Simplifying (12s^2 - 15s) - (s - 9) A Step-by-Step Guide

by ADMIN 57 views

Simplify the expression (12s^2 - 15s) - (s - 9).

Introduction

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It's the cornerstone of solving more complex equations and understanding mathematical relationships. In this comprehensive guide, we will delve into the process of simplifying the expression (12s^2 - 15s) - (s - 9). This seemingly simple expression involves several key concepts, including the distributive property, combining like terms, and understanding the order of operations. By mastering these concepts, you'll be well-equipped to tackle a wide range of algebraic problems.

Our journey will begin with a step-by-step breakdown of the simplification process, ensuring that each step is clearly explained and justified. We'll explore the logic behind each operation, emphasizing why we perform certain actions and the mathematical principles that underpin them. This approach will not only help you understand how to simplify this specific expression but also equip you with the skills to simplify similar expressions independently.

Furthermore, we will discuss the common pitfalls and mistakes that students often make when simplifying expressions. By highlighting these potential errors, we aim to help you avoid them and develop a more accurate and efficient approach to algebraic manipulation. We'll also provide tips and tricks for checking your work, ensuring that your final answer is correct.

Finally, we'll explore the broader context of algebraic simplification, discussing its importance in various fields of mathematics and its applications in real-world scenarios. This will provide you with a deeper appreciation for the value of this skill and motivate you to continue honing your algebraic abilities.

So, let's embark on this journey of mathematical discovery and unlock the secrets of simplifying algebraic expressions!

Step-by-Step Simplification Process

1. Understanding the Expression

Before diving into the simplification process, it's crucial to understand the expression we're working with: (12s^2 - 15s) - (s - 9). This expression consists of two groups of terms enclosed in parentheses, separated by a subtraction sign. The first group, (12s^2 - 15s), contains two terms: 12s^2 and -15s. The second group, (s - 9), also contains two terms: s and -9. The variable in this expression is s, and the terms involve different powers of s (specifically, s^2 and s).

The subtraction sign between the two groups of parentheses indicates that we need to subtract the second group from the first. This is where the distributive property comes into play, as we'll see in the next step. It's also important to remember the order of operations (often remembered by the acronym PEMDAS or BODMAS), which dictates that we should perform operations within parentheses first, followed by exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). In this case, we'll be focusing on the subtraction operation and the simplification of terms within and between the parentheses.

2. Applying the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms inside parentheses. In this case, we have a negative sign in front of the second group of parentheses, which can be thought of as multiplying the entire group by -1. This is a critical step in simplifying the expression because it removes the parentheses and allows us to combine like terms.

To apply the distributive property, we multiply -1 by each term inside the second set of parentheses: -(s - 9) = -1 * s + (-1) * (-9) = -s + 9. Notice that the sign of each term inside the parentheses changes when we multiply by -1. This is a crucial detail to remember when dealing with subtraction outside parentheses. The expression now becomes: 12s^2 - 15s - s + 9. This form is much easier to work with because we have eliminated the parentheses and can now focus on combining the like terms.

3. Combining Like Terms

Like terms are terms that have the same variable raised to the same power. In our expression, 12s^2 - 15s - s + 9, the like terms are -15s and -s. The term 12s^2 is unique because it's the only term with s^2, and the term 9 is a constant term, meaning it doesn't have any variable. Combining like terms involves adding or subtracting their coefficients (the numerical part of the term). In this case, we need to combine -15s and -s.

To combine -15s and -s, we add their coefficients: -15 + (-1) = -16. Therefore, -15s - s simplifies to -16s. The expression now becomes: 12s^2 - 16s + 9. This is the simplified form of the expression because there are no more like terms to combine.

4. Final Simplified Expression

After applying the distributive property and combining like terms, we arrive at the simplified expression: 12s^2 - 16s + 9. This expression is in its simplest form because it contains no parentheses and no like terms that can be combined. The terms are arranged in descending order of the powers of the variable s, which is a standard practice in algebra. This makes the expression easier to read and understand.

