Completing The Square What Number To Add To X² - 20x = 5

Completing the square is a powerful technique in algebra that allows us to rewrite quadratic equations into a more manageable form, making it easier to solve for the unknown variable. In this comprehensive guide, we will delve into the intricacies of completing the square, focusing on the specific equation x² - 20x = 5. We will meticulously explore the steps involved, unraveling the underlying concepts and providing clear, concise explanations to ensure a thorough understanding. By the end of this exploration, you will be equipped with the knowledge and skills to confidently tackle similar quadratic equations and master the art of completing the square.

Understanding Completing the Square

Before we dive into the specifics of our equation, let's first establish a solid foundation by understanding the core concept of completing the square. The essence of this technique lies in transforming a quadratic expression of the form ax² + bx + c into a perfect square trinomial, which can then be factored into the form (x + k)² or (x - k)², where k is a constant. This transformation is crucial because it allows us to isolate the variable x and solve for its value. When we talk about completing the square, we're essentially aiming to manipulate a quadratic equation so that one side becomes a perfect square trinomial. This involves adding a specific constant to both sides of the equation, carefully chosen to create the desired perfect square. By converting a quadratic equation into this form, we unlock a straightforward path to finding the solutions or roots of the equation. The process not only simplifies the equation but also reveals key properties of the quadratic function, such as the vertex of its parabolic graph.

The Magic Number: (b/2)²

The key to completing the square lies in identifying the correct number to add to both sides of the equation. This number, often referred to as the "magic number," is calculated using the coefficient of the x term (denoted as 'b' in the general quadratic form ax² + bx + c). The formula for this magic number is (b/2)². This seemingly simple formula holds the key to transforming a quadratic expression into a perfect square trinomial. Let's break down why this formula works. When you expand (x + k)², you get x² + 2kx + k². Notice that the coefficient of the x term is 2k, and the constant term is k². To find the value of k that will complete the square, we take half of the coefficient of the x term (which is b/2) and square it (resulting in (b/2)²). This ensures that the constant term we add creates a perfect square trinomial that can be easily factored. Understanding the origin and purpose of this formula is crucial for mastering the technique of completing the square. It's not just about memorizing a formula; it's about grasping the underlying algebraic principles that make the transformation possible.

Applying the Concept to x² - 20x = 5

Now that we have a firm grasp of the fundamental principles of completing the square, let's apply this knowledge to our specific equation: x² - 20x = 5. Our mission is to determine the number we need to add to both sides of this equation to transform the left-hand side into a perfect square trinomial. This will allow us to rewrite the equation in a form that is easier to solve for x. Before we start, it's important to recognize the structure of the equation and identify the coefficient of the x term. In this case, the coefficient is -20, which will play a crucial role in our calculations. By focusing on this coefficient and applying the formula we discussed earlier, we can systematically find the missing piece of the puzzle that will complete the square.

Step-by-Step Calculation

To find the magic number, we first identify the coefficient of our x term, which is -20. We then divide this coefficient by 2, resulting in -10. Next, we square this result: (-10)² = 100. Therefore, the number we need to add to both sides of the equation to complete the square is 100. This seemingly simple calculation is the cornerstone of the entire process. By adding 100 to both sides, we're not just arbitrarily changing the equation; we're strategically transforming it into a form that reveals its underlying structure and simplifies the process of finding the solutions. This step-by-step approach ensures accuracy and clarity, allowing you to confidently apply the technique to other quadratic equations.

Completing the Square: The Detailed Process

Now that we've identified the magic number, 100, let's proceed with the actual process of completing the square. We'll start by adding 100 to both sides of our original equation, x² - 20x = 5. This crucial step maintains the balance of the equation while simultaneously setting the stage for transforming the left-hand side into a perfect square trinomial. By carefully executing this step, we're not just manipulating the numbers; we're strategically reshaping the equation to reveal its hidden structure and make it easier to solve. This process requires attention to detail and a clear understanding of the algebraic principles at play. Once we've added 100 to both sides, we'll be ready to factor the left-hand side and continue our journey towards finding the solutions for x.

Adding 100 to Both Sides

Adding 100 to both sides of the equation x² - 20x = 5 gives us: x² - 20x + 100 = 5 + 100. This step is critical because it sets the stage for creating a perfect square trinomial on the left-hand side. By adding the same value to both sides, we maintain the equality of the equation while strategically transforming its form. The addition of 100 is not arbitrary; it's a calculated move based on our earlier determination of the magic number (b/2)². This step demonstrates the power of algebraic manipulation in simplifying complex equations. It's not just about performing operations; it's about making informed decisions that lead us closer to the solution.

Factoring the Perfect Square Trinomial

The left-hand side of the equation, x² - 20x + 100, is now a perfect square trinomial. This means it can be factored into the form (x - k)², where k is a constant. In this case, the expression factors into (x - 10)². This factorization is a direct result of our careful choice of the number 100 to add to both sides of the equation. The perfect square trinomial structure allows us to condense the expression into a simpler, more manageable form. Factoring is a fundamental skill in algebra, and its application here highlights the elegance and efficiency of completing the square. By recognizing and utilizing the perfect square trinomial pattern, we can significantly simplify the process of solving quadratic equations.

Simplifying the Right-Hand Side

On the right-hand side of the equation, we have 5 + 100, which simplifies to 105. This step is a straightforward arithmetic operation, but it's important to ensure accuracy in the calculations. Simplifying the right-hand side allows us to present the equation in its most concise form, making it easier to proceed with the remaining steps. This simplification is not just about tidying up the equation; it's about creating a clear and uncluttered path towards the solution. By keeping the equation as simple as possible, we reduce the chances of making errors and enhance our understanding of the relationships between the variables.

The Transformed Equation: (x - 10)² = 105

After completing the steps of adding 100 to both sides, factoring the perfect square trinomial, and simplifying the right-hand side, our equation has been transformed into (x - 10)² = 105. This transformed equation is a significant milestone in our journey to solve for x. It's now in a form that allows us to easily isolate x by taking the square root of both sides. The process of completing the square has effectively converted a complex quadratic equation into a simpler, more manageable form. This transformation is the essence of the technique, and it demonstrates the power of algebraic manipulation in solving problems.

Solving for x (Optional)

While our primary focus was on completing the square, let's briefly outline the steps to solve for x in the transformed equation (x - 10)² = 105. To do this, we would take the square root of both sides, remembering to consider both positive and negative roots. This would give us x - 10 = ±√105. Finally, we would add 10 to both sides to isolate x, resulting in x = 10 ± √105. This final step demonstrates the ultimate goal of completing the square: to find the solutions or roots of the quadratic equation. By transforming the equation into a form where x can be easily isolated, we unlock the path to finding these solutions.

Conclusion

In conclusion, to complete the square for the equation x² - 20x = 5, you would add 100 to both sides. This process transforms the equation into a form where the left-hand side is a perfect square trinomial, making it easier to solve for x. Completing the square is a fundamental technique in algebra with applications in various areas of mathematics and beyond. By mastering this technique, you gain a powerful tool for solving quadratic equations and understanding the properties of quadratic functions. This exploration has provided a comprehensive understanding of the process, from the initial concept to the final transformed equation. With this knowledge, you are well-equipped to tackle similar problems and further explore the fascinating world of algebra.