Solve For M If 2x^2+3x+1=(x+2)(2x-1)+m.

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Introduction

In this article, we will delve into the algebraic equation 2x^2 + 3x + 1 = (x + 2)(2x - 1) + m and meticulously solve for the unknown variable m. This equation combines polynomial expressions and requires a systematic approach to simplify and isolate m. Understanding how to solve such equations is a fundamental skill in algebra, often encountered in various mathematical contexts and standardized tests like the CBSE board exams. This detailed exploration will provide a step-by-step guide, ensuring clarity and comprehension for students and enthusiasts alike. Mastering these techniques is crucial for advancing in mathematics and tackling more complex problems. The ability to manipulate algebraic expressions and solve for unknowns is a cornerstone of mathematical proficiency. Let's embark on this algebraic journey to uncover the value of m.

Step-by-Step Solution

Expanding the Right Side

To begin, we focus on the equation 2x^2 + 3x + 1 = (x + 2)(2x - 1) + m. The first critical step is to expand the right side of the equation, specifically the product (x + 2)(2x - 1). We use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to ensure every term in the first parenthesis is multiplied by every term in the second parenthesis. This process involves multiplying x by 2x and -1, and then multiplying 2 by 2x and -1. This meticulous expansion is vital for simplifying the equation and moving closer to isolating m. Neglecting this step or performing it incorrectly will lead to an inaccurate solution. By correctly expanding the product, we set the stage for further simplification and ultimately, solving for m. This foundational algebraic manipulation is key to unraveling the equation. Once we have expanded the product, we can combine like terms and continue our journey towards finding the value of m.

Expanding (x + 2)(2x - 1):

  • First: x * 2x = 2x^2
  • Outer: x * -1 = -x
  • Inner: 2 * 2x = 4x
  • Last: 2 * -1 = -2

Combining these terms, we get 2x^2 - x + 4x - 2. Simplifying further, we combine the like terms -x and 4x to get 3x. Thus, the expanded form is 2x^2 + 3x - 2. This expansion is a critical step in solving the equation, as it allows us to rewrite the original equation in a more manageable form. The correct application of the distributive property and careful combination of like terms ensures we maintain the equation's integrity. With this expanded form, we are now ready to substitute it back into the original equation and proceed with isolating m. This meticulous approach to algebraic manipulation is essential for achieving accurate results.

Substituting and Simplifying

After expanding the right side, our equation transforms from 2x^2 + 3x + 1 = (x + 2)(2x - 1) + m to 2x^2 + 3x + 1 = 2x^2 + 3x - 2 + m. This substitution is a pivotal step as it replaces the complex product with its simplified form, making the equation easier to manipulate. The next crucial action is to simplify the equation by identifying and canceling out like terms on both sides. Notice that the terms 2x^2 and 3x appear on both the left and right sides of the equation. These identical terms can be eliminated by subtracting them from both sides, a valid algebraic operation that maintains the equation's balance. This simplification process is vital for isolating m, the variable we are solving for. By removing the common terms, we streamline the equation and reduce it to a more manageable form. This step demonstrates a key principle in algebra: simplifying expressions to make problem-solving more efficient. With the common terms canceled, we are one step closer to isolating m and determining its value.

Canceling out the like terms (2x^2 and 3x) from both sides, the equation simplifies dramatically. This process involves subtracting 2x^2 from both sides, which eliminates the 2x^2 term. Similarly, subtracting 3x from both sides eliminates the 3x term. The equation now appears much simpler, making it easier to isolate m. This simplification highlights the power of algebraic manipulation in reducing complex equations to their essential components. By carefully applying the principles of algebraic equality, we ensure that the equation remains balanced while we progress towards the solution. This step is not only crucial for solving this specific problem but also illustrates a fundamental technique applicable to a wide range of algebraic equations. The simplified equation sets the stage for the final step in solving for m, which involves isolating m on one side of the equation.

Isolating m

Following the simplification, our equation stands as 1 = -2 + m. This concise form makes isolating m a straightforward task. To isolate m, we need to eliminate the -2 from the right side of the equation. We achieve this by performing the inverse operation: adding 2 to both sides of the equation. This is a fundamental principle in algebra – performing the same operation on both sides maintains the equation's balance and allows us to isolate the variable of interest. Adding 2 to both sides effectively cancels out the -2 on the right side, leaving m isolated. This step showcases the elegance and efficiency of algebraic manipulation. By applying the appropriate operation, we directly arrive at the solution for m. This final step underscores the importance of understanding inverse operations in solving equations. With m now isolated, we can clearly state its value, completing the solution to the problem. The process of isolating m is a culmination of the previous steps, demonstrating the logical progression inherent in algebraic problem-solving.

Adding 2 to both sides:

  • 1 + 2 = -2 + m + 2
  • 3 = m

Thus, we find that m = 3. This is the solution to the equation, and it represents the value of m that makes the original equation true. This final answer is the result of a careful and methodical application of algebraic principles, including expansion, simplification, and isolation of the variable. The process demonstrates the power of algebra in solving for unknowns and highlights the importance of precision and attention to detail in mathematical problem-solving. The solution m = 3 not only answers the specific question but also reinforces the understanding of algebraic techniques that can be applied to a wide variety of mathematical problems. This step-by-step solution provides a clear and comprehensive guide for anyone seeking to understand and master algebraic equation solving.

Conclusion

In summary, we successfully solved the equation 2x^2 + 3x + 1 = (x + 2)(2x - 1) + m by meticulously following algebraic principles. Our journey began with expanding the product (x + 2)(2x - 1), which transformed the equation into a more manageable form. Subsequently, we simplified the equation by canceling out like terms on both sides, a critical step in isolating m. Finally, we isolated m by adding 2 to both sides of the equation, revealing the solution m = 3. This step-by-step process demonstrates the systematic approach required for solving algebraic equations. Each step was carefully executed to maintain the equation's balance and ensure an accurate solution. The techniques used in this solution, such as expansion, simplification, and isolation of the variable, are fundamental to algebra and applicable to a wide range of mathematical problems. Understanding and mastering these techniques is essential for success in algebra and related fields. The solution m = 3 not only provides an answer to the specific question but also reinforces the importance of precision and methodical problem-solving in mathematics.

The solution to this equation exemplifies the power and elegance of algebraic manipulation. By breaking down the problem into smaller, manageable steps, we were able to systematically solve for the unknown variable m. The ability to solve such equations is a crucial skill in mathematics, with applications ranging from basic algebra to advanced calculus and beyond. The principles demonstrated in this solution, such as the distributive property, combining like terms, and maintaining equation balance, are cornerstones of mathematical problem-solving. Furthermore, this exercise highlights the importance of clarity and organization in mathematical work. By presenting the solution in a step-by-step manner, we ensure that the logic and reasoning are transparent and easy to follow. This approach not only aids in understanding the solution but also promotes good mathematical practice. In conclusion, the solution to 2x^2 + 3x + 1 = (x + 2)(2x - 1) + m underscores the significance of algebraic skills and the value of a systematic approach to problem-solving.