The simplified expression 12s^2 - 16s + 9 is a quadratic expression, which is a polynomial of degree 2. The term 12s^2 is the quadratic term, -16s is the linear term, and 9 is the constant term. Understanding the structure of this expression is crucial for further algebraic manipulations, such as factoring or solving for the roots of the equation. The process of simplifying algebraic expressions is a fundamental skill that is used extensively in various branches of mathematics, including calculus, trigonometry, and linear algebra. Mastering this skill is essential for success in higher-level mathematics courses.

Common Mistakes and How to Avoid Them

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes that students often make and how to avoid them:

  1. Forgetting to Distribute the Negative Sign: This is one of the most common errors. When subtracting a group of terms in parentheses, it's crucial to distribute the negative sign (or think of it as multiplying by -1) to each term inside the parentheses. Failing to do so will result in an incorrect expression. To avoid this, always write out the step where you distribute the negative sign explicitly. For example, instead of directly writing - (s - 9) = -s - 9, write - (s - 9) = -1 * s + (-1) * (-9) = -s + 9. This will help you visualize the distribution and avoid sign errors.

  2. Combining Unlike Terms: Another common mistake is combining terms that are not like terms. Remember, like terms must have the same variable raised to the same power. For example, you cannot combine 12s^2 and -16s because they have different powers of s. To avoid this, carefully identify the terms with the same variable and power before combining them. Underline or circle like terms to help you keep track of them.

  3. Arithmetic Errors: Simple arithmetic mistakes, such as adding or subtracting coefficients incorrectly, can also lead to errors in simplification. To avoid this, double-check your arithmetic calculations, especially when dealing with negative numbers. Use a calculator if necessary.

  4. Order of Operations: Forgetting the order of operations (PEMDAS/BODMAS) can also cause errors. In this specific example, the distributive property takes precedence over combining like terms. To avoid this, always follow the order of operations meticulously. If you're unsure, write out each step clearly and in the correct order.

  5. Not Simplifying Completely: Sometimes, students stop simplifying before the expression is in its simplest form. Make sure you have combined all like terms and that there are no parentheses left. To avoid this, review your simplified expression carefully to ensure that there are no further simplifications possible.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and efficiency in simplifying algebraic expressions.

Tips and Tricks for Checking Your Work

Ensuring the correctness of your simplified expression is as crucial as the simplification process itself. Here are some valuable tips and tricks to help you check your work and minimize errors:

  1. Substitute a Value for the Variable: A powerful technique is to substitute a numerical value for the variable in both the original expression and your simplified expression. If both expressions yield the same result, it's a strong indication that your simplification is correct. For instance, let's use s = 2 in our example. The original expression (12s^2 - 15s) - (s - 9) becomes (12 * 2^2 - 15 * 2) - (2 - 9) = (48 - 30) - (-7) = 18 + 7 = 25. The simplified expression 12s^2 - 16s + 9 becomes 12 * 2^2 - 16 * 2 + 9 = 48 - 32 + 9 = 25. Since both expressions evaluate to 25, our simplification is likely correct. Remember to choose a value that is easy to work with and avoids making the calculations too complex.

  2. Work Backwards: Another helpful strategy is to reverse the simplification steps. Start with your simplified expression and apply the opposite operations to see if you can arrive back at the original expression. For example, if you combined like terms, try separating them again. If you distributed a negative sign, try factoring it out. If you can successfully reverse the steps and obtain the original expression, your simplification is likely correct.

  3. Use a Computer Algebra System (CAS): Tools like Wolfram Alpha, Symbolab, or Desmos can be invaluable for checking your work. Simply input both the original expression and your simplified expression into the CAS and see if they are equivalent. These systems use sophisticated algorithms to simplify and compare expressions, providing a reliable way to verify your results. However, it's essential to understand the simplification process yourself and not solely rely on these tools, as they may not always show the steps involved.

  4. Ask a Peer or Teacher: Discussing your work with a peer or teacher can provide a fresh perspective and help you identify any errors you may have overlooked. Explaining your steps to someone else can also solidify your understanding of the process.

