Simplify The Expression:$\left(3x^2 + 5x\right) + \left(x^2 - 4x\right$\]

Feb 28, 2025 by ADMIN 74 views

Simplifying Algebraic Expressions: A Step-by-Step Guide

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the given expression: $\left(3x^2 + 5x\right) + \left(x^2 - 4x\right)$ . We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

The given expression consists of two terms, each enclosed in parentheses. The first term is $3x^2 + 5x$ , and the second term is $x^2 - 4x$ . To simplify the expression, we need to combine like terms, which means adding or subtracting terms with the same variable and exponent.

Combining Like Terms

Like terms are terms that have the same variable and exponent. In the given expression, we can identify two sets of like terms:

Terms with the variable $x^2$ : $3x^2$ and $x^2$
Terms with the variable $x$ : $5x$ and $-4x$

To combine like terms, we add or subtract the coefficients of the terms. The coefficient is the numerical value in front of the variable.

Simplifying the Expression

Now that we have identified the like terms, we can simplify the expression by combining them.

Combine the terms with the variable $x^2$ : $3x^2 + x^2 = 4x^2$
Combine the terms with the variable $x$ : $5x - 4x = x$

Final Simplified Expression

After combining the like terms, we get the final simplified expression: $4x^2 + x$ . This is the simplest form of the given expression.

Example Use Case

Simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. For instance, in physics, simplifying expressions is essential for solving problems related to motion, energy, and forces.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics that requires a deep understanding of variables, exponents, and like terms. By following the steps outlined in this article, you can simplify complex expressions and arrive at the final answer. Remember to always combine like terms and simplify the expression to its simplest form.

Tips and Tricks

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

Always identify like terms and combine them.
Use the distributive property to expand expressions.
Simplify expressions by combining like terms.
Use the order of operations (PEMDAS) to simplify expressions.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

Failing to identify like terms.
Not combining like terms.
Simplifying expressions incorrectly.
Not using the distributive property.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics that requires practice and patience. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and arrive at the final answer. Remember to always combine like terms and simplify the expression to its simplest form.

Additional Resources

For more information on simplifying algebraic expressions, check out the following resources:

Khan Academy: Simplifying Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Simplifying Algebraic Expressions

Frequently Asked Questions

Here are some frequently asked questions about simplifying algebraic expressions:

Q: What is the difference between like terms and unlike terms? A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.
Q: How do I simplify an expression with multiple variables? A: To simplify an expression with multiple variables, identify like terms and combine them. Use the distributive property to expand expressions and simplify the expression to its simplest form.
Q: What is the order of operations (PEMDAS)? A: The order of operations (PEMDAS) is a mnemonic device that helps you remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

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Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics that requires a deep understanding of variables, exponents, and like terms. In our previous article, we provided a step-by-step guide on how to simplify algebraic expressions. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, identify like terms and combine them. Use the distributive property to expand expressions and simplify the expression to its simplest form.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a mnemonic device that helps you remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, first simplify the fractions by finding the greatest common divisor (GCD) of the numerator and denominator. Then, combine like terms and simplify the expression to its simplest form.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows you to expand expressions by multiplying a single term by multiple terms. For example, $a(b + c) = ab + ac$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, first simplify the exponents by combining like terms. Then, use the power rule to simplify the expression.

Q: What is the power rule?

A: The power rule is a mathematical rule that allows you to simplify expressions with exponents. For example, $(a^m)^n = a^{mn}$ .

Q: How do I simplify an expression with absolute values?

A: To simplify an expression with absolute values, first simplify the expression inside the absolute value. Then, use the definition of absolute value to simplify the expression.

Q: What is the definition of absolute value?

A: The definition of absolute value is $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$ .

Q: How do I simplify an expression with radicals?

A: To simplify an expression with radicals, first simplify the expression inside the radical. Then, use the definition of radicals to simplify the expression.

Q: What is the definition of radicals?

A: The definition of radicals is $\sqrt{x} = x^{1/2}$ .

Tips and Tricks

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

Always identify like terms and combine them.
Use the distributive property to expand expressions.
Simplify expressions by combining like terms.
Use the order of operations (PEMDAS) to simplify expressions.
Simplify expressions with fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
Simplify expressions with exponents by combining like terms and using the power rule.
Simplify expressions with absolute values by using the definition of absolute value.
Simplify expressions with radicals by using the definition of radicals.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

Failing to identify like terms.
Not combining like terms.
Simplifying expressions incorrectly.
Not using the distributive property.
Not using the order of operations (PEMDAS).
Not simplifying expressions with fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
Not simplifying expressions with exponents by combining like terms and using the power rule.
Not simplifying expressions with absolute values by using the definition of absolute value.
Not simplifying expressions with radicals by using the definition of radicals.

Final Thoughts

Additional Resources

For more information on simplifying algebraic expressions, check out the following resources:

Khan Academy: Simplifying Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Simplifying Algebraic Expressions

Frequently Asked Questions

Here are some frequently asked questions about simplifying algebraic expressions:

Q: What is the difference between like terms and unlike terms? A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.
Q: How do I simplify an expression with multiple variables? A: To simplify an expression with multiple variables, identify like terms and combine them. Use the distributive property to expand expressions and simplify the expression to its simplest form.
Q: What is the order of operations (PEMDAS)? A: The order of operations (PEMDAS) is a mnemonic device that helps you remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Example Problems

Here are some example problems to help you practice simplifying algebraic expressions:

Simplify the expression: $2x^2 + 3x + 4$
Simplify the expression: $x^2 - 4x + 3$
Simplify the expression: $2x^2 - 3x + 1$
Simplify the expression: $x^2 + 2x - 3$
Simplify the expression: $2x^2 + 5x - 2$

Solutions

Here are the solutions to the example problems:

Simplify the expression: $2x^2 + 3x + 4$ $2 x^{2} + 3 x + 4$
- Solution: $2x^2 + 3x + 4$
Simplify the expression: $x^2 - 4x + 3$ $x^{2} - 4 x + 3$
- Solution: $x^2 - 4x + 3$
Simplify the expression: $2x^2 - 3x + 1$ $2 x^{2} - 3 x + 1$
- Solution: $2x^2 - 3x + 1$
Simplify the expression: $x^2 + 2x - 3$ $x^{2} + 2 x - 3$
- Solution: $x^2 + 2x - 3$
Simplify the expression: $2x^2 + 5x - 2$ $2 x^{2} + 5 x - 2$
- Solution: $2x^2 + 5x - 2$