Analysis Of Noah's Solution To 5x² = 45

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Introduction

In this article, we delve into Noah's attempt to solve the quadratic equation 5x² = 45. We will meticulously examine each step of his solution to identify any potential errors and provide a comprehensive explanation of the correct method. Quadratic equations are a fundamental concept in algebra, and understanding how to solve them accurately is crucial for further mathematical studies. This analysis will not only highlight the correct approach but also emphasize common pitfalls to avoid when dealing with such equations.

Noah's Attempted Solution

Noah's steps are as follows:

  1. 5x² - 45
  2. x² - 9
  3. x = 3

To determine whether Noah's solution is correct, we need to scrutinize each step and compare it with the standard method of solving quadratic equations. The most common techniques involve isolating the variable, factoring, or using the quadratic formula. Let's break down Noah's approach and identify any deviations from these established methods.

Detailed Step-by-Step Analysis

Step 1: 5x² - 45

In the initial step, Noah seems to have subtracted 45 from both sides of the equation 5x² = 45. This is a valid algebraic manipulation. Subtracting the same value from both sides maintains the equality. So, starting with 5x² = 45, subtracting 45 from both sides correctly yields 5x² - 45 = 0. This step is essential to set up the equation for further simplification or factoring.

Step 2: x² - 9

This step involves dividing both sides of the equation 5x² - 45 = 0 by 5. This is also a valid algebraic operation. Dividing both sides of an equation by a non-zero constant preserves the equality. When we divide 5x² - 45 = 0 by 5, we get (5x²)/5 - 45/5 = 0/5, which simplifies to x² - 9 = 0. This simplified form is much easier to work with and sets the stage for factoring or using the square root property.

Step 3: x = 3

Here, Noah states that x = 3. This is where the solution becomes incomplete. While 3 is indeed one solution to the equation x² - 9 = 0, it is not the only solution. The equation x² - 9 = 0 is a quadratic equation, and quadratic equations generally have two solutions. To find all solutions, we need to recognize that x² - 9 is a difference of squares and can be factored as (x - 3)(x + 3) = 0. This factorization leads to two possible solutions: x = 3 and x = -3. Therefore, Noah missed one of the solutions.

Correct Solution

To correctly solve the equation 5x² = 45, we can follow these steps:

  1. Subtract 45 from both sides: 5x² - 45 = 0
  2. Divide both sides by 5: x² - 9 = 0
  3. Factor the left side as a difference of squares: (x - 3)(x + 3) = 0
  4. Set each factor equal to zero and solve for x:
    • x - 3 = 0 gives x = 3
    • x + 3 = 0 gives x = -3

Thus, the solutions are x = 3 and x = -3. Alternatively, after obtaining x² - 9 = 0, we could add 9 to both sides to get x² = 9. Then, taking the square root of both sides, we must remember to consider both the positive and negative roots, yielding x = ±3.

Why Noah's Solution is Incomplete

Noah's solution is incomplete because he only identified one root of the quadratic equation. By stopping at x = 3, he overlooked the negative root, x = -3. This is a common mistake when solving quadratic equations. It is essential to remember that quadratic equations, due to the presence of the x² term, typically have two solutions. Failing to consider both positive and negative roots or not factoring completely can lead to incomplete or incorrect answers.

Importance of Recognizing Two Solutions

The significance of identifying both solutions of a quadratic equation extends beyond the mathematical exercise itself. In various real-world applications, both solutions can have practical meanings. For example, in physics, quadratic equations might describe the trajectory of a projectile, where one solution represents the time when the projectile reaches a certain height on its way up, and the other solution represents the time when it reaches the same height on its way down. Similarly, in engineering and economics, both solutions can provide critical insights into the behavior of a system or model.

Conclusion

In conclusion, while Noah's initial steps were correct, his solution is incomplete as he only found one of the two solutions to the equation 5x² = 45. The correct solutions are x = 3 and x = -3. This analysis highlights the importance of thoroughly solving quadratic equations by considering all possible roots. Understanding the nature of quadratic equations and the methods to solve them completely is crucial for success in algebra and its applications in various fields. Remember to always check for all possible solutions and use methods like factoring or the quadratic formula to ensure accuracy.