How To Simplify The Expression -6x^2 - 7 + 10x + 11x^2 - 14x

Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to reduce complex equations into more manageable forms, making them easier to understand and solve. This article delves into the process of simplifying a specific algebraic expression: $-6x^2 - 7 + 10x + 11x^2 - 14x$. By combining like terms and rearranging the expression, we can arrive at a simplified form that retains the original’s mathematical integrity while being more concise and clear. This process is crucial in various areas of mathematics, from basic algebra to more advanced calculus and beyond. Let's embark on this mathematical journey and unravel the steps involved in simplifying this expression.

Understanding the Expression

The given expression is $-6x^2 - 7 + 10x + 11x^2 - 14x$. To simplify this expression effectively, it's essential to first understand its components. This algebraic expression comprises several terms, each with its own characteristics. We have terms with $x^2$, terms with $x$, and constant terms. Recognizing these components is the first step toward simplification. The terms $-6x^2$ and $11x^2$ are quadratic terms, meaning they involve the variable x raised to the power of 2. The terms $10x$ and $-14x$ are linear terms, which have x raised to the power of 1. Lastly, we have the constant term $-7$, which does not involve any variables. By categorizing these terms, we can see how they can be combined to simplify the expression. This initial analysis provides a roadmap for the subsequent steps in the simplification process. Understanding the nature and types of terms present is pivotal for efficient algebraic manipulation. Without this foundational understanding, the process of simplification can become convoluted and prone to errors. Therefore, a thorough initial assessment is always recommended.

Combining Like Terms

The core of simplifying algebraic expressions lies in the principle of combining like terms. Like terms are those that contain the same variable raised to the same power. In our expression, $-6x^2 - 7 + 10x + 11x^2 - 14x$, we identify two sets of like terms: the quadratic terms ($-6x^2$ and $11x^2$) and the linear terms ($10x$ and $-14x$). To combine these terms, we simply add their coefficients. For the quadratic terms, we have $-6x^2 + 11x^2$. Adding the coefficients -6 and 11 gives us 5, so the combined term is $5x^2$. Similarly, for the linear terms, we have $10x - 14x$. Adding the coefficients 10 and -14 results in -4, so the combined term is $-4x$. The constant term, -7, does not have any like terms in the expression, so it remains as is. This process of combining like terms is a fundamental aspect of algebraic simplification. It allows us to reduce the number of terms in an expression, making it more concise and easier to work with. By focusing on the coefficients and keeping the variables and their powers consistent, we can accurately combine like terms and move closer to the simplified form of the expression.

Rearranging the Expression

After combining like terms, the expression becomes $5x^2 - 4x - 7$. While this is a simplified form, it's often beneficial to rearrange the terms to follow a standard convention. In mathematics, expressions are typically written in descending order of the variable's power. This means the term with the highest power of x comes first, followed by terms with lower powers, and finally the constant term. In our case, the term with the highest power of x is $5x^2$, followed by $-4x$, and then the constant term $-7$. Therefore, the expression is already arranged in the standard form: $5x^2 - 4x - 7$. Rearranging expressions in this manner makes them easier to read and compare with other expressions. It also aligns with the common practice in mathematical notation, which enhances clarity and reduces the potential for misinterpretation. This step, while seemingly simple, plays a significant role in the overall presentation and understanding of algebraic expressions. By adhering to the standard convention, we ensure that the expression is not only simplified but also presented in the most accessible and mathematically sound way.

Final Simplified Form

Having combined like terms and rearranged the expression, we arrive at the final simplified form: $5x^2 - 4x - 7$. This is the most concise representation of the original expression, $-6x^2 - 7 + 10x + 11x^2 - 14x$. The simplified form retains all the mathematical properties of the original expression but is much easier to work with. It clearly shows the quadratic, linear, and constant components of the expression, making it straightforward to analyze and use in further calculations or problem-solving scenarios. The process of simplifying expressions is not just about making them shorter; it's about making them clearer and more manageable. In this case, we have successfully transformed a seemingly complex expression into a simple quadratic trinomial. This skill is invaluable in various areas of mathematics, from solving equations to graphing functions and beyond. The final simplified form, $5x^2 - 4x - 7$, is the result of careful application of algebraic principles and demonstrates the power of simplification in mathematics.

Conclusion

In conclusion, we have successfully simplified the expression $-6x^2 - 7 + 10x + 11x^2 - 14x$ to its final form, $5x^2 - 4x - 7$. This process involved several key steps, including understanding the components of the expression, combining like terms, and rearranging the expression in standard form. Each of these steps is crucial in simplifying algebraic expressions and is a fundamental skill in mathematics. The ability to simplify expressions is not only about obtaining a shorter form but also about gaining a clearer understanding of the underlying mathematical relationships. The simplified expression, $5x^2 - 4x - 7$, is easier to analyze, interpret, and use in further mathematical operations. This exercise highlights the importance of algebraic manipulation in mathematics and demonstrates how complex expressions can be reduced to simpler, more manageable forms. By mastering these simplification techniques, students and practitioners of mathematics can approach more complex problems with confidence and clarity. The journey from the initial expression to the simplified form underscores the elegance and power of mathematical simplification.