Analysis Of Marcus's Solution To X² - 10x + 25 = 0

#h1 Analyzing Marcus's Solution to the Quadratic Equation

In this article, we will dissect Marcus's attempt to solve the quadratic equation x² - 10x + 25 = 0. Marcus employed the quadratic formula, a fundamental tool in algebra, to find the roots of this equation. We will meticulously examine each step of his work, pinpointing any errors or inconsistencies, and ultimately determining the accuracy of his solution. Understanding the correct application of the quadratic formula is crucial for anyone studying algebra, and this analysis will provide valuable insights into the process. This article aims to help readers grasp the nuances of the quadratic formula and enhance their problem-solving skills in algebra. Marcus's attempt serves as a great learning opportunity to highlight common mistakes and solidify the correct methodology for solving quadratic equations.

The quadratic formula is a cornerstone of algebra, providing a universal method for solving quadratic equations of the form ax² + bx + c = 0. It states that the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

This formula is derived by completing the square in the general quadratic equation and is applicable regardless of whether the quadratic equation can be factored easily. The term inside the square root, b² - 4ac, is known as the discriminant, which provides critical information about the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. Mastering the quadratic formula is essential for students and professionals in various fields, including mathematics, physics, engineering, and computer science. The formula's versatility and broad applicability make it an indispensable tool in problem-solving. Understanding its derivation and the significance of the discriminant further enhances one's algebraic proficiency.

When applying the quadratic formula, careful attention must be paid to the signs and values of the coefficients a, b, and c. A common mistake is misidentifying these coefficients or incorrectly substituting them into the formula. The order of operations must also be strictly followed to ensure accurate calculations. Furthermore, the simplification of the square root and the final fraction requires a thorough understanding of algebraic manipulations. For instance, if the discriminant is a perfect square, the square root can be simplified to an integer, leading to rational roots. If the discriminant is not a perfect square, the roots will involve radicals, and further simplification may be possible. Accurate application of the quadratic formula demands both precision and a solid foundation in algebraic principles. Through practice and careful analysis, one can develop the proficiency needed to solve quadratic equations effectively.

Step 1: The Quadratic Formula

Marcus correctly stated the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

This is the standard formula used to solve quadratic equations of the form ax² + bx + c = 0. This initial step demonstrates that Marcus understands the fundamental tool needed to solve the problem. The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0 and provides a direct method for finding the roots, regardless of whether the equation can be easily factored. The correct statement of the quadratic formula is a crucial first step, as any error here would propagate through the rest of the solution. This formula is a cornerstone of algebra, and its accurate recall and application are essential skills for students and professionals alike.

The quadratic formula is not just a memorized equation; it represents a powerful and versatile technique. Its derivation provides insight into why it works, and understanding this derivation can enhance one's confidence in using the formula. The formula's components, particularly the discriminant (b² - 4ac), offer valuable information about the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots. The ability to interpret the discriminant is a key aspect of mastering the quadratic formula. Furthermore, the quadratic formula serves as a foundation for more advanced algebraic concepts, highlighting its fundamental importance in mathematics.

To effectively use the quadratic formula, it is crucial to identify the coefficients a, b, and c correctly from the quadratic equation. These coefficients correspond to the numerical values multiplying the x², x, and constant terms, respectively. Careful attention must be paid to the signs of these coefficients, as errors in sign can lead to incorrect solutions. For instance, in the equation 2x² - 5x + 3 = 0, the coefficients are a = 2, b = -5, and c = 3. Accurate identification of the coefficients is a critical prerequisite for the correct application of the quadratic formula. This skill requires practice and attention to detail, but it is essential for successfully solving quadratic equations using this method.

Step 2: Substitution into the Formula

In the given quadratic equation x² - 10x + 25 = 0, the coefficients are a = 1, b = -10, and c = 25. Marcus's substitution in the second step is:

x = (-(-10) ± √((-10)² - 4(1)(25))) / 2(1)

Here, we observe a critical error. Marcus wrote +(10)² instead of (-10)² inside the square root. While the result of squaring either 10 or -10 is the same (100), the initial substitution reflects a misunderstanding of the formula's direct application. This type of error, although seemingly minor, can lead to significant discrepancies in the final solution. Correct substitution is paramount when using the quadratic formula, as it ensures that the subsequent calculations are based on accurate values. This step highlights the importance of careful attention to detail and a thorough understanding of the formula's structure.

To avoid errors in substitution, it is helpful to write out the quadratic formula explicitly with the coefficients labeled before plugging in the numerical values. This systematic approach reduces the chances of misplacing signs or substituting values incorrectly. For example, one might write: x = (-b ± √(b² - 4ac)) / 2a, then identify a = 1, b = -10, c = 25, and finally substitute these values into the formula. This method provides a clear and organized way to ensure accurate substitution. Using a systematic approach can significantly improve one's success rate in solving quadratic equations with the quadratic formula.

Furthermore, it is beneficial to double-check the substituted values against the original equation to ensure accuracy. This verification process can help catch any subtle errors that might have been overlooked during the initial substitution. For instance, one might ask,