Simplifying Algebraic Expressions Finding The Equivalent Of 9x - (x-8)(x+1) + 10x

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#h1 Understanding the Expression and Simplification Process

In this comprehensive guide, we will dissect the mathematical expression 9x - (x-8)(x+1) + 10x step-by-step to determine its equivalent form. This is a fundamental skill in algebra, often encountered in standardized tests and higher-level mathematics. Our primary goal is to simplify the given expression by expanding the terms, combining like terms, and ultimately arriving at a simplified quadratic expression. This involves applying the distributive property, understanding polynomial multiplication, and being meticulous with arithmetic operations. The ability to simplify algebraic expressions is crucial for solving equations, analyzing functions, and various other mathematical tasks. Let’s dive into the intricacies of this expression and unravel its simplified form together.

Our journey begins by acknowledging the expression at hand: 9x - (x-8)(x+1) + 10x. The first crucial step involves expanding the product of the two binomials, (x-8) and (x+1). This is achieved by applying the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last), which ensures that each term in the first binomial is multiplied by each term in the second binomial. Thus, we multiply x by x, x by 1, -8 by x, and -8 by 1. This meticulous expansion is the cornerstone of simplifying the expression. Next, we need to carefully handle the negative sign preceding the product. It’s a common mistake to forget to distribute the negative sign across all terms resulting from the expansion. After distributing the negative sign, we combine like terms, which involves adding or subtracting the coefficients of terms with the same variable and exponent. In our case, this means grouping the x² terms, the x terms, and the constant terms separately. The final step is to arrange the terms in the standard form of a quadratic expression, which is ax² + bx + c, where a, b, and c are constants. This allows us to easily identify the coefficients and the constant term, leading us to the equivalent expression. Through this process, we transform the original expression into its most simplified and easily understandable form.

#h2 Step-by-Step Breakdown of the Simplification

The core of solving this problem lies in meticulously following the order of operations and applying algebraic principles correctly. We start with the original expression: 9x - (x-8)(x+1) + 10x. The first step is to expand the product of the two binomials, (x-8)(x+1). Applying the FOIL method, we multiply x by x to get x², x by 1 to get x, -8 by x to get -8x, and -8 by 1 to get -8. Combining these terms, we have x² + x - 8x - 8. This simplifies to x² - 7x - 8. Now, we substitute this back into the original expression, being mindful of the negative sign in front of the parentheses. This gives us 9x - (x² - 7x - 8) + 10x. The next crucial step is to distribute the negative sign across the terms inside the parentheses, which means changing the sign of each term. This transforms the expression to 9x - x² + 7x + 8 + 10x. A common error here is to forget to distribute the negative sign to all terms, which can lead to an incorrect result. Now, we have a series of terms that we can combine. We group the like terms together: the x² terms, the x terms, and the constant terms. This gives us -x² + 9x + 7x + 10x + 8. Combining the x terms, we add their coefficients: 9 + 7 + 10 = 26. Thus, the expression simplifies to -x² + 26x + 8. This final expression is the simplified equivalent of the original expression, and it represents the same mathematical relationship in a more concise form.

This step-by-step simplification highlights the importance of careful attention to detail in algebraic manipulations. Each step, from expanding the binomials to distributing the negative sign and combining like terms, requires precision to arrive at the correct answer. Understanding and mastering these techniques is crucial for success in algebra and beyond.

#h3 Detailed Expansion of (x-8)(x+1)

To truly grasp the simplification process, let's delve into the detailed expansion of the binomial product (x-8)(x+1). This is a crucial component of the larger expression, and a thorough understanding of this step is vital. As mentioned earlier, we employ the FOIL method (First, Outer, Inner, Last) to ensure each term in the first binomial is multiplied by each term in the second binomial. This method is a systematic approach to polynomial multiplication, preventing us from missing any terms and ensuring accuracy. The FOIL method stands for:

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of the binomials.
  • Inner: Multiply the inner terms of the binomials.
  • Last: Multiply the last terms of each binomial.

Applying this method to (x-8)(x+1):

  1. First: Multiply the first terms, which are x and x. This gives us x * x = x². This term represents the product of the leading terms in each binomial and sets the stage for the quadratic component of the result.
  2. Outer: Multiply the outer terms, which are x and 1. This gives us x * 1 = x. This term contributes to the linear component of the resulting expression.
  3. Inner: Multiply the inner terms, which are -8 and x. This gives us -8 * x = -8x. This term also contributes to the linear component and highlights the importance of keeping track of signs.
  4. Last: Multiply the last terms, which are -8 and 1. This gives us -8 * 1 = -8. This term is the constant term of the resulting expression.

