Solving $x^2 + 8x = 33$ By Completing The Square A Step-by-Step Guide

Introduction

In this article, we will delve into solving the quadratic equation $x^2 + 8x = 33$ using the method of completing the square. This technique is a powerful tool in algebra for solving quadratic equations and rewriting them in a more convenient form. We will go through each step meticulously, providing a detailed explanation to ensure a clear understanding of the process. By the end of this guide, you will not only be able to solve this specific equation but also grasp the general method of completing the square, enabling you to tackle a wide range of quadratic equations. Understanding the method of completing the square is a foundational skill in algebra, paving the way for more advanced mathematical concepts and problem-solving techniques. The method involves transforming the quadratic equation into a perfect square trinomial, which then allows us to easily solve for the variable x. This method is especially useful when factoring is not straightforward, or when dealing with equations that do not have rational roots. We'll explore each step, from setting up the equation to isolating x and finding the solution set. This comprehensive guide will help you master this valuable algebraic technique and confidently apply it to various quadratic equations.

Understanding Quadratic Equations

Before we dive into the specifics of completing the square, it's crucial to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is given by:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions can be found using various methods, including factoring, using the quadratic formula, and, of course, completing the square. In our given equation, $x^2 + 8x = 33$, we can identify a = 1, b = 8, and c = -33 (after rearranging the equation to the standard form). The method of solving quadratic equations often involves manipulating the equation to isolate the variable x, which can be achieved through various algebraic techniques. Factoring, for example, relies on expressing the quadratic expression as a product of two binomials, while the quadratic formula provides a direct solution using the coefficients a, b, and c. However, completing the square offers a systematic approach that not only yields the solutions but also provides valuable insight into the structure of quadratic equations and their relationship to perfect square trinomials. This understanding is crucial for more advanced topics in algebra and calculus, where manipulating quadratic expressions is a common task. By grasping the underlying principles of quadratic equations, you'll be better equipped to handle various mathematical problems and applications involving these fundamental algebraic expressions. The ability to solve quadratic equations is a fundamental skill in mathematics, essential for various applications in physics, engineering, economics, and computer science.

Steps to Complete the Square

The method of completing the square involves transforming a quadratic equation into a perfect square trinomial, making it easier to solve. Let's break down the steps using our equation, $x^2 + 8x = 33$, as an example:

Step 1: Ensure the Coefficient of $x^2$ is 1

In our equation, the coefficient of $x^2$ is already 1, so we can proceed to the next step. If the coefficient were not 1, we would need to divide the entire equation by that coefficient. This step is crucial because the method of completing the square relies on having a leading coefficient of 1. When the coefficient of $x^2$ is not 1, it complicates the process of creating a perfect square trinomial. Dividing the entire equation by the leading coefficient ensures that we are working with a simplified form that is amenable to completing the square. This initial step sets the stage for the subsequent steps, making the process more straightforward and less prone to errors. By ensuring the coefficient of $x^2$ is 1, we can focus on the remaining terms and complete the square efficiently. If we were dealing with an equation like $2x^2 + 16x = 66$, we would first divide the entire equation by 2 to obtain $x^2 + 8x = 33$, which is the equation we are working with. This preliminary step is essential for the method to work effectively and provides a clear pathway to finding the solutions of the quadratic equation. The importance of this step cannot be overstated, as it simplifies the equation and allows us to proceed with the remaining steps without additional complexities.

Step 2: Move the Constant Term to the Right Side

Our equation is already in the form $x^2 + 8x = 33$, where the constant term is isolated on the right side. This step is crucial because it sets up the equation for the next step, where we will add a value to both sides to complete the square. Isolating the constant term allows us to focus on the quadratic and linear terms on the left side and manipulate them to form a perfect square trinomial. Without isolating the constant term, it would be difficult to determine the value needed to complete the square. This separation of terms is a fundamental step in the process and ensures that we can proceed methodically towards finding the solutions of the equation. By keeping the constant term separate, we maintain clarity and avoid unnecessary complications in the algebraic manipulations. For example, if the equation were $x^2 + 8x - 33 = 0$, we would add 33 to both sides to obtain $x^2 + 8x = 33$. This rearrangement is a key step in the process of solving quadratic equations by completing the square and prepares the equation for the subsequent steps. The isolation of the constant term is a simple yet essential manipulation that paves the way for the rest of the solution process.

Step 3: Complete the Square

This is the core of the method. To complete the square, we take half of the coefficient of the x term (which is 8), square it, and add the result to both sides of the equation. Half of 8 is 4, and $4^2$ is 16. So, we add 16 to both sides:

$x^2 + 8x + 16 = 33 + 16$

This step is the heart of completing the square, and it's where we transform the left side of the equation into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. By adding the appropriate value to both sides, we ensure that the left side can be expressed in this form. The value we add is always the square of half the coefficient of the x term, which guarantees the formation of a perfect square trinomial. This manipulation allows us to rewrite the equation in a form that is easily solvable. On the left side, $x^2 + 8x + 16$ is now a perfect square trinomial, which can be factored as $(x + 4)^2$. On the right side, we have $33 + 16 = 49$. The equation now becomes $(x + 4)^2 = 49$, which is much easier to solve than the original equation. This step demonstrates the power and elegance of completing the square as a method for solving quadratic equations. The ability to recognize and create perfect square trinomials is a valuable skill in algebra and is essential for mastering this technique. By completing the square, we transform the equation into a more manageable form, making it straightforward to find the solutions.

