Simplify The Expression:${ -\left(15x^2 - 20 + 4x + 2x^2\right) }$

Mar 13, 2025 by ADMIN 68 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

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Introduction

Algebraic expressions can be complex and daunting, but with the right techniques, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying the given expression: $-\left(15x^2 - 20 + 4x + 2x^2\right)$ . We will break down the expression into manageable parts, apply the rules of algebra, and arrive at a simplified form.

Understanding the Expression

The given expression is a combination of terms, each containing a variable $x$ raised to a power, and constants. The expression is enclosed in parentheses and preceded by a negative sign. To simplify the expression, we need to apply the rules of algebra, which include combining like terms, removing parentheses, and rearranging the terms.

Step 1: Remove the Negative Sign

The first step in simplifying the expression is to remove the negative sign that precedes the parentheses. This can be done by multiplying each term inside the parentheses by $-1$ . The expression becomes:

\left(15x^2 - 20 + 4x + 2x^2\right)

Step 2: Combine Like Terms

The next step is to combine like terms, which are terms that have the same variable raised to the same power. In this expression, we have two terms with $x^2$ , one term with $x$ , and one constant term. We can combine the $x^2$ terms by adding their coefficients:

15x^2 + 2x^2 = 17x^2

The $x$ term remains the same, and the constant term also remains unchanged. The expression now becomes:

17x^2 + 4x - 20

Step 3: Rearrange the Terms

The final step is to rearrange the terms in the expression to put the terms with the highest degree of $x$ first. In this case, we have a term with $x^2$ , a term with $x$ , and a constant term. We can rearrange the terms as follows:

17x^2 + 4x - 20

becomes

\boxed{17x^2 + 4x - 20}

Conclusion

Simplifying the given expression involved removing the negative sign, combining like terms, and rearranging the terms. By applying the rules of algebra, we were able to arrive at a simplified form of the expression. This process demonstrates the importance of algebraic manipulation in revealing the underlying structure of complex expressions.

Tips and Tricks

When simplifying expressions, it's essential to identify like terms and combine them.
Removing parentheses and rearranging terms can help simplify the expression.
Algebraic manipulation is a crucial skill in mathematics, and practice is key to mastering it.

Real-World Applications

Simplifying expressions has numerous real-world applications, including:

Science and Engineering: Algebraic manipulation is used to model and solve complex problems in physics, engineering, and other scientific fields.
Computer Science: Algebraic manipulation is used in computer programming to simplify complex expressions and improve code efficiency.
Finance: Algebraic manipulation is used in finance to simplify complex financial models and make informed investment decisions.

Final Thoughts

Simplifying expressions is a fundamental skill in mathematics that has numerous real-world applications. By applying the rules of algebra, we can simplify complex expressions and reveal their underlying structure. With practice and patience, anyone can master the art of algebraic manipulation and apply it to real-world problems.

Additional Resources

For further practice and exploration, we recommend the following resources:

Algebra textbooks: "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
Practice problems: Try simplifying expressions on your own using online resources or practice problems.

Frequently Asked Questions

Q: What is the difference between combining like terms and rearranging terms? A: Combining like terms involves adding or subtracting terms with the same variable raised to the same power. Rearranging terms involves reordering the terms in the expression to put the terms with the highest degree of $x$ first.
Q: How do I know when to remove parentheses and when to leave them in? A: Remove parentheses when the expression inside the parentheses can be simplified or combined with other terms. Leave parentheses when the expression inside is complex or cannot be simplified.
Q: Can I use algebraic manipulation to solve equations? A: Yes, algebraic manipulation can be used to solve equations by simplifying the equation and isolating the variable.

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Introduction

Algebraic manipulation is a crucial skill in mathematics that has numerous real-world applications. In our previous article, we explored the process of simplifying expressions using algebraic manipulation. In this article, we will address some of the most frequently asked questions about algebraic manipulation and provide answers to help you better understand this important topic.

Q&A

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of using mathematical operations to simplify or transform algebraic expressions. This involves combining like terms, removing parentheses, and rearranging terms to reveal the underlying structure of the expression.

Q: Why is algebraic manipulation important?

A: Algebraic manipulation is essential in mathematics because it allows us to simplify complex expressions and reveal their underlying structure. This is crucial in solving equations, graphing functions, and modeling real-world problems.

Q: How do I know when to use algebraic manipulation?

A: You should use algebraic manipulation whenever you need to simplify an expression or solve an equation. This can involve combining like terms, removing parentheses, and rearranging terms to reveal the underlying structure of the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, $2x^2 + 5x^2 = 7x^2$ .

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

A: Combining like terms involves adding or subtracting terms with the same variable raised to the same power. Rearranging terms involves reordering the terms in the expression to put the terms with the highest degree of $x$ first.

Q: How do I remove parentheses?

A: To remove parentheses, you need to multiply each term inside the parentheses by the coefficient of the parentheses. For example, $-2(3x^2 + 4x - 5) = -6x^2 - 8x + 10$ .

Q: What is the order of operations in algebraic manipulation?

A: The order of operations in algebraic manipulation is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use algebraic manipulation to solve equations?

A: Yes, algebraic manipulation can be used to solve equations by simplifying the equation and isolating the variable.

Q: What are some common mistakes to avoid in algebraic manipulation?

A: Some common mistakes to avoid in algebraic manipulation include:

Forgetting to combine like terms
Not removing parentheses when necessary
Not following the order of operations
Making errors when simplifying expressions

Conclusion

Algebraic manipulation is a crucial skill in mathematics that has numerous real-world applications. By understanding the basics of algebraic manipulation, you can simplify complex expressions and solve equations with ease. Remember to combine like terms, remove parentheses, and rearrange terms to reveal the underlying structure of the expression.

Tips and Tricks

Practice, practice, practice: The more you practice algebraic manipulation, the more comfortable you will become with the process.
Use online resources: There are many online resources available to help you learn algebraic manipulation, including video tutorials and practice problems.
Break down complex expressions: When simplifying complex expressions, break them down into smaller parts and simplify each part separately.

Final Thoughts

Algebraic manipulation is a powerful tool in mathematics that can help you simplify complex expressions and solve equations with ease. By understanding the basics of algebraic manipulation, you can apply it to real-world problems and become a more confident and proficient mathematician.

Additional Resources

For further practice and exploration, we recommend the following resources:

Algebra textbooks: "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
Practice problems: Try simplifying expressions on your own using online resources or practice problems.

Frequently Asked Questions

Q: What is the difference between algebraic manipulation and algebraic solving? A: Algebraic manipulation involves simplifying expressions, while algebraic solving involves solving equations.
Q: Can I use algebraic manipulation to solve systems of equations? A: Yes, algebraic manipulation can be used to solve systems of equations by simplifying the equations and isolating the variables.
Q: What are some real-world applications of algebraic manipulation? A: Algebraic manipulation has numerous real-world applications, including science, engineering, computer science, and finance.