7x−3)=□{(7x - 8) + (-5x^2 - 7x - 3) = \square} as an example and broke down the steps involved in simplifying it. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has given you the confidence to tackle more complex expressions.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Combine like terms: Like terms are terms that have the same variable raised to the same power. Combine like terms by adding their coefficients.
  • Use the distributive property: The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. Use the distributive property to expand expressions and simplify them.
  • Simplify fractions: Fractions can be simplified by dividing the numerator and denominator by their greatest common divisor (GCD).
  • Use algebraic identities: Algebraic identities are formulas that can be used to simplify expressions. For example, the identity (a + b)^2 = a^2 + 2ab + b^2 can be used to simplify expressions involving squares.

Common Algebraic Identities

Here are some common algebraic identities that can be used to simplify expressions:

  • (a + b)^2 = a^2 + 2ab + b^2
  • (a - b)^2 = a^2 - 2ab + b^2
  • (a + b)(a - b) = a^2 - b^2
  • (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are a few examples:

  • Physics: Algebraic expressions are used to describe the motion of objects in physics. Simplifying these expressions can help physicists understand the behavior of objects and make predictions about their motion.
  • Engineering: Algebraic expressions are used to design and optimize systems in engineering. Simplifying these expressions can help engineers make more efficient designs and improve the performance of systems.
  • Computer Science: Algebraic expressions are used to describe algorithms and data structures in computer science. Simplifying these expressions can help computer scientists understand the behavior of algorithms and make more efficient code.

Conclusion

Introduction

In our previous article, we explored how to simplify algebraic expressions by combining like terms and performing the indicated operation. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, if you have the expression 2x + 4x, you can combine the like terms by adding their coefficients: 2x + 4x = 6x.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses. For example, the expression (a + b) * (c + d) can be expanded using the distributive property as follows:

(a + b) * (c + d) = ac + ad + bc + bd

Q: How do I simplify fractions?

A: To simplify fractions, you need to divide the numerator and denominator by their greatest common divisor (GCD). For example, the fraction 6/8 can be simplified by dividing both the numerator and denominator by 2:

6/8 = 3/4

Q: What are algebraic identities?

A: Algebraic identities are formulas that can be used to simplify expressions. For example, the identity (a + b)^2 = a^2 + 2ab + b^2 can be used to simplify expressions involving squares.

Q: How do I use algebraic identities to simplify expressions?

A: To use algebraic identities to simplify expressions, you need to identify the type of expression you are working with and then apply the corresponding identity. For example, if you have the expression (a + b)^2, you can use the identity (a + b)^2 = a^2 + 2ab + b^2 to simplify it as follows:

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

Q: What are some common algebraic identities?

A: Some common algebraic identities include:

  • (a + b)^2 = a^2 + 2ab + b^2
  • (a - b)^2 = a^2 - 2ab + b^2
  • (a + b)(a - b) = a^2 - b^2
  • (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Q: How do I apply algebraic identities to simplify expressions?

A: To apply algebraic identities to simplify expressions, you need to identify the type of expression you are working with and then apply the corresponding identity. For example, if you have the expression (a + b)^2, you can use the identity (a + b)^2 = a^2 + 2ab + b^2 to simplify it as follows:

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

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

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

  • Physics: Algebraic expressions are used to describe the motion of objects in physics. Simplifying these expressions can help physicists understand the behavior of objects and make predictions about their motion.
  • Engineering: Algebraic expressions are used to design and optimize systems in engineering. Simplifying these expressions can help engineers make more efficient designs and improve the performance of systems.
  • Computer Science: Algebraic expressions are used to describe algorithms and data structures in computer science. Simplifying these expressions can help computer scientists understand the behavior of algorithms and make more efficient code.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By combining like terms and performing the indicated operation, we can simplify complex expressions and make them easier to work with. In this article, we answered some frequently asked questions about simplifying algebraic expressions and provided examples of how to apply algebraic identities to simplify expressions. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has given you the confidence to tackle more complex expressions.