Solving 6x² - 15x = 0 By Factoring A Step-by-Step Guide

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Introduction

In the realm of mathematics, solving equations is a fundamental skill, and quadratic equations hold a significant place. These equations, characterized by the presence of a squared term, often require specific techniques to find their solutions. Among these techniques, factoring stands out as a powerful and elegant method. This comprehensive guide delves into the intricacies of solving quadratic equations by factoring, providing a step-by-step approach and illustrating the process with a detailed example. We will explore the underlying principles, discuss the importance of showing your work, and highlight the benefits of mastering this essential skill.

Factoring, in its essence, is the process of breaking down a complex expression into simpler components, typically by expressing it as a product of factors. When applied to quadratic equations, factoring allows us to rewrite the equation in a form where the solutions become readily apparent. The factored form reveals the roots of the equation, which are the values of the variable that make the equation true. This method relies on the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. By setting each factor equal to zero, we can obtain the solutions to the quadratic equation.

Understanding the concept of factoring is crucial not only for solving quadratic equations but also for a wide range of mathematical applications. It forms the basis for simplifying algebraic expressions, solving higher-degree polynomials, and tackling problems in calculus and other advanced topics. Moreover, the ability to factor efficiently can significantly enhance problem-solving skills and provide a deeper understanding of mathematical relationships. This article aims to equip you with the knowledge and skills necessary to confidently solve quadratic equations by factoring, emphasizing the importance of showing your steps and presenting the factored form as part of your solution. Let's embark on this journey of mathematical exploration and unlock the power of factoring!

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros, and they represent the values of x that satisfy the equation. These roots can be real or complex numbers, and a quadratic equation can have up to two distinct roots.

The significance of quadratic equations extends far beyond the realm of pure mathematics. They appear in various scientific and engineering disciplines, modeling phenomena such as projectile motion, the trajectory of a ball thrown into the air, and the curvature of suspension cables on bridges. In physics, quadratic equations are used to describe the motion of objects under constant acceleration. In engineering, they are employed in the design of structures, circuits, and control systems. The ability to solve quadratic equations is therefore an essential skill for anyone pursuing a career in these fields.

Several methods exist for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and disadvantages, and the choice of method often depends on the specific form of the equation. Factoring is particularly effective when the quadratic expression can be easily factored into two linear factors. Completing the square is a more general method that can be used to solve any quadratic equation, but it can be more time-consuming than factoring. The quadratic formula provides a direct solution for any quadratic equation, regardless of whether it can be factored easily. However, it is important to understand the underlying principles behind each method to choose the most appropriate approach for a given problem.

In this guide, we will focus on the factoring method, as it provides a powerful and insightful way to solve quadratic equations. By understanding the process of factoring, you will gain a deeper appreciation for the structure of quadratic equations and the relationships between their coefficients and roots. We will walk through the steps involved in factoring, illustrate the process with examples, and emphasize the importance of showing your work to ensure accuracy and clarity. Mastering this skill will not only enable you to solve quadratic equations but also enhance your overall problem-solving abilities in mathematics.

The Factoring Method: A Step-by-Step Approach

To solve quadratic equations by factoring, we rely on the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property allows us to transform a quadratic equation into a set of simpler equations that can be easily solved.

The general strategy for solving a quadratic equation by factoring involves the following steps:

  1. Write the equation in standard form: The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. If the equation is not already in this form, rearrange the terms to bring it into the standard form. This step ensures that all the terms are on one side of the equation, and the other side is equal to zero.
  2. Factor the quadratic expression: The next step is to factor the quadratic expression ax² + bx + c. Factoring involves finding two binomials that, when multiplied together, give the original quadratic expression. Several techniques can be used for factoring, including trial and error, the AC method, and recognizing special patterns such as the difference of squares or perfect square trinomials. The choice of technique depends on the specific form of the quadratic expression.
  3. Set each factor equal to zero: Once the quadratic expression is factored, we obtain an equation of the form (px + q) (rx + s) = 0, where p, q, r, and s are constants. According to the zero-product property, this equation is true if and only if at least one of the factors is equal to zero. Therefore, we set each factor equal to zero, resulting in two linear equations: px + q = 0 and rx + s = 0.
  4. Solve each linear equation: The final step is to solve each of the linear equations obtained in the previous step. This involves isolating the variable x in each equation, which can be done by performing algebraic operations such as adding, subtracting, multiplying, or dividing both sides of the equation by a constant. The solutions to these linear equations are the roots of the original quadratic equation.

Throughout this process, it is crucial to show your work clearly and systematically. Writing down each step not only helps to prevent errors but also allows you to track your progress and understand the reasoning behind each operation. Moreover, showing your work is essential for demonstrating your understanding of the factoring method and for receiving full credit on assignments and exams.

Example: Solving 6x² - 15x = 0

Let's illustrate the factoring method with the example equation 6x² - 15x = 0. This example will provide a concrete application of the steps outlined in the previous section, demonstrating how to solve a quadratic equation by factoring.