Now, we combine these terms: x² + x - 8x - 8. To simplify further, we combine the like terms, which are the x terms. Adding x and -8x, we get x - 8x = -7x. So, the expanded form of (x-8)(x+1) is x² - 7x - 8. This meticulous breakdown of the FOIL method demonstrates how each term is derived and combined to form the resulting quadratic expression. This expanded form is then substituted back into the original expression, where the negative sign and the remaining terms are handled to arrive at the final simplified form. Understanding this expansion is not only crucial for this problem but also for various other algebraic manipulations involving polynomial multiplication.

#h3 Handling the Negative Sign Correctly

A critical aspect of simplifying the expression 9x - (x-8)(x+1) + 10x is handling the negative sign that precedes the expanded form of (x-8)(x+1). This step is a common source of errors, and it's imperative to understand and apply the distributive property correctly to avoid mistakes. After expanding (x-8)(x+1) to x² - 7x - 8, we substitute this back into the original expression: 9x - (x² - 7x - 8) + 10x. The negative sign in front of the parentheses means we are subtracting the entire expression inside the parentheses. To do this correctly, we must distribute the negative sign to each term within the parentheses. This is equivalent to multiplying each term inside the parentheses by -1. Therefore, we multiply -1 by x², -1 by -7x, and -1 by -8.

  • Multiplying -1 by x² gives us -x². This changes the sign of the x² term from positive to negative, reflecting the subtraction operation.
  • Multiplying -1 by -7x gives us +7x. The negative sign is crucial here; a common mistake is to overlook it and incorrectly write -7x. This term now becomes positive, indicating that we are adding 7x rather than subtracting it.
  • Multiplying -1 by -8 gives us +8. Similarly, the constant term changes its sign from negative to positive. This means we are adding 8 to the expression.

After distributing the negative sign, the expression becomes 9x - x² + 7x + 8 + 10x. This step is vital because it sets the stage for accurately combining like terms. Failing to distribute the negative sign correctly can lead to an entirely different result. For instance, if we incorrectly left the expression as 9x - x² - 7x - 8 + 10x, the subsequent combination of like terms would yield an incorrect simplified expression. This underscores the significance of being meticulous with each step in the simplification process, especially when dealing with negative signs and parentheses. The correct distribution of the negative sign ensures that we are performing the intended mathematical operations, leading us to the accurate equivalent expression. By mastering this step, we strengthen our ability to tackle more complex algebraic problems with confidence.

#h3 Combining Like Terms for Final Simplification

Once we have expanded the binomial product and correctly distributed the negative sign, the next pivotal step is to combine like terms. This process involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. In our expression, 9x - x² + 7x + 8 + 10x, we have three types of terms: the x² term, the x terms, and the constant term. The x² term is -x². Since there are no other x² terms in the expression, it remains as -x² in the simplified form. Next, we identify the x terms: 9x, 7x, and 10x. These terms have the same variable (x) raised to the same power (1), which means they are like terms and can be combined. To combine them, we add their coefficients: 9 + 7 + 10. This gives us 26. Therefore, the combined x term is 26x. Lastly, we have the constant term, which is 8. Since there are no other constant terms in the expression, it remains as 8 in the simplified form. Now, we assemble the simplified expression by writing the combined terms in the standard form of a quadratic expression, which is ax² + bx + c, where a, b, and c are constants. In our case, a = -1, b = 26, and c = 8. Thus, the simplified expression is -x² + 26x + 8. This is the final equivalent expression, representing the same mathematical relationship as the original expression but in a more concise and easily understandable form. Combining like terms is a fundamental skill in algebra, essential for simplifying expressions, solving equations, and performing various other mathematical operations. It allows us to consolidate terms and reduce the complexity of an expression, making it easier to work with. Mastery of this technique is crucial for success in algebra and beyond. By carefully identifying and combining like terms, we ensure accuracy and arrive at the correct simplified expression.

#h2 Conclusion: The Equivalent Expression

In summary, after meticulously expanding, distributing, and combining like terms in the expression 9x - (x-8)(x+1) + 10x, we arrive at the equivalent expression -x² + 26x + 8. This detailed walkthrough highlights the importance of each step in the simplification process, from the FOIL method to handling the negative sign and combining like terms. The correct application of these techniques is crucial for accurate algebraic manipulation. The equivalent expression, -x² + 26x + 8, represents the same mathematical relationship as the original expression but in a simplified, standard quadratic form. This simplified form is not only easier to work with but also provides valuable insights into the behavior and properties of the expression. Understanding how to simplify algebraic expressions is a fundamental skill in mathematics, essential for solving equations, analyzing functions, and tackling more advanced mathematical concepts. By mastering these techniques, students and practitioners can approach complex problems with confidence and accuracy. This example serves as a testament to the power of algebraic simplification and its role in making mathematical expressions more manageable and understandable.