Evaluate M^2 + 8m - 3 When M = 0 A Step-by-Step Guide

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In mathematics, evaluating expressions is a fundamental skill. It involves substituting given values for variables within an expression and then simplifying the expression using the order of operations. This process is crucial in various areas of mathematics, including algebra, calculus, and beyond. In this comprehensive guide, we will delve into the process of evaluating the expression m² + 8m - 3 when m = 0. We will break down each step with clarity, ensuring a thorough understanding of the underlying concepts. Understanding how to substitute values into expressions correctly and perform the necessary arithmetic operations is a cornerstone of mathematical proficiency. This skill not only helps in solving equations and simplifying complex formulas but also builds a strong foundation for advanced mathematical studies. By mastering the art of evaluating expressions, you equip yourself with a powerful tool for tackling mathematical challenges.

The expression we are tasked with evaluating is m² + 8m - 3. This is a polynomial expression in the variable m. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Our expression contains three terms: m², 8m, and -3. The first term, m², represents m multiplied by itself. The second term, 8m, represents 8 times m. The third term, -3, is a constant term. Understanding the structure of the expression is key to accurately evaluating it. Each term plays a specific role, and the order of operations dictates how we combine these terms. Recognizing the variable (m) and the constant terms allows us to substitute the given value correctly. In this case, we are given that m = 0, which simplifies the evaluation process significantly. By identifying the components of the expression, we set the stage for a systematic approach to finding its value.

The core of evaluating an expression is the substitution step. We are given that m = 0, and our goal is to replace every instance of m in the expression m² + 8m - 3 with the value 0. This process involves careful attention to detail to ensure that all occurrences of the variable are properly replaced. When we substitute m = 0, the expression transforms as follows:

  • Original expression: m² + 8m - 3
  • After substitution: (0)² + 8(0) - 3

Notice that we've replaced each m with a 0, using parentheses to maintain clarity, especially in the term 8m. The parentheses emphasize that 8 is being multiplied by 0. This substitution is a critical step because it converts the algebraic expression into a numerical expression, which we can then simplify using arithmetic operations. Accurate substitution is essential for obtaining the correct result. A mistake in this step can lead to an incorrect final answer. By meticulously replacing each variable with its given value, we lay the groundwork for the subsequent simplification.

After substituting m = 0 into the expression, we now have the numerical expression (0)² + 8(0) - 3. The next step is to simplify this expression using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following PEMDAS ensures that we perform operations in the correct sequence, leading to the accurate result.

  1. Exponents: First, we address the exponent. (0)² means 0 raised to the power of 2, which is 0 * 0 = 0. So, (0)² simplifies to 0.
  2. Multiplication: Next, we perform the multiplication. We have 8(0), which means 8 multiplied by 0. Any number multiplied by 0 equals 0. Thus, 8(0) simplifies to 0.
  3. Addition and Subtraction: Now, we are left with 0 + 0 - 3. Adding 0 and 0 gives us 0. Then, subtracting 3 from 0 results in -3.

Therefore, the simplified expression is -3. This step-by-step simplification demonstrates how the order of operations guides us to the final value. Each operation is performed in its proper sequence, ensuring that we arrive at the correct answer. By carefully applying PEMDAS, we can confidently simplify complex numerical expressions.

After substituting m = 0 into the expression m² + 8m - 3 and simplifying, we arrive at the final answer: -3. This means that when m is equal to 0, the value of the expression m² + 8m - 3 is -3. The process of reaching this answer involved several key steps: understanding the expression, substituting the value of the variable, and simplifying the resulting numerical expression using the order of operations. Each step is crucial for accuracy, and a thorough understanding of these steps is essential for successfully evaluating algebraic expressions. The final answer, -3, represents the specific value of the expression under the given condition. This result is not just a number; it's the outcome of a systematic mathematical process. By mastering this process, you can confidently tackle a wide range of algebraic evaluation problems.

To further solidify your understanding of evaluating expressions, let's explore some additional examples. These examples will showcase different expressions and values, reinforcing the key steps of substitution and simplification.

Example 1: Evaluate the expression 2x² - 5x + 1 when x = 2.