  1. Write the equation in standard form: The given equation, 6x² - 15x = 0, is already in standard form, where a = 6, b = -15, and c = 0. This step is often straightforward, but it's important to check that the equation is indeed in the correct form before proceeding.
  2. Factor the quadratic expression: To factor the quadratic expression 6x² - 15x, we look for common factors in the terms. Both terms have a common factor of 3x. Factoring out 3x, we get: 3x(2x - 5) = 0. This step is crucial as it transforms the quadratic expression into a product of simpler factors.
  3. Set each factor equal to zero: Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. We set each factor, 3x and (2x - 5), equal to zero: 3x = 0 and 2x - 5 = 0. This step breaks down the original quadratic equation into two linear equations.
  4. Solve each linear equation: We solve each linear equation separately. For the first equation, 3x = 0, we divide both sides by 3 to get x = 0. For the second equation, 2x - 5 = 0, we add 5 to both sides to get 2x = 5, and then divide both sides by 2 to get x = 5/2. These are the two solutions to the quadratic equation.

Therefore, the solutions to the equation 6x² - 15x = 0 are x = 0 and x = 5/2. The factored form of the equation is 3x(2x - 5) = 0. This example illustrates the power of factoring in solving quadratic equations. By breaking down the equation into simpler factors, we can easily find the roots of the equation. It also highlights the importance of showing each step in the process, as it ensures accuracy and clarity.

The Importance of Showing Your Work

In mathematics, showing your work is as important as arriving at the correct answer. It demonstrates your understanding of the underlying concepts and the steps involved in solving a problem. When solving quadratic equations by factoring, showing your work provides a clear and logical progression from the initial equation to the final solutions. It allows you, as well as anyone reviewing your work, to follow your reasoning and identify any potential errors.

Specifically, when solving quadratic equations by factoring, you should show the following steps:

  • The original equation: Write down the given quadratic equation in its original form. This serves as a starting point and ensures that everyone is working with the same problem.
  • The factored form: Clearly show the factored form of the quadratic expression. This is a crucial step as it demonstrates your ability to factor the expression correctly. It also provides a visual representation of the factors that lead to the solutions.
  • Setting each factor equal to zero: Write out the equations obtained by setting each factor equal to zero. This step explicitly shows the application of the zero-product property, which is the foundation of the factoring method.
  • Solving each linear equation: Show the steps involved in solving each linear equation. This includes any algebraic operations performed, such as adding, subtracting, multiplying, or dividing both sides of the equation by a constant. This step demonstrates your ability to solve linear equations, which is a fundamental skill in algebra.
  • The solutions: Clearly state the solutions to the quadratic equation. These are the values of the variable that satisfy the equation. It is important to present the solutions in a clear and concise manner.

Showing your work not only helps you to avoid errors but also enhances your problem-solving skills. By breaking down a complex problem into smaller, manageable steps, you can develop a systematic approach to problem-solving. This approach can be applied to a wide range of mathematical problems, not just quadratic equations. Moreover, showing your work allows you to learn from your mistakes. If you make an error, you can easily identify where the error occurred by reviewing your steps. This allows you to correct your understanding and avoid making the same mistake in the future.

In addition to the benefits for the individual student, showing your work is also important for communication in mathematics. Mathematics is a language, and showing your work allows you to communicate your ideas clearly and effectively. It enables others to understand your reasoning and to evaluate the correctness of your solution. This is particularly important in collaborative settings, where individuals need to be able to understand each other's work to solve complex problems.

Conclusion

In conclusion, solving quadratic equations by factoring is a powerful and essential skill in mathematics. This method provides a direct and insightful way to find the roots of a quadratic equation, revealing the values of the variable that satisfy the equation. By understanding the underlying principles of factoring and the zero-product property, you can effectively solve a wide range of quadratic equations.

Throughout this guide, we have explored the step-by-step approach to solving quadratic equations by factoring. We have emphasized the importance of writing the equation in standard form, factoring the quadratic expression, setting each factor equal to zero, and solving the resulting linear equations. We have also illustrated the process with a detailed example, demonstrating how to apply these steps in practice.

Furthermore, we have highlighted the critical importance of showing your work. Showing each step in the solution process not only helps to prevent errors but also demonstrates your understanding of the underlying concepts. It allows you to track your progress, identify any mistakes, and communicate your reasoning clearly and effectively. Showing your work is an essential part of mathematical problem-solving and is crucial for success in mathematics.

The ability to solve quadratic equations by factoring is not only valuable in mathematics but also in various scientific and engineering disciplines. Quadratic equations appear in models of projectile motion, electrical circuits, and many other real-world phenomena. Mastering this skill will therefore provide you with a solid foundation for further studies in these fields.

As you continue your mathematical journey, remember the power of factoring and the importance of showing your work. Practice solving quadratic equations by factoring, and you will develop confidence and proficiency in this essential skill. With a solid understanding of factoring, you will be well-equipped to tackle more advanced mathematical concepts and problems. Embrace the challenge, and unlock the power of factoring in your mathematical endeavors.