  5. Practice Regularly: The more you practice simplifying expressions, the better you'll become at recognizing patterns and avoiding mistakes. Regularly working through different types of problems will build your confidence and improve your accuracy.

By incorporating these tips and tricks into your problem-solving routine, you can significantly enhance your ability to check your work and ensure the correctness of your simplified expressions.

Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is not just a mathematical exercise; it's a fundamental skill with widespread applications in various fields. Understanding its importance can motivate you to master this skill and appreciate its value in both academic and real-world contexts.

  1. Foundation for Higher-Level Mathematics: Simplifying expressions is a building block for more advanced mathematical concepts. It's essential for solving equations, inequalities, and systems of equations. It also plays a crucial role in calculus, trigonometry, and linear algebra. Without a solid grasp of simplification techniques, you'll struggle to succeed in these higher-level courses. For example, in calculus, you often need to simplify expressions before you can differentiate or integrate them. Similarly, in linear algebra, simplifying matrix expressions is a common task.

  2. Problem-Solving: Simplifying expressions is a key step in solving many mathematical problems. By reducing complex expressions to their simplest form, you can often reveal underlying patterns and relationships that make the problem easier to solve. In physics, for instance, simplifying equations of motion can help you determine the trajectory of a projectile. In economics, simplifying cost functions can help businesses make informed decisions about pricing and production.

  3. Real-World Applications: Algebraic simplification has numerous applications in real-world scenarios. It's used in engineering, computer science, finance, and many other fields. For example, engineers use simplification techniques to design circuits, optimize structures, and analyze systems. Computer scientists use simplification to write efficient algorithms and optimize code. Financial analysts use simplification to model financial markets and make investment decisions.

  4. Efficiency and Clarity: Simplified expressions are easier to work with and understand. They reduce the risk of errors and make it easier to communicate mathematical ideas. A simplified expression provides a clear and concise representation of the relationship between variables. Imagine trying to solve a complex equation with dozens of terms versus solving a simplified version with just a few terms. The simplified version is much easier to handle and less prone to errors.

  5. Critical Thinking: The process of simplifying expressions involves critical thinking and logical reasoning. It requires you to analyze the expression, identify the appropriate steps, and apply them correctly. These skills are valuable not only in mathematics but also in many other areas of life. When you simplify an expression, you're essentially breaking down a complex problem into smaller, more manageable parts. This is a valuable skill that can be applied to a wide range of situations.

In conclusion, simplifying algebraic expressions is a fundamental skill with far-reaching implications. Mastering this skill will not only help you succeed in mathematics but also equip you with valuable problem-solving and critical-thinking skills that are applicable in various aspects of life.

Conclusion

In this comprehensive guide, we have explored the process of simplifying the algebraic expression (12s^2 - 15s) - (s - 9). We broke down the process into manageable steps, including understanding the expression, applying the distributive property, combining like terms, and arriving at the final simplified expression: 12s^2 - 16s + 9.

We also discussed common mistakes that students often make when simplifying expressions, such as forgetting to distribute the negative sign, combining unlike terms, making arithmetic errors, and not following the order of operations. By understanding these pitfalls, you can develop strategies to avoid them and improve your accuracy.

Furthermore, we provided valuable tips and tricks for checking your work, including substituting a value for the variable, working backwards, using a computer algebra system, asking a peer or teacher, and practicing regularly. These techniques will empower you to verify your solutions and build confidence in your algebraic skills.

Finally, we emphasized the importance of simplifying algebraic expressions as a fundamental skill with wide-ranging applications in mathematics and various real-world scenarios. Mastering this skill is essential for success in higher-level mathematics courses and for solving problems in fields such as engineering, computer science, and finance.

Simplifying algebraic expressions is a journey of mathematical discovery. By understanding the underlying principles, practicing regularly, and employing effective checking techniques, you can unlock the power of algebraic manipulation and excel in your mathematical endeavors. Embrace the challenge, persevere through difficulties, and celebrate your successes along the way. The world of mathematics awaits your exploration!