Simplify Algebraic Expressions Removing Grouping Symbols And Combining Like Terms

In mathematics, simplifying expressions is a fundamental skill that allows us to manipulate and understand complex equations more easily. This often involves removing grouping symbols such as parentheses, brackets, and braces, and then combining like terms. Let's delve into a step-by-step guide on how to simplify the algebraic expression $-x^2 - [3x - (12x^2 - 7x)] - [2 + (-2x - 2)]$, writing the terms from the highest to the lowest power of the variable.

Understanding the Basics

Before we jump into the specifics, let's clarify some key concepts. Simplifying expressions is about rewriting an expression in a more concise and manageable form. This doesn't change the value of the expression, but it makes it easier to work with. Grouping symbols are used to indicate the order of operations. Parentheses ( ) are the innermost grouping symbols, followed by brackets [ ], and then braces { }. When simplifying, we work from the inside out. Like terms are terms that have the same variable raised to the same power. For example, 3x and -7x are like terms, while 12x^2 and 3x are not. Combining like terms involves adding or subtracting their coefficients (the numerical part of the term).

Step 1: Remove the Innermost Grouping Symbols

Our expression is: $-x^2 - [3x - (12x^2 - 7x)] - [2 + (-2x - 2)]$. We begin by addressing the innermost parentheses. In this case, we have two sets of parentheses: (12x^2 - 7x) and (-2x - 2). The first set is preceded by a negative sign, so we need to distribute the negative sign across the terms inside the parentheses. The second set is being added to 2, which simplifies the process.

The expression now becomes:

$-x^2 - [3x - 12x^2 + 7x] - [2 - 2x - 2]$

Notice how the - (12x^2 - 7x) became - 12x^2 + 7x after distributing the negative sign. In the second set of parentheses, we simply removed the parentheses since they were preceded by a + sign.

Step 2: Combine Like Terms Within Brackets

Next, we focus on the expressions within the brackets: [3x - 12x^2 + 7x] and [2 - 2x - 2]. We need to combine the like terms within each set of brackets.

In the first bracket, 3x and 7x are like terms. Adding them together gives us 10x. So, the expression within the first bracket simplifies to -12x^2 + 10x. In the second bracket, 2 and -2 are like terms, and they cancel each other out, leaving us with -2x.

The expression now looks like this:

$-x^2 - [-12x^2 + 10x] - [-2x]$

Step 3: Remove the Brackets

Now we need to remove the brackets. Similar to removing parentheses, we distribute the negative sign if there is a negative sign in front of the bracket. In our case, both sets of brackets are preceded by a negative sign.

Distributing the negative sign across the first bracket [-12x^2 + 10x] changes the signs of the terms inside, giving us 12x^2 - 10x. Distributing the negative sign across the second bracket [-2x] changes the sign, giving us +2x.

The expression now becomes:

$-x^2 + 12x^2 - 10x + 2x$

Step 4: Combine Like Terms

The final step is to combine all the like terms in the expression. We have two terms with x^2: -x^2 and 12x^2. Combining them gives us 11x^2. We also have two terms with x: -10x and 2x. Combining them gives us -8x.

Therefore, the simplified expression is:

$11x^2 - 8x$

This is the final simplified form of the expression, with the terms written from the highest power of the variable (x^2) to the lowest power of the variable (x).

Additional Examples and Practice

To solidify your understanding, let's consider a few more examples:

1. Simplify: $2(3y - 4) + 5(y + 2)$

   • Distribute: $6y - 8 + 5y + 10$
   • Combine like terms: $11y + 2$

2. Simplify: $3[2a - (b - a)] + 4b$

   • Remove inner parentheses: $3[2a - b + a] + 4b$
   • Combine like terms within brackets: $3[3a - b] + 4b$
   • Distribute: $9a - 3b + 4b$
   • Combine like terms: $9a + b$

3. Simplify: $-{4c + 2[c - 3(c + 1)]}$

   • Remove innermost parentheses: $-{4c + 2[c - 3c - 3]}$
   • Combine like terms within brackets: $-{4c + 2[-2c - 3]}$
   • Distribute within braces: $-{4c - 4c - 6}$
   • Combine like terms within braces: $-{-6}$
   • Simplify: $6$

Common Mistakes to Avoid

When simplifying algebraic expressions, it's crucial to be mindful of common mistakes. Here are a few to watch out for:

• Forgetting to distribute the negative sign: This is a very common error. Remember that when a negative sign is in front of a grouping symbol, you need to multiply each term inside the grouping symbol by -1.
• Combining unlike terms: Only terms with the same variable and exponent can be combined. For example, you cannot combine x^2 and x.
• Incorrectly applying the order of operations: Always follow the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Making arithmetic errors: Double-check your arithmetic, especially when dealing with negative numbers.

The Importance of Simplification

Simplifying algebraic expressions is not just an exercise in following rules; it's a fundamental skill that has significant applications in various areas of mathematics and beyond. Here are some key reasons why simplification is important:

• Solving equations: Many equations are easier to solve once they have been simplified. By removing grouping symbols and combining like terms, you can often isolate the variable you're trying to solve for.
• Graphing functions: Simplifying a function's equation can make it easier to graph. You can identify key features of the graph, such as intercepts and turning points, more readily.
• Calculus: Simplification is crucial in calculus, where you often need to manipulate complex expressions when finding derivatives and integrals.
• Real-world applications: Many real-world problems can be modeled using algebraic expressions. Simplifying these expressions can help you find solutions and make predictions.

For instance, consider a business scenario where you need to calculate the profit from selling a product. The profit might be represented by a complex expression involving revenue and costs. Simplifying this expression can give you a clearer understanding of the factors that affect profit and help you make informed business decisions.

Tips for Success

To become proficient at simplifying algebraic expressions, keep these tips in mind:

• Practice regularly: The more you practice, the more comfortable you'll become with the process.
• Show your work: Writing out each step can help you avoid mistakes and make it easier to track your progress.
• Check your answers: After simplifying an expression, you can often check your answer by substituting a numerical value for the variable in both the original expression and the simplified expression. If the results are the same, your simplification is likely correct.
• Seek help when needed: If you're struggling with a particular concept or problem, don't hesitate to ask for help from a teacher, tutor, or online resource.

In conclusion, simplifying algebraic expressions by removing grouping symbols and combining like terms is a crucial skill in mathematics. By mastering this skill, you'll be well-equipped to tackle more complex problems and gain a deeper understanding of mathematical concepts. Remember to work systematically, pay attention to detail, and practice regularly. With dedication and effort, you can become proficient at simplifying expressions and unlock the power of algebra.

By following the steps outlined in this guide and practicing regularly, you can master the art of simplifying algebraic expressions and build a strong foundation for future mathematical endeavors. Remember, simplification is not just about getting the right answer; it's about developing a deeper understanding of mathematical principles and enhancing your problem-solving skills. So, embrace the challenge, and enjoy the journey of learning and mastering algebra!

#Simplify the Expression: A Step-by-Step Guide

Repair Input Keyword

Simplify the expression $-x^2-[3x-(12x^2-7x)]-[2+(-2x-2)]$ by removing grouping symbols and combining like terms. Write the terms from the highest to the lowest power of the variable.

SEO Title

Simplify Algebraic Expressions: Removing Grouping Symbols and Combining Like Terms