Solving X^2 + 12x + 6 = 0 By Completing The Square A Step-by-Step Guide

The completing-the-square method is a powerful technique for solving quadratic equations. It transforms the equation into a perfect square trinomial, making it easier to isolate the variable x. In this comprehensive guide, we will walk you through the steps to solve the quadratic equation $x^2 + 12x + 6 = 0$ using the completing-the-square method. We will also discuss the underlying principles and provide clear explanations to ensure you grasp the concept thoroughly. By the end of this article, you'll not only be able to solve this specific equation but also apply the method to other quadratic equations with confidence. Understanding the completing-the-square method is crucial for mastering algebra and solving various mathematical problems. This method provides a solid foundation for further studies in mathematics and related fields. Let's dive into the step-by-step solution and unlock the power of completing the square!

Understanding the Completing-the-Square Method

The completing-the-square method is a technique used to rewrite a quadratic equation in the form of $(x + a)^2 = b$, where a and b are constants. This form allows us to easily solve for x by taking the square root of both sides. The key idea behind this method is to manipulate the quadratic equation by adding a constant term to both sides, which transforms the left side into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, $x^2 + 6x + 9$ is a perfect square trinomial because it can be factored as $(x + 3)^2$. To complete the square, we need to find the constant term that, when added to the quadratic expression, will create a perfect square trinomial. This constant term is found by taking half of the coefficient of the x term, squaring it, and adding it to both sides of the equation. The power of the completing-the-square method lies in its ability to convert any quadratic equation into a form that can be easily solved. This is particularly useful when the quadratic equation cannot be easily factored using traditional methods. Furthermore, understanding this method provides a deeper understanding of the structure of quadratic equations and their solutions. It's a fundamental tool in algebra and has applications in various areas of mathematics, including calculus and analytic geometry. By mastering the completing-the-square method, you'll enhance your problem-solving skills and gain a valuable technique for tackling quadratic equations.

Step-by-Step Solution for $x^2 + 12x + 6 = 0$

Let's apply the completing-the-square method to solve the equation $x^2 + 12x + 6 = 0$. We'll break down the process into clear, manageable steps to ensure you understand each stage thoroughly. This step-by-step approach will not only help you solve this specific equation but also equip you with the skills to tackle similar problems in the future. By carefully following each step, you'll gain confidence in your ability to manipulate quadratic equations and find their solutions. Each step is designed to build upon the previous one, leading you to the final answer in a logical and systematic manner. Remember, practice is key to mastering any mathematical technique, so make sure to work through this example and try others on your own. The goal is not just to find the solution but also to understand the underlying principles and the reasoning behind each step. So, let's begin our journey to solve this equation using the completing-the-square method.

Step 1: Move the Constant Term to the Right Side

First, we need to isolate the terms with x on one side of the equation. To do this, we subtract the constant term, 6, from both sides of the equation. This gives us:

$x^2 + 12x + 6 - 6 = 0 - 6$

Simplifying, we get:

$x^2 + 12x = -6$

This step is crucial because it sets the stage for completing the square. By moving the constant term, we create space on the left side to manipulate the equation into a perfect square trinomial. This initial rearrangement is a fundamental aspect of the completing-the-square method, ensuring that we can focus on the quadratic and linear terms. Understanding this step is vital for successfully applying the method to any quadratic equation. It's a simple yet essential manipulation that paves the way for the subsequent steps. Remember, the goal is to isolate the x terms, making it easier to transform the left side into a perfect square. This first step is the foundation upon which the rest of the solution is built.

Step 2: Complete the Square

This is the heart of the completing-the-square method. We need to find a constant to add to both sides of the equation that will make the left side a perfect square trinomial. To find this constant, we take half of the coefficient of the x term, which is 12, and square it. Half of 12 is 6, and 6 squared is 36. Therefore, we need to add 36 to both sides of the equation:

$x^2 + 12x + 36 = -6 + 36$

The left side, $x^2 + 12x + 36$, is now a perfect square trinomial. It can be factored as $(x + 6)^2$. On the right side, we simplify $-6 + 36$ to get 30. So, our equation becomes:

$(x + 6)^2 = 30$

This step is the core of the completing-the-square technique. By adding the appropriate constant, we transform the quadratic expression into a perfect square, making it much easier to solve. The process of taking half of the x coefficient and squaring it is crucial for ensuring that the resulting trinomial is indeed a perfect square. This manipulation allows us to rewrite the equation in a form where we can easily isolate x. Understanding this step thoroughly is essential for mastering the completing-the-square method. It's the key to unlocking the solution of many quadratic equations.

Step 3: Take the Square Root of Both Sides

Now that we have $(x + 6)^2 = 30$, we can take the square root of both sides of the equation. Remember that when we take the square root, we need to consider both the positive and negative roots:

$\sqrt{(x + 6)^2} = \{ \pm \sqrt{30} \}$

This simplifies to:

$x + 6 = \pm \sqrt{30}$

This step is a direct application of the properties of square roots and is crucial for isolating x. By taking the square root of both sides, we undo the squaring operation, bringing us closer to the solution. The inclusion of both positive and negative roots is essential because both values, when squared, will result in 30. Understanding this step requires a solid grasp of square root properties and their application in solving equations. It's a fundamental step in the completing-the-square method and a key technique in algebra.

Step 4: Solve for x

To isolate x, we subtract 6 from both sides of the equation:

$x + 6 - 6 = -6 \pm \sqrt{30}$

This gives us the solutions:

$x = -6 \pm \sqrt{30}$

Therefore, the solutions are $x = -6 + \sqrt{30}$ and $x = -6 - \sqrt{30}$.

This final step completes the solution process. By isolating x, we arrive at the roots of the quadratic equation. The plus-minus symbol indicates that there are two solutions, one with addition and one with subtraction. This step demonstrates the power of the completing-the-square method in providing a clear and systematic way to find the solutions of quadratic equations. Understanding this step is essential for interpreting the results and expressing the solutions in their final form. It's the culmination of all the previous steps, leading to the answer and showcasing the effectiveness of the method.

Final Answer

The solutions to the equation $x^2 + 12x + 6 = 0$ are $x = -6 + \sqrt{30}$ and $x = -6 - \sqrt{30}$. This corresponds to option D. $x = -6 \pm \sqrt{30}$.

Conclusion

In this article, we've demonstrated how to solve the quadratic equation $x^2 + 12x + 6 = 0$ using the completing-the-square method. We've broken down the process into clear, manageable steps, explaining the underlying principles at each stage. The completing-the-square method is a valuable technique for solving quadratic equations, particularly those that are difficult to factor. By mastering this method, you'll gain a deeper understanding of quadratic equations and enhance your problem-solving skills. This method not only helps in finding solutions but also provides insights into the structure and properties of quadratic equations. Furthermore, the skills learned through completing the square are transferable to other areas of mathematics, making it a fundamental technique to master. We encourage you to practice this method with various quadratic equations to solidify your understanding and build confidence in your abilities. The more you practice, the more proficient you'll become in applying this powerful technique. Remember, mathematics is a skill that improves with practice, and the completing-the-square method is a valuable tool in your mathematical arsenal.