Sunil's Next Step Solving Quadratic Equations By Completing The Square
The completing the square method is a powerful technique used to solve quadratic equations, which are equations of the form ax² + bx + c = 0. This method transforms the quadratic equation into a perfect square trinomial, making it easier to isolate the variable x and find the solutions. While it may seem a bit intricate at first, mastering completing the square opens doors to a deeper understanding of quadratic equations and their applications in various fields, from physics and engineering to economics and computer science. In this comprehensive guide, we'll walk through the process step-by-step, illustrating each stage with examples and highlighting the key concepts involved. We'll also delve into the underlying principles that make completing the square work, providing you with a solid foundation for tackling quadratic equations with confidence.
Initial Setup: Preparing the Equation
Before diving into the core steps of completing the square, it's crucial to ensure the quadratic equation is in the correct format. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. The first step often involves rearranging the equation to isolate the terms containing x on one side and the constant term on the other. This sets the stage for the subsequent steps. This preparation is essential because completing the square relies on manipulating the equation to create a perfect square trinomial, which is a trinomial that can be factored into the form (px + q)². By isolating the constant term, we create the space needed to add a specific value that will transform the left-hand side of the equation into a perfect square trinomial. For instance, if you start with an equation like 2x² + 8x + 3 = 0, the initial setup would involve subtracting 3 from both sides to get 2x² + 8x = -3. This rearrangement allows us to focus on the x terms and manipulate them to form the desired perfect square. This foundational step is not merely a cosmetic change; it is a strategic move that simplifies the entire process of solving the quadratic equation.
Sunil's Initial Steps: A Closer Look
In the specific problem Sunil is tackling, the equation is 6x² + 24x + 5 = 0. Sunil has already taken the crucial first step by isolating the constant term. He has correctly transformed the equation to 6x² + 24x = -5. This initial move is a fundamental part of the completing the square method, as it sets the stage for creating a perfect square trinomial on the left-hand side of the equation. By moving the constant term to the right side, Sunil has effectively cleared the way to focus on manipulating the terms involving x. This isolation is not just about aesthetics; it's a strategic maneuver that allows us to concentrate on the core of the completing the square process. With the constant term out of the way, we can now turn our attention to the coefficients of the x² and x terms, which are the key ingredients in forming the perfect square. Sunil's decision to isolate the constant demonstrates a solid understanding of the initial requirements for completing the square, and it positions him well for the next steps in the solution. This careful setup is a hallmark of a methodical approach to problem-solving, ensuring that each subsequent step builds logically upon the previous one. Therefore, Sunil's initial steps showcase a commendable grasp of the foundational principles of completing the square.
The Next Critical Step: Addressing the Leading Coefficient
After isolating the constant term, the next pivotal step in completing the square is to ensure that the coefficient of the x² term is equal to 1. This is a crucial requirement because the method relies on creating a perfect square trinomial, and having a leading coefficient other than 1 complicates this process. To achieve this, you typically divide both sides of the equation by the current coefficient of x². This step might seem like a minor detail, but it significantly simplifies the subsequent steps, particularly when determining the value needed to complete the square. For instance, in the equation 6x² + 24x = -5, the coefficient of x² is 6. To make it 1, we need to divide every term in the equation by 6. This transforms the equation into x² + 4x = -5/6. Now, the coefficient of x² is 1, and we can proceed with the core steps of completing the square. Failing to address the leading coefficient can lead to errors and make the process unnecessarily complex. Therefore, this step is not just a matter of convenience; it is an essential prerequisite for successful completion of the square. It ensures that the subsequent manipulations are mathematically sound and that the resulting perfect square trinomial is accurate.
Why Isolating x² is Crucial
Isolating the x² term, as mentioned earlier, is a pivotal step in the process of completing the square. This step is crucial because the method relies on transforming the quadratic expression into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as (x + a)². The coefficient of x² must be 1 to easily determine the constant term needed to complete the square. When the coefficient of x² is not 1, it complicates the process of finding the correct constant to add to both sides of the equation. By dividing both sides of the equation by the coefficient of x², we ensure that we are working with a simplified quadratic expression where the coefficient of x² is 1. This simplification makes it straightforward to calculate the value needed to complete the square, which is half the coefficient of the x term, squared. This value is then added to both sides of the equation to maintain balance and create the perfect square trinomial. Isolating x² is, therefore, not just a preliminary step; it is a fundamental requirement for the correct application of the completing the square method. It streamlines the subsequent algebraic manipulations and ensures that the resulting solutions are accurate.
Sunil's Next Move: The Correct Path
Given the equation 6x² + 24x = -5, Sunil's next logical step should be to isolate the x² term. This means dividing both sides of the equation by 6. This step is essential because it sets the stage for creating a perfect square trinomial, which is the heart of the completing the square method. By dividing by 6, Sunil will transform the equation into x² + 4x = -5/6. Now, the coefficient of x² is 1, which is a prerequisite for easily completing the square. With the leading coefficient addressed, Sunil can now focus on finding the value needed to complete the square, which involves taking half of the coefficient of the x term (which is 4), squaring it, and adding it to both sides of the equation. This process will create a perfect square trinomial on the left side, allowing Sunil to rewrite the equation in a form that can be easily solved for x. Choosing to isolate x² at this stage demonstrates a clear understanding of the procedural steps involved in completing the square, and it ensures that Sunil is on the right track to finding the solutions to the quadratic equation. This step is not just a matter of following a recipe; it's a strategic decision that simplifies the remaining steps and increases the likelihood of a successful outcome.
Why Other Options Are Incorrect
While isolating the constant term is a necessary initial step, it's not the next step after Sunil has already done so. Option B, "Isolate the constant," is therefore incorrect because Sunil has already performed this action. Similarly, option C, "Factor 6 out," might seem like a plausible approach, but it doesn't directly lead to completing the square. Factoring out 6 would give 6(x² + 4x) = -5, but this still leaves the issue of completing the square within the parentheses. The key is to have a leading coefficient of 1 for the x² term before proceeding with the square-completing steps. Therefore, factoring out the 6, while not inherently wrong, doesn't advance the process of completing the square in the most efficient way. The most direct and correct next step is to isolate the x² term by dividing both sides of the equation by 6. This prepares the equation for the core steps of completing the square and ensures that the subsequent manipulations are mathematically sound. Understanding why these other options are not the most suitable choice is just as important as knowing the correct step. It reinforces the logical sequence of the completing the square method and highlights the importance of each step in the overall process. This nuanced understanding is what separates rote memorization from true mastery of the technique.
Conclusion: Mastering Completing the Square
Completing the square is a valuable technique for solving quadratic equations, and understanding each step is crucial for success. Sunil's situation highlights the importance of following the correct sequence of actions to efficiently solve the equation. The next step for Sunil should be to isolate the x² term by dividing both sides of the equation by 6. This will set the stage for creating a perfect square trinomial and ultimately finding the solutions to the equation. By mastering this method, students gain a deeper understanding of quadratic equations and their properties, which is essential for success in higher-level mathematics and related fields. The ability to complete the square is not just about solving equations; it's about developing a problem-solving mindset and a systematic approach to mathematical challenges. It teaches the importance of breaking down complex problems into smaller, manageable steps and executing each step with precision. This methodical approach is a valuable skill that extends far beyond the realm of mathematics, benefiting individuals in various aspects of their lives and careers. Therefore, investing time and effort in mastering completing the square is an investment in one's overall mathematical proficiency and problem-solving abilities.
