Solving M = √(a - 3bc) Value And Integer Range

This article delves into the intricacies of solving a mathematical problem involving square roots and algebraic expressions. We'll break down each step, ensuring clarity and understanding for readers of all backgrounds. Our focus will be on question 11.1, which presents a scenario where m is defined as the square root of (a - 3bc), mathematically expressed as m = √(a - 3bc). We will tackle two sub-questions: firstly, determining the value of m given specific values for a, b, and c, and secondly, identifying the two consecutive integers between which the value of m lies without explicitly calculating m. This comprehensive guide aims to provide a step-by-step solution, enhancing your problem-solving skills in mathematics.

1.1.1 Determining the Value of m when a = -6, b = -2, and c = 5

In this section, we embark on a journey to pinpoint the exact value of m when provided with specific numerical values for the variables a, b, and c. The values in question are a = -6, b = -2, and c = 5. Our primary tool is the given equation: m = √(a - 3bc). The core concept here involves substituting the provided values into the equation and simplifying the expression to arrive at a final answer. This process highlights the fundamental algebraic principle of substitution, a cornerstone of mathematical problem-solving. By meticulously replacing each variable with its corresponding numerical value, we transform the abstract equation into a concrete arithmetic problem, paving the way for a straightforward calculation. The challenge lies not only in the substitution itself but also in ensuring the correct order of operations, a critical aspect of mathematical accuracy. The BODMAS or PEMDAS rule dictates the sequence in which we perform calculations: Parentheses/Brackets, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Ignoring this order can lead to erroneous results. In our case, we must first address the multiplication within the expression (a - 3bc) before we can tackle the subtraction and ultimately the square root. This meticulous approach ensures that we arrive at the correct value of m, expressed in its simplest root form. The process of simplification may involve identifying perfect square factors within the radicand (the expression under the square root) and extracting them to reduce the root to its simplest form. This skill is crucial for presenting mathematical answers in their most elegant and concise manner, adhering to the conventions of mathematical notation and communication. Thus, this sub-question serves not only as an exercise in substitution but also as a reinforcement of the importance of order of operations and the techniques for simplifying radical expressions. By mastering these skills, we build a solid foundation for tackling more complex mathematical problems in the future.

Let's begin by substituting the given values into the equation:

m = √(-6 - 3 * (-2) * 5)

Now, we follow the order of operations (PEMDAS/BODMAS). First, we perform the multiplication:

m = √(-6 - (-30))

Next, we simplify the subtraction:

m = √(-6 + 30)

m = √24

Finally, we simplify the square root by finding the largest perfect square factor of 24, which is 4:

m = √(4 * 6)

m = 2√6

Therefore, the value of m in its simplest root form is 2√6.

1.1.2 Determining Consecutive Integers Between Which m Lies

Now, let's shift our focus to the second part of the problem, where we aim to identify the two consecutive integers that bracket the value of m without actually calculating its exact decimal representation. We know from the previous section that m = 2√6. The core strategy here revolves around understanding the approximate value of √6 and then using that approximation to estimate the value of 2√6. This approach emphasizes the importance of estimation skills in mathematics, a valuable tool for quick checks and for gaining a sense of the magnitude of a result. We can begin by recognizing that 6 lies between two perfect squares: 4 and 9. This observation is crucial because we know the square roots of 4 and 9 are 2 and 3, respectively. Therefore, √6 must lie between 2 and 3. This establishes a range within which our square root resides, giving us a starting point for our estimation. Furthermore, since 6 is closer to 4 than it is to 9, we can infer that √6 is closer to 2 than it is to 3. This nuanced understanding allows us to refine our estimate, moving beyond a simple range and towards a more precise approximation. Now, we need to consider the multiplication by 2. Multiplying our estimated range by 2 will give us the range within which 2√6 lies. If √6 is between 2 and 3, then 2√6 will be between 22 and 23, which translates to between 4 and 6. However, we need to be more precise to identify the consecutive integers. To do this, we can refine our estimate of √6 further. Since √6 is closer to 2 than 3, let's consider a value slightly above 2, say 2.4. Multiplying this by 2 gives us 4.8. This suggests that 2√6 is greater than 4 but less than 5. To solidify our conclusion, we can also consider a slightly higher estimate for √6, say 2.5. Multiplying this by 2 gives us 5, indicating that 2√6 is less than 5. Thus, by carefully considering the position of 6 between perfect squares and by refining our estimations, we can confidently pinpoint the two consecutive integers between which m lies. This process highlights the power of estimation in mathematics, demonstrating how we can arrive at a reasonable answer without resorting to exact calculations. It also reinforces the importance of understanding the properties of square roots and how they relate to the integers.

We know that √4 = 2 and √9 = 3. Since 6 lies between 4 and 9, √6 lies between 2 and 3. We can estimate that √6 is approximately 2.4 (a more precise value is about 2.449).

Therefore, m = 2√6 is approximately 2 * 2.4 = 4.8.

Thus, the value of m lies between the consecutive integers 4 and 5.

In this comprehensive guide, we've meticulously dissected question 11.1, demonstrating the process of solving for m in the equation m = √(a - 3bc). We successfully determined the value of m when a = -6, b = -2, and c = 5, arriving at the simplified root form of 2√6. Furthermore, we skillfully identified the two consecutive integers (4 and 5) between which m lies, employing estimation techniques and a solid understanding of square roots. This exercise highlights the importance of several key mathematical concepts, including substitution, order of operations, simplification of radicals, and estimation skills. By mastering these concepts, we equip ourselves with the tools necessary to tackle a wide range of mathematical challenges. The ability to substitute values into equations, adhering to the correct order of operations, is fundamental to algebraic manipulation. Simplifying radicals allows us to express mathematical answers in their most concise and elegant form, adhering to mathematical conventions. Estimation skills, on the other hand, provide a valuable tool for quick checks and for gaining a sense of the magnitude of a result. These skills are not only essential for academic success but also for real-world problem-solving. Whether we're calculating dimensions for a construction project or estimating costs for a budget, mathematical proficiency is a valuable asset. This article serves as a testament to the power of structured problem-solving, demonstrating how a complex question can be broken down into manageable steps. By approaching mathematical problems with a clear methodology and a solid understanding of fundamental concepts, we can unlock solutions and enhance our mathematical abilities. We encourage readers to practice these techniques and to apply them to other mathematical problems, further solidifying their understanding and building their confidence. The journey of mathematical learning is a continuous process of exploration and discovery, and each problem solved is a step forward on that path.