Solving $x^2 + 12x = -20$ By Completing The Square A Step By Step Guide

Solving quadratic equations can sometimes feel like navigating a maze, but with the right techniques, it becomes a manageable task. One powerful method for solving these equations is called completing the square. This method allows us to rewrite a quadratic equation in a form that makes it easy to find the solutions. In this comprehensive guide, we will walk through the process of solving the quadratic equation $x^2 + 12x = -20$ by completing the square, providing a detailed, step-by-step explanation that will help you master this technique.

1. Understanding the Concept of Completing the Square

Before we dive into the specifics of solving our equation, let's first grasp the underlying concept of completing the square. At its core, this method involves transforming a quadratic expression of the form $ax^2 + bx + c$ into a perfect square trinomial, which can then be factored into the form $(x + k)^2$ or $(x - k)^2$. This transformation simplifies the equation and allows us to isolate the variable $x$ more easily.

Completing the square is a valuable technique because it provides a systematic way to solve any quadratic equation, regardless of whether it can be easily factored using other methods. It's especially useful when dealing with equations that have irrational or complex solutions. The key idea is to manipulate the equation algebraically so that one side becomes a perfect square trinomial. This involves adding a specific constant term to both sides of the equation, which is derived from the coefficient of the linear term (the term with $x$).

The process of completing the square not only helps in finding solutions to quadratic equations but also provides insights into the structure and properties of quadratic functions. For instance, by completing the square, we can easily determine the vertex of a parabola represented by a quadratic function, which is crucial for understanding the function's behavior and graph. Furthermore, this method is fundamental in deriving the quadratic formula, a universal solution for quadratic equations.

Therefore, mastering completing the square is not just about solving equations; it's about developing a deeper understanding of quadratic expressions and their applications in various mathematical contexts. It's a technique that empowers you to tackle a wide range of problems and builds a solid foundation for more advanced mathematical concepts. So, let's delve into the steps involved and see how this powerful method works in practice.

2. Step 1: Adding $(\frac{b}{2})^2$ to Both Sides

The first crucial step in completing the square involves adding $(\frac{b}{2})^2$ to both sides of the equation. This ensures that we maintain the equality while transforming the left side into a perfect square trinomial. In our equation, $x^2 + 12x = -20$, we need to identify the value of $b$, which is the coefficient of the $x$ term. In this case, $b = 12$. Now, let's calculate $(\frac{b}{2})^2$:

$\qquad (\frac{b}{2})^2 = (\frac{12}{2})^2 = (6)^2 = 36$

Therefore, the value of $(\frac{b}{2})^2$ in this equation is 36. We will add this value to both sides of the equation to proceed with completing the square. Adding 36 to both sides gives us:

$\qquad x^2 + 12x + 36 = -20 + 36$

This step is critical because it sets the stage for rewriting the left side of the equation as a perfect square. The addition of $(\frac{b}{2})^2$ is precisely what makes this transformation possible. By adding this specific value, we ensure that the resulting trinomial on the left side can be factored into a binomial squared, which is the essence of completing the square.

The importance of this step cannot be overstated. It's the foundation upon which the entire method rests. Without adding the correct value to both sides, we wouldn't be able to create the perfect square trinomial, and the subsequent steps would be rendered ineffective. So, it's essential to understand why we add $(\frac{b}{2})^2$ and how it contributes to the overall goal of solving the quadratic equation.

In essence, adding $(\frac{b}{2})^2$ is like finding the missing piece of a puzzle. It's the key to unlocking the perfect square trinomial that allows us to simplify the equation and ultimately find the solutions. Once we've added this value, we can move on to the next step, which involves writing the left side of the equation as a binomial squared. This is where the magic of completing the square truly comes to life, as we transform a complex expression into a simpler, more manageable form.

3. Step 2: Writing the Left Side as a Binomial Squared

After adding $(\frac{b}{2})^2$ to both sides, our equation now reads $x^2 + 12x + 36 = -20 + 36$. The next step is to rewrite the left side of the equation as a binomial squared. This is where the concept of a perfect square trinomial comes into play. A perfect square trinomial is a trinomial that can be factored into the form $(x + k)^2$ or $(x - k)^2$. In our case, the left side, $x^2 + 12x + 36$, is indeed a perfect square trinomial.