  1. Substitution: Replace x with 2: 2(2)² - 5(2) + 1
  2. Exponents: Calculate 2²: 2(4) - 5(2) + 1
  3. Multiplication: Perform the multiplications: 8 - 10 + 1
  4. Addition and Subtraction: Simplify from left to right: -2 + 1 = -1

The final answer is -1.

Example 2: Evaluate the expression y³ + 3y - 4 when y = -1.

  1. Substitution: Replace y with -1: (-1)³ + 3(-1) - 4
  2. Exponents: Calculate (-1)³: -1 + 3(-1) - 4
  3. Multiplication: Perform the multiplication: -1 - 3 - 4
  4. Addition and Subtraction: Simplify from left to right: -4 - 4 = -8

The final answer is -8.

Example 3: Evaluate the expression a² - 4b + 6 when a = 3 and b = 1.

  1. Substitution: Replace a with 3 and b with 1: (3)² - 4(1) + 6
  2. Exponents: Calculate 3²: 9 - 4(1) + 6
  3. Multiplication: Perform the multiplication: 9 - 4 + 6
  4. Addition and Subtraction: Simplify from left to right: 5 + 6 = 11

The final answer is 11.

These examples illustrate the consistent application of substitution and simplification in evaluating expressions. By working through various examples, you can develop confidence and proficiency in this fundamental mathematical skill.

When evaluating expressions, it's essential to be aware of common pitfalls that can lead to incorrect answers. Identifying and avoiding these common mistakes can significantly improve your accuracy and understanding.

  1. Incorrect Substitution: A frequent error is misplacing or mis-substituting values. Always double-check that you've replaced the variable with the correct number and that you haven't missed any instances of the variable in the expression. For example, if the expression is 2x² + 3x - 1 and x = 4, ensure you substitute 4 for x in both the 2x² and 3x terms.
  2. Order of Operations Errors: Not following the order of operations (PEMDAS) is a common mistake. For instance, in the expression 5 + 2 * 3, you must multiply 2 and 3 before adding 5. Failing to do so can lead to a wrong answer. Always remember to address Parentheses, Exponents, Multiplication and Division (from left to right), and then Addition and Subtraction (from left to right).
  3. Sign Errors: Dealing with negative numbers can be tricky. A simple sign error can change the entire outcome. For example, (-2)² is 4, but -2² is -4 because the exponent only applies to the 2, not the negative sign. Pay close attention to the signs and use parentheses when necessary to avoid confusion.
  4. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition, subtraction, multiplication, or division, can occur. Always double-check your calculations, especially in multi-step problems. It can be helpful to write out each step clearly and use a calculator for more complex calculations.
  5. Forgetting the Distributive Property: When an expression involves parentheses and multiplication, remember to distribute. For example, in 3(x + 2), you need to multiply both x and 2 by 3. Forgetting to distribute can lead to an incorrect simplification.

By being mindful of these common mistakes, you can significantly improve your accuracy in evaluating expressions. Practice and attention to detail are key to mastering this skill.

In conclusion, evaluating the expression m² + 8m - 3 when m = 0 is a fundamental exercise in algebra that highlights several critical mathematical concepts. Throughout this guide, we've systematically walked through the process, emphasizing the importance of each step. We began by understanding the expression, recognizing its components, and identifying the variable and constant terms. Next, we meticulously substituted the given value, m = 0, into the expression, replacing every instance of m with 0. This substitution transformed the algebraic expression into a numerical one, which we then simplified. The simplification process involved applying the order of operations (PEMDAS) to ensure accuracy. We addressed exponents, multiplication, and finally, addition and subtraction, leading us to the final answer of -3. Moreover, we explored additional examples to reinforce the concepts and provide a broader understanding of evaluating expressions in various contexts. These examples demonstrated the consistent application of substitution and simplification, showcasing the versatility of these skills. We also discussed common mistakes to avoid, such as incorrect substitution, errors in the order of operations, sign errors, arithmetic errors, and neglecting the distributive property. Recognizing these pitfalls can significantly enhance your accuracy and confidence in evaluating expressions. Mastering the evaluation of expressions is not just about arriving at the correct answer; it's about developing a systematic approach to problem-solving. This skill is a building block for more advanced mathematical topics and is essential for success in algebra and beyond. By practicing and applying the principles outlined in this guide, you can confidently tackle a wide range of algebraic challenges.