Penelope's Quadratic Equation Solution Correcting Mistakes In Completing The Square

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In the realm of mathematics, solving quadratic functions is a fundamental skill. Among the various methods available, completing the square stands out as a powerful technique for transforming a quadratic equation into a more manageable form. This method not only aids in finding the solutions (or roots) of the equation but also reveals the vertex of the parabola represented by the quadratic function. In this article, we delve into the journey of Penelope, a diligent mathematician, as she tackles a quadratic function using the method of completing the square. We will dissect her steps, identify any potential missteps, and provide a comprehensive guide to ensure accuracy and understanding. This exploration will serve as a valuable resource for students and enthusiasts alike, aiming to master this essential algebraic technique.

Penelope's Approach to Solving the Quadratic Function

Penelope embarks on her mathematical journey with the quadratic function f(x) = 4x² + 8x + 1. Her goal is to determine the solutions, or roots, of this function by employing the method of completing the square. Let's meticulously examine her steps to understand her approach and identify any areas for improvement.

Step 1: Isolating the Constant Term

Penelope initiates the process by isolating the constant term on one side of the equation. She subtracts 1 from both sides, resulting in the equation: -1 = 4x² + 8x. This step is crucial as it sets the stage for manipulating the quadratic expression to complete the square.

Step 2: Factoring out the Leading Coefficient

Next, Penelope focuses on the terms containing the variable x. She factors out the leading coefficient, which is 4 in this case, from the right side of the equation. This yields: -1 = 4(x² + 2x). Factoring out the leading coefficient is a pivotal step in the completing the square method, as it ensures that the coefficient of the x² term inside the parenthesis is 1, a necessary condition for completing the square.

Step 3: Completing the Square

This is where the core of the method comes into play. Penelope aims to complete the square inside the parenthesis. To do this, she takes half of the coefficient of the x term (which is 2), squares it (resulting in 1), and adds it inside the parenthesis. However, she makes a critical error here. She adds 1 inside the parenthesis but fails to account for the factor of 4 outside the parenthesis. To maintain the equation's balance, she must add 4 times 1, which is 4, to the left side of the equation. Penelope's equation becomes: -1 + 1 = 4(x² + 2x + 1). This step is where the first mistake appears. She added 1 to the left side, but she should have added 4 to compensate for the factor of 4 outside the parenthesis.

Step 4: Expressing as a Perfect Square

Penelope proceeds to express the quadratic expression inside the parenthesis as a perfect square. The expression x² + 2x + 1 can be written as (x + 1)². Thus, the equation becomes: 0 = 4(x + 1)². This step correctly applies the principle of completing the square, transforming the quadratic expression into a squared term, which is essential for solving the equation.

Step 5: Solving for x

Penelope continues by dividing both sides of the equation by 4, which gives: 0 = (x + 1)². Then, she takes the square root of both sides, leading to: 0 = x + 1. Finally, she subtracts 1 from both sides to solve for x, obtaining: x = -1. This final step correctly isolates x and provides a potential solution for the quadratic equation. However, due to the earlier error in Step 3, this solution needs to be verified and corrected.

Identifying and Rectifying the Error

The critical error in Penelope's solution arises in Step 3, where she fails to account for the factor of 4 when adding to both sides of the equation. To correctly complete the square, she needed to add 4 to the left side, not 1. Let's rectify this error and provide the correct solution.

Corrected Step 3: Completing the Square (Corrected)

To balance the equation, we need to add 4 times the value added inside the parenthesis to the left side. So, the correct equation should be: -1 + 4 = 4(x² + 2x + 1). This simplifies to 3 = 4(x + 1)².

Corrected Step 4: Expressing as a Perfect Square (Corrected)

As before, the quadratic expression inside the parenthesis can be written as a perfect square: 3 = 4(x + 1)².

Corrected Step 5: Solving for x (Corrected)

  1. Divide both sides by 4: 3/4 = (x + 1)²
  2. Take the square root of both sides: ±√(3/4) = x + 1, which simplifies to ±√3 / 2 = x + 1
  3. Subtract 1 from both sides: x = -1 ± √3 / 2

Therefore, the correct solutions for the quadratic equation are x = -1 + √3 / 2 and x = -1 - √3 / 2. This correction demonstrates the importance of meticulous attention to detail when completing the square.

The Significance of Completing the Square

Completing the square is not merely a method for solving quadratic equations; it is a gateway to understanding the structure and properties of these equations. This technique transforms a quadratic function from its standard form (ax² + bx + c) into vertex form (a(x - h)² + k), where (h, k) represents the vertex of the parabola. The vertex is a critical point as it indicates the maximum or minimum value of the function.

Applications Beyond Solving Equations

Beyond finding solutions and identifying the vertex, completing the square has applications in various mathematical contexts:

  • Graphing Quadratic Functions: The vertex form directly reveals the vertex, axis of symmetry, and the direction of the parabola's opening.
  • Deriving the Quadratic Formula: The quadratic formula itself is derived using the method of completing the square.
  • Optimization Problems: Finding maximum or minimum values in real-world scenarios often involves completing the square.

Mastering the Technique

To master completing the square, one must focus on the underlying principles and practice consistently. The key is to understand why each step is performed and how it contributes to the overall goal. Common mistakes, like the one Penelope made, often stem from a lack of attention to detail or a misunderstanding of the balancing act required in algebraic manipulations. By carefully working through examples and paying close attention to each step, learners can develop a strong command of this technique.

Conclusion: Lessons from Penelope's Journey

Penelope's journey through solving the quadratic function by completing the square provides valuable insights into the process and the importance of accuracy. While she made an initial error in balancing the equation, her perseverance and the subsequent correction highlight the essence of mathematical problem-solving: meticulous attention to detail and a willingness to learn from mistakes.

Completing the square is a powerful technique with far-reaching applications. By understanding its principles and practicing diligently, anyone can master this method and unlock a deeper understanding of quadratic functions. Penelope's experience serves as a reminder that even in mathematics, the journey of discovery is as important as the destination. The ability to identify errors, correct them, and learn from them is a hallmark of a proficient mathematician. This exploration not only clarifies the method of completing the square but also emphasizes the broader principles of mathematical rigor and problem-solving acumen.