Step 4: Factor the Left Side and Simplify the Right Side

The left side, $x^2 + 8x + 16$, is now a perfect square trinomial and can be factored as $(x + 4)^2$. The right side simplifies to $33 + 16 = 49$. So, our equation becomes:

$(x + 4)^2 = 49$

This step is a direct result of the previous step, where we added a value to both sides of the equation to complete the square. Factoring the left side into a squared binomial is a crucial step in isolating the variable x. The perfect square trinomial is designed to factor neatly into the square of a binomial, making this step straightforward. Simplifying the right side by adding the constants together prepares the equation for the next step, where we will take the square root of both sides. The equation $(x + 4)^2 = 49$ is now in a form that is easily solvable, with the squared binomial on one side and a constant on the other. This step highlights the elegance of completing the square, as it transforms the original equation into a much simpler form. The ability to factor the perfect square trinomial and simplify the right side is essential for progressing towards the solutions of the quadratic equation. This step showcases the power of algebraic manipulation in simplifying complex equations and making them more accessible to solve. The factored form of the equation allows us to directly apply the square root property, which is the next step in our solution process.

Step 5: Take the Square Root of Both Sides

Taking the square root of both sides of $(x + 4)^2 = 49$, we get:

$x + 4 = ±

This step is crucial for isolating x and finding the solutions to the equation. When we take the square root of both sides, we must remember to consider both the positive and negative square roots, as both values will satisfy the equation when squared. This is why we introduce the ± symbol. The square root of 49 is 7, so we have two possible equations: $x + 4 = 7$ and $x + 4 = -7$. This step effectively undoes the squaring operation on the left side, bringing us closer to isolating x. By considering both the positive and negative roots, we ensure that we find all possible solutions to the quadratic equation. The ± symbol is a reminder that there are typically two solutions to a quadratic equation, and both must be accounted for. Taking the square root of both sides is a fundamental step in solving quadratic equations that have been transformed into this form. It allows us to break down the equation into two simpler linear equations, which are then easily solved for x. The result of this step sets the stage for the final step, where we will isolate x and determine the solution set.

Step 6: Solve for $x$

Now we have two equations to solve:

1. $x + 4 = 7$

Subtracting 4 from both sides gives $x = 7 - 4 = 3$.

2. $x + 4 = -7$

Subtracting 4 from both sides gives $x = -7 - 4 = -11$.

This is the final step in solving the quadratic equation, where we isolate x in each of the two equations resulting from the square root operation. By subtracting 4 from both sides of each equation, we find the two solutions for x. The first equation, $x + 4 = 7$, yields $x = 3$, while the second equation, $x + 4 = -7$, yields $x = -11$. These two values are the roots of the quadratic equation and represent the points where the parabola intersects the x-axis. This step demonstrates the importance of considering both the positive and negative square roots in the previous step, as each leads to a distinct solution. The ability to solve these simple linear equations is a fundamental skill in algebra and is essential for completing the process of completing the square. The solutions $x = 3$ and $x = -11$ are the values that satisfy the original equation, and they form the solution set. This final step brings closure to the process and provides the answer to the problem. The accuracy and precision in this step are crucial to ensure that the correct solutions are obtained.

Solution Set

The solutions for the equation $x^2 + 8x = 33$ are $x = 3$ and $x = -11$. Therefore, the solution set is {-11, 3}.

The solution set is the set of all values that satisfy the given equation. In this case, we found two values, $x = 3$ and $x = -11$, that make the equation $x^2 + 8x = 33$ true. The solution set is typically represented using curly braces, and the values are listed inside the braces. This step provides a clear and concise answer to the problem, summarizing the results of the previous steps. The solution set {-11, 3} indicates that these are the only two values of x that will satisfy the equation. This final step is crucial for clarity and completeness, as it presents the final answer in a standardized format. The solution set is a fundamental concept in algebra and is used to represent the complete set of solutions for any equation or system of equations. By stating the solution set, we provide a clear and unambiguous answer that leaves no room for misinterpretation. This step ensures that the entire solution process is concluded with a well-defined and accurate answer. The solution set is the ultimate goal of solving any equation, and its correct determination is a testament to the accuracy of the solution process.

Conclusion

We have successfully solved the quadratic equation $x^2 + 8x = 33$ by completing the square. The solution set is {-11, 3}. This method provides a systematic approach to solving quadratic equations and is a valuable tool in algebra. By mastering the steps outlined in this guide, you can confidently tackle a variety of quadratic equations and deepen your understanding of algebraic techniques. Completing the square is not just a method for finding solutions; it also provides insight into the structure of quadratic equations and their properties. This understanding is crucial for more advanced mathematical concepts and applications. The ability to complete the square is a foundational skill that opens doors to various areas of mathematics, including calculus, complex numbers, and more. It also enhances problem-solving skills and logical thinking, which are valuable in many fields beyond mathematics. By practicing and applying this method, you will develop a deeper appreciation for the elegance and power of algebra. The journey through solving this equation has demonstrated the effectiveness of completing the square as a reliable method for finding solutions and has reinforced the importance of each step in the process. The skills and knowledge gained from this exercise will undoubtedly be beneficial in future mathematical endeavors.