To factor it into a binomial squared, we need to identify the value of $k$ such that $(x + k)^2 = x^2 + 2kx + k^2$. By comparing this with our trinomial, $x^2 + 12x + 36$, we can see that $2k = 12$ and $k^2 = 36$. Solving for $k$, we find that $k = 6$. Therefore, we can rewrite the left side as:

$\qquad x^2 + 12x + 36 = (x + 6)^2$

This step is crucial because it simplifies the equation significantly. By expressing the left side as a binomial squared, we've essentially transformed a complex trinomial into a much simpler expression. This makes it easier to isolate the variable $x$ and solve for its values. The ability to recognize and factor perfect square trinomials is a fundamental skill in algebra, and completing the square provides a practical application of this skill.

The process of writing the left side as a binomial squared is not just about factoring; it's about recognizing the underlying structure of the quadratic expression. By understanding that $x^2 + 12x + 36$ is a perfect square trinomial, we can quickly and efficiently rewrite it in its factored form. This step highlights the power of algebraic manipulation in simplifying complex expressions and making them more manageable.

Now that we've rewritten the left side as a binomial squared, our equation becomes $(x + 6)^2 = -20 + 36$. We can further simplify the right side by performing the addition, which gives us $(x + 6)^2 = 16$. This simplified equation is now in a form that we can easily solve by taking the square root of both sides. The next steps will involve isolating $x$ and finding the solutions to the equation. So, let's proceed with the final steps to complete the solution.

4. Solving for x

With the equation now in the form $(x + 6)^2 = 16$, we're in the final stretch of solving for $x$. The next step involves taking the square root of both sides of the equation. Remember that when we take the square root of a number, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will result in the same positive number.

Taking the square root of both sides, we get:

$\qquad \sqrt{(x + 6)^2} = \pm\sqrt{16}$

This simplifies to:

$\qquad x + 6 = \pm 4$

Now we have two separate equations to solve:

1. $x + 6 = 4$
2. $x + 6 = -4$

Let's solve each equation individually.

For the first equation, $x + 6 = 4$, we subtract 6 from both sides:

$\qquad x = 4 - 6$ $\qquad x = -2$

For the second equation, $x + 6 = -4$, we also subtract 6 from both sides:

$\qquad x = -4 - 6$ $\qquad x = -10$

Therefore, the solutions to the quadratic equation $x^2 + 12x = -20$ are $x = -2$ and $x = -10$. These are the values of $x$ that satisfy the original equation. We have successfully solved the equation by completing the square, demonstrating the power and effectiveness of this method.

This final step of solving for $x$ highlights the importance of considering both positive and negative roots when taking the square root of an equation. It's a common mistake to overlook the negative root, which can lead to incomplete solutions. By carefully considering both possibilities, we ensure that we find all the solutions to the quadratic equation.

In conclusion, we have walked through the entire process of solving the quadratic equation $x^2 + 12x = -20$ by completing the square. From adding $(\frac{b}{2})^2$ to both sides to writing the left side as a binomial squared and finally solving for $x$, each step has been explained in detail. This method provides a systematic approach to solving quadratic equations and is a valuable tool in any mathematician's arsenal.

5. Conclusion

In this comprehensive guide, we've meticulously demonstrated how to solve the quadratic equation $x^2 + 12x = -20$ by completing the square. We began by understanding the fundamental concept of completing the square, which involves transforming a quadratic expression into a perfect square trinomial. This transformation is crucial as it allows us to rewrite the equation in a form that is easier to solve.

We then delved into the step-by-step process, starting with adding $(\frac{b}{2})^2$ to both sides of the equation. This step is the cornerstone of the method, as it sets the stage for creating the perfect square trinomial. We calculated $(\frac{b}{2})^2$ to be 36 and added it to both sides, ensuring that the equation remained balanced.

Next, we focused on writing the left side of the equation as a binomial squared. We recognized that $x^2 + 12x + 36$ is a perfect square trinomial and factored it into $(x + 6)^2$. This simplification is a key advantage of completing the square, as it transforms a complex expression into a more manageable form.

Finally, we tackled the task of solving for $x$. We took the square root of both sides of the equation, remembering to consider both positive and negative roots. This led us to two separate equations, which we solved to find the solutions $x = -2$ and $x = -10$. These are the values of $x$ that satisfy the original quadratic equation.

Completing the square is not just a method for solving quadratic equations; it's a powerful technique that enhances our understanding of quadratic expressions and their properties. It provides a systematic approach that can be applied to any quadratic equation, regardless of its complexity. Moreover, it lays the foundation for more advanced mathematical concepts, such as deriving the quadratic formula and analyzing the graphs of quadratic functions.

By mastering completing the square, you equip yourself with a valuable tool for tackling a wide range of mathematical problems. It's a skill that not only helps in solving equations but also fosters a deeper appreciation for the elegance and structure of algebra. So, practice this method, explore its applications, and you'll find yourself well-equipped to conquer quadratic equations and beyond.