Simplify The Expression (3 - 13x - 7x²) - (5x² + 12x - 10)

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Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic form, making them easier to understand and manipulate. This article delves into the process of simplifying a given algebraic expression. Our primary focus will be on the expression (3 - 13x - 7x²) - (5x² + 12x - 10), where we aim to find its most simplified equivalent. This involves combining like terms, which is a core concept in algebra. By understanding how to simplify such expressions, we can tackle more complex mathematical problems with greater confidence and accuracy. This skill is not only crucial for academic pursuits but also has practical applications in various fields, including engineering, finance, and computer science. So, let's embark on this mathematical journey to simplify the expression and uncover its true essence.

Understanding the Expression: A Step-by-Step Breakdown

At the heart of this mathematical problem lies the expression (3 - 13x - 7x²) - (5x² + 12x - 10). To effectively simplify this, we must first dissect it into its constituent parts and understand the operations involved. The expression is composed of two polynomials, each containing terms with different powers of the variable x. The first polynomial is (3 - 13x - 7x²), which includes a constant term (3), a linear term (-13x), and a quadratic term (-7x²). The second polynomial is (5x² + 12x - 10), which also contains a quadratic term (5x²), a linear term (12x), and a constant term (-10). The operation connecting these two polynomials is subtraction, which is a crucial aspect to consider during simplification. Understanding the structure of the expression and the role of each term is the foundation for the subsequent steps in the simplification process. We need to pay close attention to the signs of each term, as they play a significant role when we combine like terms. This initial breakdown allows us to approach the problem methodically and avoid common errors in algebraic manipulation. By carefully examining each component, we set the stage for a successful simplification.

Step 1: Distributing the Negative Sign

The crucial first step in simplifying the expression (3 - 13x - 7x²) - (5x² + 12x - 10) involves addressing the subtraction operation between the two polynomials. Specifically, we need to distribute the negative sign in front of the second polynomial, (5x² + 12x - 10). This means that we are essentially multiplying each term inside the parentheses of the second polynomial by -1. This process is vital because it changes the signs of each term within the second polynomial, which is essential for accurately combining like terms later on. When we distribute the negative sign, the expression transforms as follows:

  • -(5x²) becomes -5x²
  • -(12x) becomes -12x
  • -(-10) becomes +10

Therefore, after distributing the negative sign, the expression becomes: 3 - 13x - 7x² - 5x² - 12x + 10. This step is a critical juncture in the simplification process. A mistake in distributing the negative sign can lead to an incorrect final answer. By carefully applying this step, we ensure that the signs of the terms are correctly accounted for, paving the way for accurate combination of like terms in the subsequent steps. This meticulous attention to detail is what separates a correct solution from an incorrect one in algebraic simplification.

Step 2: Identifying and Grouping Like Terms

With the negative sign successfully distributed, the expression now stands as 3 - 13x - 7x² - 5x² - 12x + 10. The next pivotal step in the simplification process is to identify and group like terms. Like terms are those that have the same variable raised to the same power. In our expression, we can identify three categories of like terms:

  1. Constant terms: These are the terms without any variable, which in our case are 3 and +10.
  2. Linear terms: These terms contain the variable x raised to the power of 1. In our expression, these are -13x and -12x.
  3. Quadratic terms: These terms contain the variable x raised to the power of 2. In our expression, these are -7x² and -5x².

Once we have identified the like terms, we can group them together to make the simplification process clearer. This grouping does not change the value of the expression; it merely rearranges the terms to make it easier to combine them. The grouped expression looks like this: (-7x² - 5x²) + (-13x - 12x) + (3 + 10). This step is crucial for organizing the expression and preventing errors during the combination of terms. By visually grouping like terms, we can focus on each category separately, ensuring that we combine them correctly. This methodical approach is key to simplifying complex expressions accurately and efficiently.

Step 3: Combining Like Terms

Having identified and grouped the like terms in the expression (-7x² - 5x²) + (-13x - 12x) + (3 + 10), the next crucial step is to actually combine these terms. This involves performing the arithmetic operations (addition or subtraction) on the coefficients of the like terms. Let's break down the combination process for each group:

  1. Quadratic terms: We have -7x² and -5x². To combine them, we add their coefficients: -7 + (-5) = -12. Therefore, -7x² - 5x² simplifies to -12x².
  2. Linear terms: We have -13x and -12x. Adding their coefficients: -13 + (-12) = -25. So, -13x - 12x simplifies to -25x.
  3. Constant terms: We have 3 and +10. Adding them together: 3 + 10 = 13.

By combining each group of like terms, we effectively reduce the complexity of the expression. This step is where the actual simplification takes place, and it requires careful attention to the signs and coefficients of each term. Once we have combined all the like terms, we will have the simplified form of the original expression. This process of combining like terms is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. The accuracy in this step directly impacts the correctness of the final simplified expression.

The Simplified Form: -12x² - 25x + 13

After meticulously combining the like terms in the expression (-7x² - 5x²) + (-13x - 12x) + (3 + 10), we arrive at the simplified form. As we calculated in the previous step:

  • The quadratic terms -7x² and -5x² combined to give -12x².
  • The linear terms -13x and -12x combined to give -25x.
  • The constant terms 3 and 10 combined to give 13.

Therefore, the simplified form of the expression (3 - 13x - 7x²) - (5x² + 12x - 10) is -12x² - 25x + 13. This final form is a more concise and manageable representation of the original expression. It contains only three terms, each with a different power of the variable x, and no further simplification is possible. This simplified expression is equivalent to the original expression, meaning it will yield the same result for any given value of x. Finding the simplified form is often the goal in algebraic manipulation, as it makes the expression easier to work with in subsequent calculations or analyses. This result demonstrates the power of algebraic simplification in reducing complexity and revealing the underlying structure of mathematical expressions. The ability to simplify expressions is a cornerstone of mathematical proficiency.

Conclusion

In conclusion, the journey of simplifying the expression (3 - 13x - 7x²) - (5x² + 12x - 10) has been a valuable exercise in algebraic manipulation. We began by understanding the structure of the expression and the operations involved. Then, we meticulously distributed the negative sign, which was a critical step in ensuring the correct signs for each term. Next, we identified and grouped the like terms, organizing the expression into categories of quadratic, linear, and constant terms. The core of the simplification process involved combining these like terms, where we added or subtracted their coefficients to reduce the expression's complexity. Finally, we arrived at the simplified form: -12x² - 25x + 13. This simplified expression is equivalent to the original but is more concise and easier to work with. This process highlights the importance of careful attention to detail and methodical application of algebraic principles. The ability to simplify expressions is a fundamental skill in mathematics, with applications in various fields. By mastering this skill, we can tackle more complex problems with greater confidence and efficiency. The steps outlined in this article provide a clear framework for simplifying similar expressions, reinforcing the importance of understanding each step in the process. Ultimately, simplifying expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical structures and relationships.